Time 1

I!!!I

. II

.

CONTENTS PAGE NO.

1. Simple Equations ."1Q

2.

"3. ~

4.

Ratio -Proporti~n

1 - Variation

14

Percentages -,Profit, & Loss - Partnerships

27

Numbers

49 ...

5.

Indices - Surds

88

6. Logarithms

100

7. Averages - Mixtures - Alligations

110

8. Quadratic Equations

127

9.

138

I

~

Progressions

1-10. Simple Interest - Compound Interest - Annuities

151

. 11. Time and Work

165

/; Ii

I II

II .11 I

..

12. Time an
182

13. Geometry and Mensuration

202

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1

Chapter-l

SIMp.LE EQUATIONS --e-e will

be linear equations of one or two ..n""OWnSinvariably in every problem. A linear !!O.a~n is one where each variable occurs only .. ~ first power and not in any higher powers. :o-e times we' get three equations in three ..l""-o'NnS. In general, we need as many

.-,

e:ions as the variables

we will have to solve

"C'"So, for solving for the values of two unknowns. .e ""eedtwo equations (or two conditions given in !"e =-oblem)and for solving for the vall!es of three cwns. we need three equations (and hence ~ :roblem should give three conditions from ""c:- we can frame three equations). Solving the ~-::ons by itself is not a difficult task. The most -CO'1ant part of the problem is framing the ~equations. Once the equations are ~. solving them is very easy. In this chapter. -= ,,""1deal with problems involving as many ~-::ons (of first degree) as the number of ""--owns. Later on. we will look at equations of e:::rd degree (Quadratic Equations) and linear ~ where the number of equations will be ess ;."1anthat of the number of variables (under ~ =apter Special Equations).

OI'I EQUATION IN ONE UNKNOWN

is called a system of. simultaneous equations in two unknowns. Here. we have two variables (or unknowns) x and y whose values we have to find out. This can be done using the two given equations. The steps for this are as follows:

Step.!:

Step II: Solve for the value of one variable from . the equation(in one unknown)obtained from Step I above. Therefore. y = 2.

Step III: Substitute this value of the variable found in one of the two equations to get the value of the second variable.

-- ~tion like2x + 4 = 26 is an equationin one AI'!I':'"'OWI1. We have only one variable x whose e fie haveto find out. The stepsin solvingthis.

Substituting the value of'i in equation (1) or equation (2). we get x = 1. Therefore the values of x and y that satisfy the given set of equations are x = 1 and y = 2.

-=

Take all quantities added to (or subtracted from) the x term (term with the unknown) to the right side with a change of sign. Le., 2x = 26:""4 = 22. ~

II: Take the co-efficient of x from left hand side and divide right hand side with this term to get the value of x. i.e.. x = 22/2 = 11. Therefore, x = 11.

1" 0 EQUATIONS IN TWO UNKNOWNS 4 ie" :-: equations like ;;.: - 3y = 8 (1) :.: - 4y = 13 (2) -..,

Using both the equations we first eliminate one variable (so that we can then have one equation in pne unknown). Far this purpose, we multiple equation (1) with 5 (the co-efficient of x in the second equation) and multiply equation (2) with 2 (the co-efficient of x in the first equation) to eliminate x. Thus we have (1) x 5 => 10x+ 15y = 40 (3) (2) x 2 => 10x + 8y = 26 (4) Now. subtracting equation (4) from equation (3) we have 7y = 14 (5) This is one equation in one unknown.

THREE EQUA TION-8 UNKNOWNS

IN THREE

A set of equations like x + 2y + 3z = 14 (1) 2x + y + 2z = 10 (2) 3x + 3y + 4z = 21 (3). is a system of three equations in three unknowns. Here we have three unknowns x, y and z which we have to solve for from the three given equations. The procedure for the same is as follows:

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Step I: Take two out of the three equations [say,

Sol: Let cost of each table be x and cost of each

eqn. (1) and (2)] and eliminate one variable (say x) so that we get an equation in two unknowns (y and z in this case).

chair be y. Then we have the. following equations from the given data. (1) 2x + 3y = 1800 3x + 4y = 2600 (2) To solve these two equations, multiply the first equation by 3 and the second by 2 and then subtract one froni the other. We get, y = 200 Substituting y in (1) we get 2x +'600 = 1800 :. x = 600 :. The cost of each "table.is Rs.600 and the cost of each chair is RS.200.

For this purpose, take equations (1) and (2). .Multiply equation (1) by 2 and subtract equation (2) from it. Equation.(1) x 2 => 2x + 4y + 6z = 28 2x + y + 2z = 10 -------------------3y + 4z = 18

(4)

Step II: Repeat Step I for two other equations [say equations (2) and (3)] and eliminate' the same variable (x in this case) so that we get one more equation in two unknowns' (y and z). For this purpose, take equations (2) and (3). Multiply equation (2) by 3, and from . that subtract equation (3) multiplied by 2. Equation (2) x 3 =>6x + 3y + 6z = 30 Equation (3) x 2 => 6x + 6y + 8z = 42 --------------------

-3y - 2z = -12

(5)

Step III: Now the equations in two unknowns that have been obtained from the above two steps have to be solved as discussed previously (in TWO EQUATIONS IN TWO UNKNOWNS) to get the values of two of the three variables (y and z in this case). In this case, solving equations (4) and (5), we get y = 2 and z = 3.

'

1.02 Siva bought 6 pencils,. 2 sharpeners and 4 erasers for Rs.16. Satyam bought 5 pencils, 3 sharpeners and 1 eraser for Rs.13.5. Sundaram bought 4 pencils, 3 sharpeners and 3 erasers for RS.12.5. Find the cost of 4 pencils, 4 sh~rpeners and 2 erasers..

Sol: Let price of each pencil, sharpener and eraser be p, sand e respectively. From the data given we get 6p + 2s + 4e:= 16 (1) 5p + 3s + 1e = 13.5 (2) 4p + 3s + 3e = 12.5 . (3) Let us take equations (2) and (3) and eliminate the variable/s, by subtraction (3) from (2) (4) p 2e = 1

-

Then take equation (1) and (2) Multiply (1) by 3 and (2) by 2. and sub~ract one from the other. We get

8p + 10e= 21 Step IV: Substitute these values of the two variables in'one of the three equations to get the value of the third variable. Substitute the value of y and z in equation (1) to get the value of x = 1. Thus the values of the three variables x, y and z that satisfy the three given equations are x = 1; y'= 2 and z = 3

Examples 1.01 If 2 tables and 3 chairs together cost Rs.1800 and 3 tables and 4 chairs together cost Rs.2600, then find the cost of each table and chair separately.

(5)

[(4) x 5] + (5) gives 13p = 26, P = Rs.2. Substituting the value of P in (4), e = Rs.0~5 Substitute these values in (1) to get s = 1.0 Using these values we find that 4 pencils, 4 sharpeners and 2 erasers cost Rs.13. 1.03 The sum of the digits of a two-digit number is 7. If the digits are interchanged, the resulting number is 27 more than the original number. .Find the original number.

Sol: Let us take the two digits as x and y. x is ten's digit and y is the unit's digit, hence the number itself is equal to (10x + y).

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since sum of the digits is 7, (1) When the digits are interchanged, y becomes the 10x + y ten's digit and x the units digit. The number then becomes (10y + x), since this r:t~mber is 27 more than the original number, we get (10y ...x) - (10x + y) = 27 ~ 9y - 9x = 27 ~ Y- x = 3 (2) On adding (1) and (2) we get y = 10/2 = 5. By substituting the value of y, we get x = 2. Thus the number is 25.

x+y=7

.~

Four years from now, Prakash's age will be 4 times his son's age. Twelve years from now he will be 2Y2times his son's age. Find their present ages.

Sol: Let Prakash and his son's present ages be x years and y years respectively. Four years from now, Prakash will be (x + 4) years and sorfwill be (y + 4) years. :. x + 4 = 4(y + 4) ~ x - 4y = 12--(1) Twelve years from now, Prakash will be x + 12 and his son y + 12. 5 :. x + 12 = -(y+12) 12 .

a

.

Sol: Let the present ages of the father and son be x years and y years respectively. We now have,

x= 10y

.-

(1)

x + 6 = 4(y + 6) = x - 4y = 18 Solving (1) and (2) we get x = 30 and y = 3 Let father's age be twice the son's age in 'p' years. Then we have 30 + p= 2(p + 3)~ p = 24 So, in 24 years the father will be twice as old as his son. ~ .06

y When both the numerator and the denominator are increasedby 2 each, we have x +2 3 -=-~ 5(x+ 2) = 3(y+2) y+2 5' ~5x-3y=-4 .-(1). When bOththe numeratorand the denominator are increasedby 1 each, we have x+1 1 -=~ 2(x + 1) = Y+ 1 y+1 2 ~2x-y=-1 -(2). [(1) - (2) x (3)] gives -x = -4 + 3 ~ x = 1. Substituting this value. in (2), y = 3 is obtained. Hence the fraction is 1/3.

1.07 Find the I(alues of x and y from the following equations. 7 6 4 14 -+-=7' - -=2 x+y x-y 'x-y x+y

Sol: Substitute p =~ 7p + 6q = 7 4q - 14p= 2

present age of father is 10 times his son's age. In 6 years time his age will be four times his son's age. In how many years. will

the father's be twice as old as his son?

Sol: Let the fraction be ~.

x+y

~ 2x - 5y = 36 -(2) [(1) x 2] - (2) gives -3y = -12, Y = 4 By substituting y = 4 in (1), we get x = 28.

..05 The

increased by ~, then the fraction becomes 1/2. Find the fraction.

If the numerator and the denominator of a certain fraction are each increased by 2, then the fraction becomes 3/5. If. the numerator and denominator are each

and q =~, we get x-y (1) and (2)

[(1) x 2] + 2 gives 16q = 16; q = 1. On substitution, we get p = 117. 1 1 p= -=~x+y=7 (1) x+y 7 1 --- (2) q=-=1; x-y=1 x-y Solving (1) and (2), we get X? 4 and y = 3

ADDITIONAL EQUA TIONS

CASES

I

IN LINEAR

1) If the number of equations is less than the number of unknowns, then we say the variables are 'indeterminate' or we have an "indeterminate" system of equations. Here, we cannot uniquely determine the values of all the variables. There will be infinite sets of solutions that satisfy the equations. For example, if we take the following two equations in three unknowns,

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j I

Concept Revisited Questions i \

1

1. Twice a number is 12 more than half the number. What is the number. 2.. Three years ago, Anurag's age was thrice that of Bharagav. Two years hence Anurag's age will be twice that of Bhargav. What is the present age of Anurag?

10. If a number is divided into two unequal parts, then the difference of the squares of the two unequal parts is 48 times the difference of the two unequal parts. What is the number?

3.

Goutam is four times as old as Girija. Four years hence, the age of Goutam will be thrice that of Girija. What is the present age of Goutam?

11. Three sharpeners and four. erasers cost Rs.25. Four sharpeners and three erasers cost RS.24. What is the cost of each sharpener and eraser. (1) Rs.4, RS.3 (2) Rs.3, Rs.4 (3) Rs.3, Rs,3 (4) Rs.4, Rs.4

4.

2 pencils and 3 pens cost Rs.8. 2 pens and 3 pencils cost RS.7. What is the cost of 5 pencils and 5 pens?

12. How many pairs of x and y satisfy the equations 2x + 4y = 8 and 3x + 6y = 10? (1) 0 (2) 1

5. Rajani'sage is 9 years more than half of her age. What is Rajani's age?

6.

One samasa and two puffs cost Rs.14. Three samosas and one puff cost Rs.17. What is the cost of 5 samosas and 5 puffs?

7.

Given that the two digits of a two-digit number differ by 4, what is the difference between the number and the number formed by reversing its digits?

8. The differencebetweena three-digitnumber and the number formed by reversing its digits, always divisible by (3) 101 (4) 99 (1) 9' (2) 11

9.

Given that the first and the last digit of a threedigit number differ by 3, what is the difference between the three-digit number and the number formed by reversing its digits?

(3)

00

(4) Noneof these

13. How many pairs of x and y satisfy the equations 2x + 6y = 12 and 3x + 9y = 18? (1) 0 . (2) 1 (4) Noneof these (3) 00 14. How many pairs of x and y satisfy the' equations 4x + 5y = 22 and 3x + 4y = 17? (1) 0 (2) 1

(3)

00

(4) Noneof these

15. 2 chocolates, 4 cakes and 3 milk shakes co~t RS.17. One chocolate and 2 cakes cost Rs.4. What is the cost of milk shake? (1) (2) (3) (4)

Rs.1 RS.2 Rs.3 Cannot be determined

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Exercise-lea) ~"'ections for questions 1 to 23: Select the :::rrectalternativefrom the given choices. Solve: 5(x + 5) + 6 (y 9 (x

-

1) +4 (y

{i) x =10, Y=-9 ~3) x =5, Y =-6

--

15 Solve: --x+2y 5

- 3) =- 4

-

2)

=4

(2) x =9, Y=-10 (4) x =6, Y =-5

11 =2 3x+4y 22' 5

=

2(x+2Y)+ 3x+4y 2' (1) x=3,y=1 (2) x=1,y=2

(3) x = 2, y = 1

(4) x = 1,Y= 3

,.'

:

A fraction becomes 2/3 if its numerator is increased by 1 and the denominator by 2. It becomes 3/4 if its numerator is increased by 4 and the denominator by 5. Find the fraction. (1) 4/5 (2) 3/5 (3) 5/7 (4) 2/7

1

Five years ago, a man was five times as old as his son. Two years hence, the man willbe three times as old as his son. What is the present age of the man? (1) 12 years (2) 35 years (3) 42 years (4) 40 years

:

Venkat takes 2 hours more than Vatsa to cover a distance of 600 km. If instead Venkat doubles his speed he would reach the destin~tion4 hours before Vatsa. FindVatsa's speed. (2) 50 km/hr (1) 100 km/hr (4) 120 km/hr (3) 60 km/hr

£ The cost of three apples, two mangoes and four oranges is Rs.43. The cost of five apples, three mangoes and six oranges is Rs.66. Find the cost of each apple. ~1) Rs.5 (2) Rs.6 ;3) RS.3 14) Cannot be determined .

Findthevalueofk ifthe equations3x+4y =24, and 15x+ 20y =8k are consistent. 1) 5 (2) 120 (3) 30 -"'\rl)hant

Institute of Management

(4) 15

8. The cost of three pens, five pencils and two books is Rs.68. The cost of six pens, seven pencils and four books is RS.121. The cost of nine pens, fifteen pencils and six books is

RS.204.Findthe cost ofeach book. (1) (2) (3) (4)

RS.7 Rs.11 RS.5 Cannot be determined

.

.

,-

9. Find the smaller of the two numbers such that their sum is 250 and the difference of their squares is 9000. (3) 107 (4) 100 (1) 109 (2) 141 10. A two-digit number is formed by either subtracting 17 from nine times the sum of the digits or by adding 21 to 13 times 'the difference ofthe digits. Find the number. (1) 37 .(2)'73 (3) 71 (4) Cannot be determined 11. Durgesh has only RS.2 and RS.5 coins with. . him. If he has in all 57 coins worth RS.150 with him. How many RS.2 coins does he have? (3) 40 (4) 45 (2) 20 (1) 12 12. Madhav started playing a card game. In the first round he doubled his amount and he gave away a certain amount p to his friend. In . the second round he tripled the amount left with him after the first round and gave away an amount 2p to his friends. In the third round he quadrupled the amount left with him after the second round and gave away an amount p to his friend and was finally left with no money. If he gave away a total of Rs.160 to his friend, then what was the money that he started with? (2) Rs.40 (1) .Rs.21 (3) Rs.24 }4) Rs.35 13. There are seven children standing in the line,

not all of whom have the same number of cakes with them. If the first child distributes his cakes to the 'remaining six children such

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that he doubles their respective number of cakes, then he will be left with four cakes. Instead, if the second child takes away two cakes from each of the remaining six children, then. he will be left with three cakes less than the number of cakes that the first child initially had. What is the total number of cakes that are there with the third child to the seventh child? (1) 10 (2) 11 (3) 12 (4) 13 14. Five years ago, the age of a person was three years more than four times the age of his son. Three years hence, the age of the person will be six years less than thrice the age of his son. After how many years from now will their combined age be 80 years? (4) 18 (1) 22 (2) 20 (3) 16 15. The difference between a three-digit number and the number formed by reversing its digits is 396. The difference of the hundreds and the units digit, is one less than the sum of the units and tens digits. Also, the hundreds digit is twice the units digit. Find the number. . (1) 418 (2)412 (3) 612 (4) 814 16. The expenses for yoga classes in a colony are partly constant and partly varying with the number of members. If there are 50 members, then each of the members has to bear Rs.220 per month and if there are 10 more members, then the share of each of the members comes down by Rs.15 per month. How many members would be there if the share of each member is Rs.160? (1) 150 (2) 130 (3) 90 (4) Cannot be determined 17. The cost of 3 pencils, 5 rulers and 7 erasers 'is Rs.49. The cost of 5 pencils, 8 rulers and 11 erasers is Rs.78. Find the cost of 1 pencil, 1 ruler and 1 eraser. (1) (2) (3) (4)

RS.8 Rs.9 RS.7 Cannot be determined

18. Nathasha was asked to calculate 4/7 of the number. She calculated 4/17 of the number and got an answer, which was 840 less than the correct answer. What is the number to be multiplied? (1) 2600 (2) 2599 (3) 2499 (4) None of these 19. The cost of four apples, six bananas, and seven oranges is p. The cost of five apples, seven ban~nas and nine oranges is q. The cost of two apples, four bananas and three oranges is r. Which of the following is true? (1) 4p-3q=2r (2) 4p-3q=r (3) 3p-2q=r (4) 5p-3q=4r 20. P, 0 and R are successive even positive integers in the ascending order. Four times R is 4 more than five times P. What is the value ofO? (1) 12 (2) 14 (3) 16 . (4) Cannot be determined 21. Sanju and Manju went to a Bakery shop. Sanju ate 4 puffs, 3 burgers and 2 cakes and used up all the money she had. Manju ate 3 puffs, 6 burgers and 4 cakes and paid 25% more than that Sanju paid. What percent of Sanju's money was spent on the puffs she ate? (1) 15% (2) 20% (3) 60% (4) Cannot be determined 22. A test has 175 questions. A candidate gets 4 marks for each correct answer and loses 2 marks for each wrong answer and loses 1 mark for leaving the question unattempted. A student scores 405. On analysing his performance he concludes that he left unattempted 35 questions. How many. questions did he mark wrong? (1) 30 (2) 25

J(8) 20 (4) Cannot be determined

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---=

= Findk

if. the given system of equations has ;.,finite number of solutions. 4x + ky = 2 + 10y and kx + 24y = 8 '1) -10 :2) 16 ,3) -16 4) Cannot be determined

~ons for questions 24 to 26: These :w:stions are based on the data given below: :~.:d goes to a fruit market to purchase bananas, £Des and oranges. He purchased more number ~

or:angesthan apples and more number of

.x, es than bananas. He purchased

at least 30

"":5of each. He purchased eVr.n number of fruits ": :actI. The total number of fnJltS he purchased is

:c:

~

each apple cost him Rs.5, each orange Rs.4

'f

and each banana Rs.2, find the minimum a..'11ountthat David spent for purchasing the ~its. <) RS.374 3) RS.372

~

}2-) 'Rs.354 (4) RS.352

~ the condition, "He purchased more number ::: oranges than apples and more number of apples than bananas. is deleted, then find the -Tiimum amount that David spent for :urchasing the fruits by using the data given.

above question.

...

. Rs.374 3 RS.350

~

(2) Rs.354 (4) Rs.352

David purchased less than 38 oranges, then

:

-cw many apples .

32

did he purchase?

(2) 34

(3) 36

(4) 30

~ons for questions 27 to 35: Select the ;d alternative from the given choices.

-

::

... a four-digit number, the sum of the digits in -e .Jnitsplace and the tens place is equal to s..r of the digits in the hundreds and the ~o.;sands places. The sum of the digits in the E""Sand hundreds places is twice the sum of -e other two digits. If the sum of the digits of -e "lumber is more than 20, then the digit in -e .;nits place cannot be (4) 5 2 (2) 3 (3) 4

-~.a-t t

-

28. A teacher asked a student to add nine twodigit numbers such that each number is ten more than the previous number. But the student has added all the numbers,which are' formed by interchangingthe digits of each of the given numbers. Surprisingly, the sum he got was the same as the sum the teacher had asked for. Which of the following is the difference of the digits in the least of the numbers the student was asked to add? (1) 2 (2) 3 (3) 4 (4) 5 29. The age of a woman is presently thrice that of her daughter. When the woman was 29 years old, her only son, who is three years younger than her only daughter, was born. What is the present age of the son? (1) 8 years (2) 9 years (3) 10 years (4) 11 years 30. In an examination 3/Sth of the students that appeared failed b9 20 marks and 1/5thof the students got 20 marks above the pass mark. Each of the remaining students got 40 marks above the pass mark. The maximum marks in the examination is 100. A total of 120 students appeared for the exam and their average mark is 62 marks. The pass mark is (4) 56 . (1) 64 (2) 66 (3) 62 31. A man had enough money to purchase 16 apples or 10 mangoes. If the man buys 4 apples, and five mangoes and is left with Rs.20, then 'what is the difference in the prices of an apple and a mango? (1) Rs.2 (2) RS.3 (3) Rs.4 (4) RS.6 32. Ninety is divided into three parts such that the sum of the first two parts exceeds the sum of the second and third parts by 18. If the smallest part is 18, then the greatest part is (1) ,,45 (2) 54 (3) 63 (4) Cannot be determined 33. Dheeraj has twice as many sisters as he has brothers. If Deepa, Dheeraj's sister has the same number of brothers as she has sisters, then Deepa has how many brothers? (1). 2 (2) 3 (3) 4 (4) Cannot be determined

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35. Once in ten years the government of a country counts all. the people living in the country. The following data about two neighbouring villages of Badasansthan and Chotasansthan were complied. Badasansthan has 5114 more males than Chotasansthan. Chotasansthan has twice as many females as males. Badasansthan has 3,004 more males than females. Badasansthan has 9118 females fewer than those of Chotasansthan. The number offemales in Badasansthan is (1) 11,228 (2) 16, 342 (3) 13,628 (4) 13,338

34. A change dispensing machine contains 1 rupee, 2 rupee and 5 rupee coins. Initially it is filled with the equal number of coins of each denomination. Then Aashish took. all five rupee coins and the total amount changed to . RS.105. Jaspal then took all one rupee coins. The total amount in the change dispensing machine at present is (1) Rs.70 (2) Rs.75 (3) Rs.35 (4) Cannot be determined

Key 1. 2. 3. 4. 5.

3 2 3 4 3

6. 3 7. 4 8. 4 9. 3 10. 2

11. 4 12.4 13.2 14. 3 15.4

16. I 17.2 18. 3 19. 3 20.2

21. 3 22.3 23.2 24.3 25. 3

26.2 27. 1 28.3 29. 3 30.3

31. 2 32. I 33.2 34. I 35. 4

I

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I

-- -

..

Exercise -1 (b) ~r-ections for questions 1 to 13: Select the ~

Solve: 5x + 8y 1) x=10,y=1

=59;

Solve:9x + 1)

11y

=

(4) x = 5, Y= 1

=53;

11x + 9y

x=2,y=3

(1) (2) (3) (4)

=47

(2) x=3,y=2

'3) x = 4, Y= 1 -

money with Sunil. If Amit gives Rs.40 to Sunil, then the amounts with them would be equal. If instead, Sunil gives Rs.10 to Amit, then Amit would have RS.100 more than that with Sunil. Find the amount with Sunil.

23 (2) x=4,y=5

2x + 3y

3) x = 7, Y= 3 .

9. There is some money with Amit and some

alternative from the given choices.

(4) x = 1, y = 4

32 -5 Solve: ~ + 4y = 11; 2 . x + {Y75} 1) x=6,y=2 (2) x = 10, Y =-8 (4) x 5, Y 4 3) x = 2, Y= 6

10. Varun's present age is thrice that of Tarun's age three years ago. Nine years hence, Varun would be thrice as old as Tarun today. Find the sum of their present ages? (1) 26 years (2) 7 years (3) 11 years (4) Cannot be determined

-

= =

to. A.two-digit number is such that the sum of its digits is five times the difference of its digits. If :.~ number exceeds the number formed by its :.~ digits by 18, then find the number. 1)

96

'(2) 75

(3) 64

(4) 42

11. The cost of 7 pencils, 5 erasers and 6 sharpeners is Rs.45, while that of 12 sharpners, 8 pencils and 10 erasers is Rs.78. Find the cost of

:. l1e sum of the ages of two friends Avish and -akhan 14 years ago was one-third of the SlIm of their ages today. If the ratio of the :>resentages of Avish and Lakhan is 4 : 3, :'~n what is the present age of Lakhan?

32years

(2) 18 years

3) 24 years

'(4) 21 years

1)

i.

5 erasers, 6 sharpeners and 3 pencils. (1) RS.12 . (2) Rs.23 (3) Rs.37 ,

(4) Cannot be determined

ne ratio of number of chocolates with Seoni and Varsha is 7 : 9. If Varsha has <4 chocolates more than Seoni, then find .the

'otalnumberof chocolateswiththem. 1) 42(2)

-

84

(3) 98

(4) 112

~ere is a total of 500 families residing in a ocality. For Ganesh festival, some families dOnated Rs.100 each and the remaining amilies donated Rs.50 each. If the total '"I"\Oneycollected is Rs.44450, what is the "'\.Imberof families, who donated RS.50each? 1) 399 (2) 389 (3) 101 (4) 111

!

Seven burgers and eight pizzas together, cost RS.780.While twelve burgers and five pizzas ::ost Rs.945. Find the cost of each pizza. 1

J Rs.60

3) Rs.45 - -.rmant

-.,

Rs.140 RS.100 Rs.120 Cannot be determined

.

12. The cost of 7 pencils, 5 pens and 2 books is Rs.39. The cost of 5 pencils, 3 pens and 4 books is Rs.39. The cost of 3 pencils, 5 pens and 7 books is RS.56. Find the cost of each pen. (1) RS.2 (2) Rs.3 (3) Rs.5 (4) Cannot be determined 13. Katrina wants to buy 6 kg of tomatoes and 7 kg of potatoes which together would cost her Rs.190. In the market as tomatoes were very good, she decided to buy 2 kg more tomatoes and 6 kg less potatoes and spent only Rs.170. What is the price of 1 kg of tomatoes?

(2) Rs.40

(1) RS.10

(2) Rs.15

(4) Cannot be determined

(3) RS.20

(4) None of these

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Directions' for questions 14 to 16: These questions are based on the data given below. A shopkeeper sold a certain number of toys all at a certain price. The number of toys that he sold is a three-digit number in which tens digit and units digit are same and are non-zero, and the price of each toy is a two-digitnumber when expressed in rupees.ay mistake he reversed the digits of both, the number of items sold and the price of each item. In doing so, he found that his stock account at the end of the day showed 792 items more than what it actuallywas. 14. What could be the actual number of toys sold? .(1) 800 (2) 119 (3) 199 (4) 911 15.. If the faulty calculations show a total sale of RS.5117, what was the actual selling price of each toy? (1) Rs.43 (2) Rs.37 (3) RS.75 (4) Rs.34 16. Using the information in the above question, what are the actual sales? (1) RS.39173

(2) Rs.30974

(3) Rs.4046

(4) Rs.4064

.

Directions for questions 17 to 35: Select the correct alternative fromthe given choices

17. A two-digitnumber is such that it is three more than six times the sum of the digits.If the number formed by reversing the digits is three less than five times' the sum of the digits,what is the tens digitof the number? (1) 2 (2) 5 (3) 7 (4) Cannot be determined 18. Rahul takes 6 hours more than Pathak to cover a distance of 540 km. If instead, Rahul doubles his speed he would reach the destination one and half hours before Pathak. Find Pathak's speed. (1) 36 kmph (2) 60 kmph (3) 45 kmph (4) 40 kmph 19. Instead of multiplyingN a two-digitnumber, by 3/4 and adding 38, a student multiplied the number formed by reversing its digits with4/3 and subtracted 38 from the product and still Triumphant

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got the same result. Find the value of N given that the sum of the digits of the number is 12. (1) 84 (2) 72 (3) 48 (4) 36 . 20. Kunal has only 25 paise and 50 paise ClJins with him. The total amount in 50 paise denomination in RS.4 more than the total amount in 25 paise denomination. The number of 25 paise coins is 20 more than the number of 50 paise coins. What is the total amount withKunal? (2) Rs.36 (1) Rs.32 (4) RS.24 (3) ~s.40 21. The cost of 6 shirts, 7 ties and 8 pairs of shoes is RS.7500 while the cost of 3 shirts, 4 ties and 5 pairs of shoes is RS.4350. Find the cost of each shirt. (1) Rs.400 (2) RS.100 (3) Rs.550 (4) Cannot be determined 22. The amount with Aarthi is RS.80 more than that with Bhargavi. Chandini has RS.50 less than the amount with Bhargavi. Divyani has Rs.120 more than the sum of the amounts with Bhargavi and Chandini. The money with them, when pooledtogether, is in denominations of Rs.10 and RS.20 only and they together have a total of Rs.500. What is the least number of RS.10 notes they can have? (1) 1 (2)' 2 (3) 3 (4) 4 I

I

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24. A certain proper fraction when expressed in ; the simplest form greater than zero has its i numerator 14 less than thrice the i denominator. If it is the least of such proper ~ fractions, what is its denominator? (4) 7 (1) 4 (2) 6 (3) 5 25. There are a total of 50 vehicles kept in the parking lot outside of a cinema theatre, of i which some are cars and the' remaining are: motorcycles. If the total number of wheels is i 160, what is the number of motorcycle that are kept in the parking lot? (Do not consider the spare tyres) (1) 10 (2) 15 (3) 20 (4) 30 DO: 958, 2nd Floor, Siddamsetty

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31. Manish was asked to find 5/6 of a number and he instead multiplied it by 6/5. As a result he got an answer, which was more than the correct answer by 649. What was the number? (1) 1870 (2) 1770 (3) 1860 (4) 1760

:E -"e present age of a woman is six times that 0' her only daugh!er, Anu. Her husband's ~ge :s two years more than seven times Anu's. ~e average age of Anu, her parents and her only brother (who is the youngest of all the 'our members) is 15 years. By how many jears is Anu elder to her brother? 1)

1

(2).2

:3) 3

32. If the following system of equations has infinite solutions, find the value of p. 3x+py=48 (p + 5)x + 2y = 96 (1) 3. (2) 2 (3) 1 (4) Cannot be determined

(4) Cannot be determined

=- At a certain stationery store, only four varieties of pens and only four varieties of pencils are sold. The .price of a set of one of each of the 'our varieties of pens is Rs.45. The price of a set of one of each of the four varieties of pencils is Rs.12. The price of the cheapest pencil is Re.1. and the prices of three of the ~rieties of pens are exactly twice the prices of the other 3 varieties of pencils. What is the price of the costliest variety of pen? '1) RS.24 (2) Rs.23 ~3) RS.22 (4) None of these

33. The expenses of a hostel are partly fixed and partly varying with the number of occupants. If the number of occupants is 60, each of the occupants has to bear Rs.1200 per month and if there are 30 more occupants, then the share of each of the occupants comes down by Rs.200 per occupant. How many occupants should be there if the share of each occupant is Rs.900? (1) 120 (2) 150 (3) 180 (4) 200

3. ~ohan has a certain amount of money and Sohan has a greater amount. If Sohan gives ~s.60 to Rohan, then the amounts with Rohan and Sohan are the' same. If Rohan gives Rs.30 to Sohan, the amount with Sohan is thrice the amount with Rohan. What is the total amount with Rohan and Sohan together? (1) Rs.540 (2) Rs.480 (3) Rs.420 (4) Rs.360 :!

34. Madhav, Munjal and Melkote bought a bike for Rs.45000. Munjal paid one-third of the total amount paid by Madhav and Melkote. Madhav paid half of the total amount paid by Munjal and Melkote. How much did Melkote pay for the bike? (2) Rs.17250 (1) RS.12750 (3) Rs.18750 (4) Rs.19250

In an organization, two-thirds of the employees are software professionals and one-fifth of them are women. If there are 240 men are software professionals, then the number of employees in the organization is (1) 450 (2) 540 (3) 750 (4) 360

35. Which one of the following conditions must a, band c satisfy so that the following system of linear equations has at least one solution, such that a + b + c,* O?

::c..The cost of each pencil is RS.2, the cost of each pen is Rs.5. There are some pencils and pens in a box costing Rs.50. If total number of pens and pencils together in the box is 16. What is the number of pencils in the box? (1) 5 (2) 8 (3) 10 (4) 15

- =a

x + 3y 4z

4x+y- 5z =b x+y-2z=c (1) 2a + 3b - 5c = 0 (2) 3a + 2b - 5c = 0 (3) 8a + 2b -11c = 0 (4) 3a+2b-11c=0

Key

- 43

-.

.

6. 4

11. 3

16.2

21. 4

26. 2

31. 2

I

7. 4 8. 3 9. 4

12. 2 13. 3 14.4

17.4 18.2 19.3

22. 2 23. 1 24.3

27. 2 28. 4 29. 1

32. 3 33. 1 34.3

2

10.4

15.4

20. I

25.3

30.3

35.4

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Chapter - 2

RATIO - PROPORTION - VARIATION RATIO If the values of two quantities A and Bare 4 and 6 respectively, then we say that they are in the ratio 4 : 6 (read as "four is to six"). Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part or parts, one quantity is of the other. The ratio of two quantities "a" and "b" is represented as a : b and read as "a is to b". Here, "a" is called antecedent, "b" is the consequent. Since the ratio expresses the number of times one quantity contains the other, it's an abstract quantity. Ratio of any number of quantities is expressed after removing any common factors that ALL the terms of the ratio have. For example, if there are two quantities having values of 4 and 6, their ratio is 4 : '6, Le:, 2 : 3 after taking the common factor 2 between them out. Similarly, if there are three quantities 6, 8 and 18, there is a common factor between all three of them. So, dividing each of the three terms by 2, we get the ratio as 3 : 4 : 9, If two quantities whose values are A and B respectively are in the ratio a : b, since we know that some common factor k(>O) would have been removed from A and B to get the ratio a : b, we can write the original values of the two quantities (Le., A and B) as ak and bk respectively. For example, if the salaries of two persons are in the ratio 7 : 5, we can write their individual salaries as 7k and 5k respectively.

From this we can find that a ratio of greater inequality is diminished and a ratio of less inequality is increased by adding same quantity to both terms, Le., in the ratio a : b, when we add the same quantity x (positive) to both the terms of the ratio, we have the following results if a < b then (a + x) : (b + x) > a : b if a > b then (a + x) : (b + x) < a : b if a = b then (fl + x) : (b + x) = a : b . This idea can also be helpful in questions on Data Interpretation when we need to .compare fractions to find the larger of two given fractions. If two quantities are in the ratio a : b, then the times the total of

the two quantities and the second quantitS'will be equal to

~ a+b

times the total of the two .

quantities.

Examples 2.01 The salaries of David and John are in the ratio of 5 : 9. The sum of their salaries is Rs.35,000. Find their individual salaries.

Sol: Since the ratio is 5 : 9. David's salary is

A ratio a : b can also be expressed as aIb. So if two items are in the ratio 2 : 3, we can say that their ratio is 2/3. If two terms are in the ratio 2, it means that they are in the ratio of 2/1, Le., 2 : 1.

~ 14

th

.

of their total salary and

.

III

John's salary is ..

~ 14

of their total salary. 5 x 35,000 = Rs.12,500 14

:. David'ssalary= -

"A ratio is said to be a ratio of greater or less inequality or of equality acCording as antecedent is greater than, less than or equal to consequent". In other words, the ratio a : b where a > b is called a ratio of greater inequality (example 3: 2) the ratio a.: b where a < b is called a ratio of less inequality (example 3 : 5) the ratio a : b where a = b is called a ratio of equality (example 1 : 1)

~ a+b

first quantity will be

9 John's salary = -x3S,OOO=Rs.22,SOO 14 2.02 If a: b = 3 : S, then find (2a + 4b) : (3a + 5b).

Sol: a: b = 3: S=> !=~ b 5

.

(2a + 4b) : (3a + Sb) = 2a +_4~ 3a+ Sb

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~ 14d = 784 :. d = 56 From this we can calculate the values of a, b and c as Rs.168, RS.224 and Rs.336 re"spectively. .

Dividing the numerator and denominator by b, we get

- 2(~)+4 - 3(~)+5

2.06 Divide Rs.4200 into four parts such that a fourth of the first part, a sixth of the second part, an eighth of the third part and a tenth of the fourth part are all equal.

Substituting the value of alb as 3/5,

- 2(~)~4

13 17

Sol: Let a, b, c and d be the four parts.

- 3(~)+5 :.::a The

then a + b + c + d = 4200 (1) We also have a bed -=-=-=-=k(say) 4 6 8 10 So, a = 4k, b = 6k, c = 8k, d = 10k Substituting these values in (1), we get

runs scored by Rahul and Ramesh are

in the ratio of 13 : 7. If Ramesh scored 48 runs less than Rahul, then find their individual runs.

28k = 4200 ~ k = 150

Scl: Since the ratio of the runs of Rahul and Ramesh is 13 : 7, we can take their individual scores as 13k and 7k respectively. So the difference between their scores will be 13k - 7k = 6k, which is given to be 48.

Hence, a = Rs.600, b = Rs.900, c = Rs.1200, d = Rs.1500 2.07 If a : b = 2 : 3 and b : c = 4 : 3, th"enfind the ratio of a : b : c.

6k = 48 ~ k = 8 :. 13k = 104 and 7k = 56 :. Rahul scored 104 runs and Ramesh scored 56 runs. :':.4 What number shall be added to or subtracted from each term of the ratio 7 : 16, so that it becomes equal to 2 : 3?

~

~

Let x be the number to be added to each term of the ratio. Then we have 7+x 2 -=~21 + 3x=32 +2x 16+x 3 :. x = 11. " So, 11 has to be added to each "term to make it 2 : 3.

"

Sol: b is common to both the ratios. Hence rewrite the two ratios such that the b term will be the same in both of them. This can be done by taking b to be the LCM of the two values pertaining to b in the two ratios. Here, these two values are 3 and 4. The LCM is 12. So, we rewrite the two ratios such that b is 12 in both of them. a: b = 2: 3 = 8: 12 (If b is made 12 by multiplying with 4 then 'a' is also multiplied by 4 to give 8) Similarly b: c = 4: 3 = 12 : 9. Hence, a : b : c = 8 : 12 : 9 2.08 If

~=~ y

:. ~ Divide Rs.784 into four parts such that 4 times the first part, three times the second part, twice the third part are each equal to twelve times the fourth part. x..

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(3x + 7y)

.

x 2 Sol: -=y 5

Let a, b, c and d be the four parts, then we !lave 4a = 3b = 2c = 12d (1) a + b + c + d = 784 (2) Substituting values of a, band c in terms of d i., (2) we get 3d + 4d + 6d + d = 784

- -.J"""phant

then find (2x + 3y)

5'

~ x = 2y 5 " By substituting value of x in 2 2y (2x+3y) (3x+7y)'

,we get

()

3

.

5"" + y = 19y;"

3 (2;)+7Y

41y

~ 40

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2.09 Two numbers are in the ratios 4 : 7. If 14 is added to each, they are in the ratio 5 : 7. Find the numbers. Sol: Let the numbers be x and y. x 4 4y -=-=>x=y 7 7 x+14 5 -==> 7(x + 14} = 5(y + 14}

Y+14 7

=> 7

(i

.

+ 14) = 5y + 70 => 4y + 98 = 5y + 70

4y 4 x 28 -=-=16 7 7 :. x = 16 and y = 28 Alternate method: Given that x : y = 4 : 7, x, y being the two numbers. Let, x = 4k and y = 7k, where k is a constant.14 is added to each; hence the new numbers are 4k + 14 and 7k + 14. Given, (4k + 14) : (7k + 14) = 5 : 7; => 5(7k + 4} = 7(4k + 14} => 7k = 28 => k = 4. Hence, the two numbers 4k and 7k are (4 x 4) and (7 x 4), Le., 16 and 28. =>y=28=>x=

Relationship Relationship Relationship Relationship Relationship DIVIDENDO.

(1) above is called INVERTENDO (2) i.scalled AL TERNENDO; (3) is called COMPONENDO; (4) is called DIVIDENDO; (5) is caliedCOMPONENDO

The last relationship, Le., COMPONENDODIVIDENDO is very helpful in simplifying problems. By this rule, whenever we know a/b = cld, then we can write (a + b) / (a - b) = (c + d) / (c - d). The converse of this is also true - whenever we know that (a'+ b) / (a - b) = (c + d)/(c - d), then we can conclude that a/b = cld. If

! = £= ! b

d

equaIt 0

then each of these ra.tios is

f

a+c+e+,.... . b+d+f+.....

If three quantities a, band c are such that a : b : : b : c, then we say that they are in CONTINUED PROPROTION. We also get b2 = ac. In such a case, c is said to be the third proportional of a and b. Also, b is said to be the mean proportional of a andc.

PROPORTION

VARIATION

When two ratios are equal, then the four quantities involved in the two ratios are said to be proportional Le., if a!b = c/d, then a, b, c and dare proportional. .

Two quantities A and B may be such that as one quantities changes in value, the other quantity also changes in value bearing certain relationship to the change in the value of the first quantity.

This is represented as a.: b : : c : d and is read as "a is to b (is) as c is to d". When a, b, c and d are in proportion, then a and d are called the EXTREMES and band c are called the MEANS. We also have the relationship Product of the MEANS = Product of the EXTREMESLe., bc:; ad If a:b = c : d then b.: a = ~d: c (1) a:c = b:d (2) (a + b) : b = (c + d) : d (3) (obtained by adding 1 to both sides of the given realationship) . (a - b) : b = (c - d) : d (4) (obtained by subtracting 1 from both sides of the

givenrelationship)

.

(a + b) : (a - b) = (c + d) : (c - d) (5) {obtainded by dividing relationship (3) above by . (4)}

DIRECT VARIATION One quantity A is said to v~rv directly as another quantity B if the two quantities depend upon each other in such a manner that if B is increased in a certain ratio, A is increased in the same ratio and. if B is decreased in a certain ratio, A is decreased

in the sameratio.

. .

This is denoted as A a B (A varies directly as B). If A a B then A = kB, where k is a constant. called a constant of proportionality.

IUs

For example. when the quantity of sugar purchased by a housewife doubles from the normal quantity, the total amount she spends on sugar also doubles, Le., the quantity and the total amount increase (or decrease) in the same ratio.

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J;rom the above definition of direct variation, we :an see that when two quantities A and B vary :::fectly with each other, then NB = k or the ratio of the two quantities is a constant. Conversely, Ifhen the ratio of two quantities is a constant, we can conclude that they vary directly with each other.

when B is constant, then A is said to vary jointly with Band C when both Band C are varying. i.e., A a B when C is constant and Aa C when B is a

constant =>A a BC

.

A a BC => A = kBC where k is. the constant of proportionality .

. X varies directly with Y 'and we have two sets of ..aluesof the variables X and Y - X1 corresponding

Examples

D Y1 and X2 corresponding 4Ie can write down

2.10 Find the value ofx, if (3x - 2) : .(2x- 1) = (4x + 8) : (7x - 2).

X1 ---

X2

Y1

Y2

X1 or --X2

to Y2, then, since XaY,

Y1

Sol:' 3x-2 = 4x+8 2x-1 7x-2

Y2

INVERSE VARIATION A quantity A is said to vary inversely as another quantity B if the two quantities depend upon each other in such a manner that if B is increased in a certain ratio, A is decreased in the same ratio and IT B is decreased in a certain ratio, then A is irjcreasedin the same ratio. i1 is the same as saying that A varies directly with 118. It is denoted as If A a 1/B i.e., A = kIB where s k a constant of proportionality. .

Cor example, as the number of men doing a certain work increases, the time taken to do the work decreases and conversely, as the number of men decreases, the time taken to do the work ncreases. ~rom the definition of inverse variation, we can see that when two quantities A and B vary 'nversely with each other, then AB = a constant, I.e., the product of the two quantities is a constant. Conversely, if the product of two quantities is a constant, we can conclude that they vary inversely with each other. If X varies inversely with Y and we have two sets of values of X and Y - X1 corresponding to Y1 and X2 corresponding to Y2, then since X and Yare inversely re"ated to each other, we can write down X1 Y2

-

X2

=-

or

X1Y1

=

X2Y2

=> (3x - 2) (7x - 2) = (2x - 1) (4x + 8) => 21~ - 20x + 4 = 8,(2+ 12x - 8 => 13~ - 32x + 12 = 0 => 13~ - 26x - 6x + 12 = 0 => 13x(x - 2) - 6(x-2) = 0 => (x-2) (13x-6) = 0 =>x = 2 or 6/13. 2.11 If x varies inversely as

i

- 6 and is equal to

2 when y = 16, find x when y = 6.

1 m . h were m IS a => x = -, y -6 y -6 constant. Since x = 2, when y = 16 . m 2= =>m=500 wheny=6 256 - 6 .

Sol: xa~

500 500 50 x= -=.'. x=36-630 3 In these types of problems on variation, there are typically three parts: the relationship between different variables is defined to frame an equation involving the variables and the constant of proportionality one set of values of all the values of all the variables to enable us find the value of the constant of proportionality the values of all but one variable are given' and we are asked to find the value of the one variable whose value is not given

Example

Y1

JOIN1' VARIATION If there are three quantities A, Band C such that A varies with B when C is constant and varies with C

2.12 The volume of a cylinder varies jointly as its height and the area of its base. When the area of the base is 64 sq.ft and the height is 10 ft, the volume is 480 cu.ft. What is the

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-- -

forms of combinations of the two variables. when x varies directly with y, we have x - = k =>x =;'ky Y Here if we have to check whether (x '+ y)2 varies with (x - y)2, we have to take the ratio of these two expressions and check if it is a constant. If the ratio is a constant, then we can conclude that they vary directly with each other.

height of the cylinder, whose volume is 360 cU.ftand the area of the base is 72 sq.ft. Sol: Let V be the volume, a = area of the base

and h = height.

.

V = mah (m is proportionality constant) we know, a = 64, h = 10 and V = 480 480 = m. (64) (10) 3ah =>m=0.75:. V=4 3x72xh Therefore, 360 = 4 h = 360x 4 3x72

(X+y)2 - (~y+y)2 - [y(k+1)]2 - (k+1)2 (X-y)2 - (ky_y)2 - ly(k-1)]2 - (k-1)2 Since this does not involve any x or y, it is constant. Hence, we can conclude that (x + y)2 varies directly with (x - yt

=>h = 20 ft 3

Hence height of the cylinder is 6% ft. Note that the there should be consistency of the units used for the variables, Le., whatever be the units used to express the vp~:ables when the constant of proportionality is being calculated, the same units should be used for different variables later on also when finding the value of the variable which we are asked to find out.

Sol: Let total expenses be T, F be the fixed part

Examples 2.13 If the incomes of A and B are in the ratio of 3 : 4 and their expenditures in the ratio of 4 : 5. Find the ratio of their savings, given that B saves a third of his income.

Sol: We

take appropriate constants of proportionality in the two cases. A B Income' . 3x 4x Expenditure 4y 5y Savings 3x - 4y 4x - 5y We have.been given that B saves a third of his income. 4x - 5y = 1/3(4x) => 8x = 15y =>xIy = 15/8 => x = 15.kand y = 8k 3x-4y 3(15k)-4(8k) --=-13k 13 =>-= 4x - 5y 4(15k)- 5(8k) 20k 20

-

:. Ratio of their savings is 13 : 20 2.14 If x varies directly with y check whether (x + y)2varies directly with (x - y)2. Sol: This is a model of a problem where a certain r~lationship is given and we are asked to check the relationship between different Triumphant

2.15 The expenses per month of Ravi's car are partly constant and partly vary with t~e number of' kilometres he travels in that month. When .he travels 100 km in a month, the total expenses come to Rs.3200. If he travels 150 km, it is RS.3800. Find the total expenses, if lie travels 250 km in a month.

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and V be the variable part. So, T = F + V The variable part varies with number of km . travelled

in the month. => V = pn; where n is the number of km travelled in a month, and p is constant of proportionality

.

:. T = F + pn From the given data, we get 3200 = F + P (100) (1) 3800 = F + p (150) (2) Solving (1) and (2), we get p = 12 and F = 2000. :. If Ram travels 250 km, total expenses are T = 2000 + 250(12) = 2000 + 3000 = RS.5,000

The problems involving ratio and proportion are just different forms of the models of the basic problems we saw above. For example, the problem we just solved above might be reframed bringing in Mangoes, Bananas, Baskets, etc. Here, practice and perseverance pay you a lot. In entrance exams, there will be direct problems on ratio or proportion or variation directly or application of these concepts just discussed to areas like Time and Work or Time and Distance.

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Concept Revisited Questions 25 chocolates are distributed in the ratio 2 : 3 between Sita and Rita respectively. How many chocolates did Rita get? (1) 10 (2) 15 (3) '12 (4) 18

=- If a : b = 2 : 7 then find 2a : 2b 1. If x : y = 3 : 7 and y : z = 7 : 4 find x : z Ratio of two numbers is 3 : 7 and their sum is 20. Find the smaller of the two numbers.

5

If a : b = 7 : 4, then find (a + b) : (a - b) (1) 11 : 3 (2) 3: 11 (3) 4: 7 (4) 7: 4 p : q = 4 : 1..If P = a + band q = a - b, then find a: b .

-

There are a total of 30 employees in a company. The ratio of male employees to the female employees is 8 : 7. How many female employees have to be recruited so that the ratio becomes 1 : 1?

S

The marks obtained by Raju in Maths, Physics and Chemistry are in the ratio 2 : 3 : 4. If the total marks that Raju obtained in these three subjects is 189, how many marks did Raju score in Maths? (1) 21 (2) 42 (3) 63 (4) 84 Salary of A varies directly with the number of working days in the mqnth. A gets on salary of Rs.10,OOO,if the number of working days in the month is 25. What salary will he get when the number of working days in the month is 26? .

.O. Quantities x and yare

inversely proportional. When x = 9, then y = 30. Find the value of y

whenx = 6.

-"tmphant

.

11. Quantity A varies directly as the product of B and C. A = 300, when B = 20 and C = 50. What is the value of B, when A = 900 and C=60? . 12. A quantity P varies directly as the sum of Q and R. If Q increases by 2 and R increases by 2, then by how much does P increase? (1) 0 (2) 2 (3) 4 (4) Cannot be determined 13. A quantity x varies inversely as the product of y and z. When y = 7, Z = 8, then x = 5. What is the value of x when y = 4 and Z = 10? 14. The salaries of A and B are in the ratio 2 : 3. The expenditures of A and B are in the ratio 3 : 4. What is the ratio of the savings of A and B? (1) 3:5 (2) 1 : 1 (3) 1 : 2 (4) Cannot be determined 15. If 8 : 27 is the triplicate ratio of a : b, then find a: b. 16. If 2 : 3 is the sub-duplicate ratio of a : b, then find the duplicate ratio of a.: b. 17. If a is 50% of b, b is 25% of C, then find a: c (1) 1 : 2 (2) 1 : 4 (3) 3: 4 (4) 1 : 8 18. The present ages of two persons are in the ratio 3 : 5. Five years hence the ratio of their ages will be 2 : 3. Find the present age of the younger person. (2) 20 years (1) 15 years (3) 25 years (4) 30 years.

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r Exercise

- 2(a)

Directions for questions 1 to 30: Select the

property more than the youngest son, find . the total value of the property of the man. (in lakhs of rupees) (1) 7.20 (2) 6.00 (3) 4.80. (4) 4.20

correct alternative from the given choices.

1. Givena: b =3:

4, find 2a+3b

3a+4b' (2) 16: 25 (4) 18: 49

(1) 18: 25 (3) 9: 25

2. The weights of Arun and Bala are in the ratio 4 : 5. If their total weight is 45 kg, find the weight of Bala. (1) 15 kg (2) 20 kg (3) 30 kg (4) 25 kg 3. A bag has coins hi the denominations of 50 p, 25 P and 20 p in the ratio 4 : 2 : 1. If the total value of the coins is Rs.54, find the number of 25 p coins in the bag. (1) 20 (2) 30 (3) 40 (4) 50

9. A party is attended by a total of 60 persons. If the ratio of men and women is 3 : 2, how many more men must join the party such that the ratio becomes 7 : 3? (4) 25 (1) 10 (2) 15 (3) 20 10. There are two variables x and y, where x varies as the cube root of y. When y 8,

=

x 2 find x, when y (1) 6 (2) 12

(2) 27

(3) 38

following must be added

(1) 9: 4 (3) 25: 16

=O.

(4) 65

.

to two numbers

m and n such that their ratio becomes

5. Find the value of a : b where aIb is an improper fraction satisfying the equation 16a2 -:-26ab + 9b2

(4) 24

(3) 18

11. There are two numbers m and n. Which of the

4. Ifx + 16 : 4x + 20 is the triplicateratio of 3 : 4, find x. (1) 11

=

=216.

(1) mx+ny y+x

(2) my+nx y-x

(3) my -nx x-y

(4) my +nx y+x

x : y?

'"

12. If a and b are two distinct positive numbers and a : b is the duplicate ratio of the positive

(2) 9: 8 (4) 16: 9

ratio (a - p) : (b - p), where a, b, p are real numbers, then find the value of p in terms of a

6. P is 60% of q, q is 25% of r, r is 40% of s. Find p: s. (1) 3: 25 (2) 3: 50 (3) 9: 40 (4) 9: 20

andb. (1) - ~a - b

(2) ~a-b

(3) - Jab

(4) Jab

7. In an exam, the ratio of Ajay's and Balu's marks is 4 : 5. If each of them had scored 36 more marks, the ratio of their marks would be 7 : 8. Find Balu's marks. (4) 36 (1) 60 (2) 56 (3) 64

13. What should be added to two numbers which are in the ratio 4 : 5 so that their ratio becomes 5 : 6? (1) a. (2) 9 (3) 12 (4) Cannot be determined

8. An old man makes a willto divide his property amohg his wife and two sons such that his wife gets half of the total amount received by the sons. His younger son gets a third of the total amount received by his wife and the elder son. If his wife gets RS.60000 worth of

14. If a - b varies directly with a + b, a2 - b2 will

vary directly with (1) a2 + b2 (2) ab (3) Both (1) and (2) (4) Neither (1) nor (2)

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'!

The volume of a cube varies directly with the cube of its side. If three cubes of sides 6 cm, 8 cm and 10 cm are taken together and melted to form a fourth cube, find the side of the fourth cube. (2) 12 em (1) 10 cm (3) 14 cm (4) 16 em

'S There are five vessels, with equal capacities, each containing some milk. The quantities of milk in the 5 vessels are in the ratio 4 : 5 : 6 : 7 : 8; such that the total of milk in the five vessels is equal to 75% of the total capacities of the 5 vessels. How many of the vessels are at least 64% full of milk? (4) 5 (1) 2 (2) 3 (3) 4

.- A number

is divided into five parts. Twice the first part, thrice the second part and four times the fourth part are equal. Twice the second part, five times the third part and six times the last part are equal. Which of the following is always true ifall the parts are integers? (1) The first part is a multiple of 72 (2) The second part is divisible by the fourth part (3) The first part is a factor of the last part (4) The product of the first and fourth parts is divisible by 30.

.~

A truck rental agency has the following terms. If a truck is rented for 8 hours or less the charge is Rs.100 per hour or RS.8 per km whichever is more. On the other hand, if the truck is rented for more, than 8 hours, the charge is Rs.80 per hour or Rs.6 per km whichever is more. A company rented a truck from the agency, and used it for 120 km and paid Rs.800. For how many hours did the

companyrent thetruck? (1) 6

(2) 8

,

(3) 10

20. The ratio of earnings to expenditure of A is 5 : 3 and that of B is 7 : 6. If the savings of A is double the that of B. Then what could be the ratio of total eamings of A and B,together to the total expenditure of A and B together? (1) 4: 3 (2) 3: 5 (3) 5: 3 (4) 2: 1 21. If heat radiated by a certain body per unit time varies directly with the square root of the excess of the temperature of the body over the ambient temperature. The heat radiated by the body in 1 second is 1i joul~s, when the temperature of the body is 34°C. Find the temperature of the body when the heat radiated in 1 second was 20 joules. (Assume the ambient temperature to be 25.C). (1) 50°C (2) 45°C (3) 40°C (4) 30.C 22. If a : b = 2 : 3 and p : q = 3 : 2, what is the value of (2a2p3+ 3b2q3):'(3abpq2+ 4a2p2q)? (1) 1 : 1 (2) 2: 3 (3) 6,: 7 (4) Cannot be d~termined 23. A quantity Q is obtained by adding three quantities. The first is a constant, the second varies as the square rq,ot of y and, the third varies as the cube root of y. When y = 1, Q = 90, when y = 64, Q = 450 and when y = 729, Q = 1270. Find the constant. (1) 5 (2) 10 (3) 15 (4) 20

(4) 12

'S. Three friends, Aravind, Bharath and Chandu are about to have their breakfast. Aravind has 7 apples, Bharath has 5 apples and Chandu has no apples but has 12 coins. He offers to pay for some apples. They agree'to share the 12 apples equally among themselves and agree that Chandu would pay 12 coins for his -'U"l)hant

share. Bharath suggests that he be paid 5 coins and Aravind be paid "7coins. Aravind says that he should get more than 7 coins. How much should Aravind get? (1) 11 coins (2) 10 coins (3) 9 coins (4) 8 coins

24. A stone is dropped from a height of one km. The distance it falls through varies directly with the square of the time taken to fall through that distance. If it travels 64 m in 4 seconds, find the distance the body covers in the 5thsecond. (2) 24 m (1) 36 m (3) 28 m (4) 44 m

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r 25. The lateral surface area of a right square pyramid varies directly with the edge of its base when the slant height (the shortestdistancefrom the vertex to anyone of its edges) is constant. Also the lateral surface area of the pyramid varies directlywith the slant height if the edge of its base is constant. When the edge of the base is 7em and slant height is 14 em the lateral surface area is 196 em2.Find the lateral surface area of the pyramid when the edge of the base is 12 em and slant height is 20 cm. (1) 240 cm2 (2) 360 cm2 (3) 480 cm2 (4) 640 cm2 26. The kinetic energy of moving bodies varies directly with their masses, when the velocity is constant and the kinetic energy of a particular body varies directly with the square of its velocity. A body having a mass of 7.2 kg and a velocity of 0.2 mls has a kinetic energy of 0.144 joules. Find the kinetic energy of a body having a mass of 3.6 kg and a velocity of . 0.8 m/s. (in joules) (1) 0.288 (2) 0.216 (3) 0.324 (4) 1.152

.

27. The time taken by a group of workers to complete a piece of work varies directly with the amount of work to be done by them when the nl!mber of workers is constant and inversely as the number of workers in the group when the amount of work is constant. If 8 workers take 1/2 a day to plough 2 acres of a field, find the time taken by 16 workers to plough 8 acres of the field. (1) 1 day (2) 1/4 days (3) 3/4 days (4) 3/5 days 28. The total surface area ofa cylinder having a certain height is the sum of two parts. One of . the

parts varies directlywith the radius and

the other parts varies directly with the square of the radius. The total surface area is 7200 units when the radius is 30 units and 3600 units when the radius is 20 units. Find

the total surface area when the radius is 10 units. (1) 1040 (3) 1260

(2) 1120 (4) None of these

29. If P + q = q + r = p + r = k , then find k. r p q (1) 1 (2) -1 (3) 2 (4) More than one of the above 30. A string is cut into two parts such that the ratio of the lengths of the complete string .and the smaller part is 20 times the ratio of the lengths of the smaller part and the larger part. Find the ratio of the length of the string and the square of the length of the smaller part (taken in cm) if the longer part is 4cm. (1) 5: 3 . (2) 5: 4 (3) 5: 2 . (4) 5: 1

Directions for questions 31 to 33: These questions are based on the data given below. . There are two villages P and Q. The population of village P exceeds that of village Q by 100. The ratio of the number of males and that of females in villages P and Q are 5 : 7 and 3.: 7 respectively. Each person in the village belongs to exactly one category

- employed,

unemployed

and studying.

The ratios of the number of the employed, the unemployed and those who are studying in the two villages P and Q are 1 : 2 : 3 and 1 : 3 : 6 respectively. The number of those who are' studying in the two villages are equal. 31. Find the population of village P. (1) 500 (2) 600 (3) 550

(4) 650

32. Find the total number of females in the two villages. (1) 400 (2) 500 (3) 600 (4) 700 33. How many more I less unemployed people are there in village Q, when compared to that in village P? (1) 50 more (2) 50 less (3) 20 more (4) 20 less

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~ons, for questions 34 and 35: Select the ="'Ed alternativefrom the given choices JI. TJ,e mean proportional between two numbers s 12. The third proportional of the same -:..mbers is 96. Find the greater of the two ~~mbers.

. 18

(2) 24. (4) None of these /

'3 12

35. The value of a diamond varies directly with the square of its weight. The diamond breaks into three pieces' whose weights are in ttie ratio 32 : 24 : 9. The loss caused due to the breakage is Rs.25.44 lakhs. Find the value of the unbroken diamond (in lakhs of rupees). (1) 33.62 (2) 16.8.1 (3) 42.25 (4) 8.405

Key .6. 2 7. 1 8. 1 9. 3 10. 1

--

~e

--""'::-198194195

11. 3 12. 3 13.4 14. 3 15.2

16.2 17.4 18. 3 19.3 20. 1

21. 1 22.3 23.2 24. 1 25.3

26.4 27. 1 28.4 29.4 30.4

31. 32. 33. 34. 35.

2 4 2 2 3

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SMIOOSOlf23

Exercise Directions for questions 1 to 15: Select the correct alternative from the given choices.

- 2(b) 8. There are three unequal quantities x, y and z in continued proportion.Which of the following equals z : x? 2

1. 6x + y : 7x + 3y equals the duplicate ratio of 2 : 3. Find x : y. (1) 3: 26 (2) 3: 14 (3) 4: 13 (4) None of these

(1) i: ~ (3) z2:

(2)

2

~

y2 - x2 (4) Allof the above

i

9. A student took 6 papers in an examination, a2 + b2

2. Ifa*cand

b2 + C2

(1) a + c (3) c - a 3.

where the maximum marks were the same

-=-=k,findk. a+b b+c (2) a - c

each

p, q, r, sand t are five integers satisfying p = 3q 4r and 2q = 5s = 12t. Which of the following pairs contains a number that can never be an integer? (1) (2p/15, q/t) (2) (pIt, 4r/t) (3) (p/4, 1"81180) (4) (p/8, sIr)

=

=

4. Find a : b : c, given 3a + 2b 7c and b =a + c (1) 3: 8: 5 (2) 1 : 6 : 5 (3) 1: 2 : 1 (4) 3: 10: 7 I

5. Anitha divided some chocolates among her three sons SLAch that for every 4 chocolates that the eldest son got, her second son got 3 chocolates. For every 2 chocolates that her second son got her third son got 3 chocolates. If her second son got 12 chocolates, find the total number of chocolates received by the other two sons. (3) 34 (1) 46 (2) 18 (4) 51 6. Number 85 is divided into two parts such that thrice the first part and twice the second part are in the ratio 18 : 5. Find the first part. (1) 60(2) 55. (3) 45 (4) 24 7. "Find a possible value of p : q given that 21p2 + pq 3pq_q2. (1) 5: 4 (3) 3: 7 Triumphant

= 15. (2) 7: 3 (4) 5: 6

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together, the candidate obtained 58% of the total marks. His marks in these papers were in the ratio of 12: 13 : 14: 15: 16: 17. Then the number of papers in which he got more than 55% is (1) 6 (2) 5, (3) 4 (4) 3

~a+c

(4)

paper.

In all papers

10. When Ganesh, was asked about his score in last classroom test in which the maximum marks were 50, he replied. "The ratio of

15 less than l"('Iyscoreto 15 morethan three times my score is 1 : 5" What is his score? (1) 35 (2) 40 (3) 45 (4) 50 11. If in an examination the scores of Shruthi and Tanuja are in the ratio 7:11, then the ratio of sum of Tanuja's score and 2 times Shruthi's score to the sum of Shruthi's score and 3 times Tanjua's score is (1) 29: 32 (2) 32: 29 (3) 5: 8 (4) 14: 33 12. If 4: 9 is the duplicate ratio of (4 + k) : (9 + k), then find a possible value of k. (1) 36 (2) 6 (3) 13 (4) 0 13. The consumption of diesel per hour of a bus varies directly as square of its speed. When the bus is travelling at 40 kmph its consumption is 1 litre per hour. If each litre cost RsAO and other expenses per hour is Rs.40, then what woufd be the minimum expenditure required to cover a distance of 400 km? (1) RS.600 (2) Rs.700 (3) RS.800 (4) Cannot be determined HO: 95B, 2'" Floor, Siddamsetty

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... .:.certain number of chocolatesare divided =rnong Ajay, Sujay a,nd Vijay such that Ajay ;e:s 1/5th of what Sujay and Vijay together ;et. Sujay gets 1/4th of what Ajay and Vijay Dgether get. If Sujay got 2 chocolates 1)'10re -:-an Ajay, then how many chocolates did ,jay get? . 22 (3) 38 (4) 48 (2) 60 <£ ~~e cost of bars of a precious metal varies as -e square of the weight of the bar. Metal bars :..
-

are two software companies in a city

- .

"!rc.'1Softcom"and "Fast Softcom".The ratio of

-

,..Imber male employees to the number of employees in 'Smart Softcom" is 5: 3 and

~ ~

Softcom has 400 employees more than the Softcom".

'-

---ow many employees are there is Fast ~ftcom? .

...~

"Fast Softcom" is 7 : 5. The ratio of number :r :-!ployees in the age groups (in completed ~ 21 to 30, 31 to 40 and 41 to 50 in "Smart iaiCom" and "Fast Softcom" are respectively 4 : 5 : 7 81: 2 : 1 : 3. The number of employees in the age JI:l-D 21 to 30 in both the companies is same and ...

'

1)

-

" ~

1600

(2) 1200

(3) 800

(4) 600

...,owmany male employees are there is two :ompanies together? 1) 1100 (2) 1300 (4) 1700 3) 1500 Nhat is V1e difference in the number of employees in the age group 41, to 50 in -Smart Softcom" and Fast Softcom? 1) 300 (2) 200 (3) 100 (4) 400 ,Vhat is t.he ratio of number of male employees ,In Fast Softcom to the number of :emale employees in Smart Softcom? 1) 7: (2) 2: 1 3) 1: 62 (4) 7: 10

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,Directions for questions 20 to 33: Select the correct altemative from the given choices. 20. A salesman for a company gets an incentive for every unit of product he sells apart from his fixed salary. He gets Rs.8,OOO and Rs.9,OOOfor 150 and 200 units he sold respectively., If he sells 400 units, what is his income per unit? (1) Rs.32.5 (2) Rs.30 (4) Rs.20 (3) RS.27.5 21. The electricity bill for a month varies as the number of units consumed. The charge per unit is Rs.1.35 upto 50 units used. If the number of units consumed is more than 50 then the cost of the additional units is Rs.2.70. If the consumption in the first month is 97 units, then what should the consumption in the second month be, such that the average for the two months combined is RS.135per month? (1) 46 . (2) 56 (3) 53 (4) 57

22. The earnings of A and B are in the ratio 3 : 7 and that of Band C is 4 : 9 and that of D and C is 7 : 6. If the sum of the earnings of A, B, C and D together is Rs.52950, then what are the earnings of D? (1) Rs.21,050 (2) Rs.22,050 (4) Rs.24,050 (3) Rs.23, 050 23. If p : q : r: s is 3 : 4 : 7 : 6 and p + q = 84, then r+ s is (2) 156 (4) 303 (1) 294 (3) 147 24. If p = 9, q = 18, r = 36, then which of the following is true? (1) p': p'" p2 : q2 (2) p: r =( 1'2+ q2) : (~ + q2) (3) p: r = q2 : ~ (4) All of these 25. The monthly telephone bill has a fixed tariff of RS.250 for upto a 50 outgoing calls. For over 50 calls, therl;~is a charge of RS.1.25 per call. The ratio of the bill paid by Aravind and Prasad for a particular month is 2 : 3 and the number of out going calls made by Aravind is 90 then what is the number of outgoing calls m~de by PrasaCl? (1) 210 (2) 250 (3) 160 (4) 180

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26. The kinetic energy that a body acquires when it falls freely for a time t varies directly with the square of t for a given mass. For bodies, of different masses for a given value of t, the kinetic energy varies directly as the mass. The kinetic ,

31. Find b : a from the equation 15a2 - 26ab + 8b2= 0 given that b/a is a proper fraction. (1) 2: 5 (2) 5: 2 (3) 4: 5 (4) 3: 4

32. The distance through which a freely falling body drops is directly proportional to tlie square of the time for which it drops. If a body falls through 320 m in 8 seconds, then find the distance that the body falls through in the next , 2 seconds. (1) 160 m (2) 180-m (3) 95 m (4) 120 m

energy of a body of mass 5 kg, which is under free fall for 10 seconds is 25 kilojoules. Find the kinetic energy of a i::>ody with mass 2.5 kg, which is free fall for 3 seconds (in kilojoules) (1) 102.5 (2) 1.125 (3) 125 (4) None of these

,

27. 18 men take 30 days to complete a piece of work working 10 hours a day. Find the time taken by 25 men to complete five times as much workworking 12 hours a day. (1) 30 days' (2) ;36days (3) 72 days (4) ,90 days 28. The cost (C) of a certain type marble varies as the square root of its surface area (A) when its thickness (T) is constant and C varies square of T when A is constant. If the cost of a piece of the marble of surface area 1600 sq.cm. and thickness 3 cm is Rs.360. what is the cost of the marble piece of area 900 .sq.cm. and thickness 4 cm? (1) Rs.360 (2) Rs.480 (3) RS.540 (4) Rs.600 29. The mass of a liquid varies directly with its volume (when all other parameters are constant). The mass of a liquid is 12' gm when its volume is 15 cm3. Find the mass of the liquid if its volume is 25 cm3.

(1) 24 gm (3) 20 gm

33. If x : y : z = 3 : 5 : 7 and a = 2x, b = 3y and c = 6z,

then a : b : c is equal to . (1) 2: 3 : 6 (2) 2: 5 : 14 (3) 5: 8 : 13 (4) 1 : 2 : 1

Directions for questions 34 and 35: These questions are based on the data given below: Thomas and Lala using cell phone services under a particular schemes in which the bill depends on . the number of out going calls made and the number of incoming calls received. For every out going call made, the charge is RS.2.50 and for every incoming call received the charge is Re.1. In a particular month, the number of calls' received by Thomas is 10 more than than that by Lala.

(2) 30 gm (4) 18 gm

30. The radii of cylinders of equal height vary directly with the square root of their volumes. The radii of cylinders of equal volume vary inversely as the square root of their heights. The radius of a cylinder is 10 cm when its volume is 1500 cm3 and height is 5 cm. Find the radius of the cylinder of volume 2400 cm3 and height 2 cm (1) 10cm (2) 15cm (3) 20 cm (4) 25 cm

34. If the number of out going calls made by Lala is 70 and the total number of calls (Le., incoming cells received and out going calls made) for Lala is 150, what is the bill for Lala? (1) RS.255 (2) RS.250 (3) RS.245 (4) Rs.260 35. If ihomas is Rs.250 and the number of calls received by Lala is 60, then what is the total number calls (Le., incoming calls received and out going calls made) for Thomas and Lala together, if the number calls made by Lala is 20 more than that of Thomas? (1) 324 (2) 314 (3) 304' (4) 294

Key l. 2. 3. 4. 5.

1 1 4 3 3

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Chapter

PERCENTAGES

-3

- PROFIT & LOSS -

~CENTAGE

PARTNERSHIPS

= Quality increase

from 1993 to 1994 x 100 Actual production of rice in 1993 = - 17 x 100 =75/9% .

~-

implies "for" every hundred". This ="='" is developed to facilitate easier ---a"'SOn of fractions by equallsing the -'::'--ators of all fractions to hundred.

..

225

Ratio between any two quantities can also be

.

}

\

...,

,

.

Whenever there is any percentage increase or decrease on a quantity, we can directly calculate the new valu~ of the quantity instead of calculating the actual increase / d~crease and then adding to / subtracting from the originalquantity.

~:ages can also be represented as decimal ~;.--s In such a case it is effectivelyequivalent. 8T:'X1ion of the originalquantity.

~

:or ~~Z'1ple,20% is the same as ~, i.e, 0.2. 100W

For example,ifthe increase on a figureof 350 is'

-

15%, the new quantity is 1.15 x 350 = 402.5 (where 1.15 = 1 + 0.15, 0.15 being the decimal equivalent of 15%).

:e-centage can be expressed as a decimal -=c,... by dividingthe percentage figure by 100

..

:::::nversely, any decimalfractioncan be

~ed

to percentage by multiplyingit by 100.

~:E\'TAGE INCREASE or DECREASE of a , ISthe ratio expressed in percentage of

- a:-..;al INCREASEo~. DECREASEof the ---, tothe originalamountofthe quantity,i.e., ~.::.ENTAGE INCREASE - .:=...a! increase x 100 :.- :;"'al quantity

~:a.'TAGE

:: ?~I ..

quantity

~ ~;Z"1ple,ifthe production of rice went up from == r- in 1993 to 242 MT in 1994. then the "-:-~-.age increase in rice production from 1993 s :..:..:.:S calculated as follows: -<:...s ~crease

.

If the production in 1994 is given as 400 MTand the increase from 1993 to 1994 is given to be 25%, then the production in 1993 willbe equal to .400/1.25

= 320

MT (where 1.25

=1

+ 0.25,

0.25

being the decimalequivalent of 25%). 4~.' . L~':Similarly, if there is a decrease of 12% on a quantity of 225, "- then the' new quantity will be equal to 225 x 0.88 (where 0.88 = 1 - 0.12,0.12 being the decimal equivalent of 12%). If the production in 1994 is given as 400 MT and it is a decrease of 13% from 1993, then the production in 1993 will be «;!qualto 400/0.87 (where 0.'37 = 1 - 0.13, 0.13 being the decimal equivalent of 13%). ,,'1

DECREASE

-=..a decrease x 100

=242 - 225 =17 MT

On the basis of percentage increase, we can write down how many times thE.\old value gives the new value. For example, if the percentage increase is 100%,thenwe can concludethat the newvalue is 2 times the old value. Ifthe pE~rcentageincrease is

~-"""4Qe increase

300%,the new value is 4 times the old value..If ---':--crt Instituteof ManagementEducationPvt. Ltd.(TJ.M.E.)HO: 958, 201Floor,SiddamsettyComplex, Secunderabad- 500003.

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~

expressed as percentage. . \. For example, if the ratio of A and 8 is 3 : 2, we can say the ratio of A : 8 is 60% : 40%.

;;W ~3"1ple" 7/11 as percentage is represented

-

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.

\

1.

~

...

..,;;'

the percentage increase is 450%, then the new' value is 5.5 times the old value. In general, if the percenta~crease is p%, then the new value is ~ + 11 100 tlrt mes the old value.

(

)

Conversely, if we know how many times the old value gives the new value, we can find out the percentage increase in the old value to get the new value. For example, if the new value is 3 times the old value, the percentage increase in the old value to get the new value is 200%. If the new value is 4.25 times the old value, then the percentage increase is 325%. In general, ,if the new value is k times the old value, then the percentage increase i~

~'X

Now, we have 100 graduates in 2000 going upto 144.375 in 2003, which is a 44.375% increase (betause the base is 100 itself). In general,

.

if there are successive increases ofp%, q% and r% in three stages, the effective percentage increase is 100 H 100 + p 100 + q -1 'X 100 100 100 {( 100

)(

)(

)}

If one or more of p, q and r are decrease percentage figures and not increase percentage, then it will be taken as a negative figure and not as a positive figure.

100. J Similarly, if the resultant figure is negative, it means it is a net decrease..

Examples 3.01 The production of rice increased by 75% from 1990 to 1995. From 1995 to 2000, there was a 100% increase. Find the percentage increase in production of rice from 1990 to , ..,~5 ,').15 2000.

.

Sol: If production of rice in 1990 is 100, 'it will be 175 in 1995. Then there was a 100% increase from 1995 to 2000: This mejins, in 2000 the rice production will be 175 + 175 =350. So, production went up from 100 in 1990 to

The same can be extended to any number of successive increase or 'decrease . " percentaQes. 3.03 The price of a Swiss watch was Rs.12,OOO in 2001. But due to devaluation of the rupee it became Rs.15,000. What is the percentage increase in the price of the Swiss watch?

Sol: Percentage increase

,

\; .'\> '"

~

:\~;)

= Final Pr ice - Initial Pr ice x 100 Initial Pr ice

350 in 2000, an increase of 250 on the

= 15,000 - 12,000 x 100 = 25% 12,000

original quantity of 100, this is a' 250% increase.

3.04 In 2000, Rahul's salary is Rs.24,000 and Reema's salary is Rs.16,000. In 1999, , ' 00 ~, Reema's salary was 25% of the sums of their salaries in 2000. What is the 1~~~) 1.4& percentage increase in Reema's salary from 1999 to 2000?

3.02 The percentage increase in graduates from a

University is as follows:

2000- 2001 ~ 10% 2001- 2002~ 25%

)0." .~:~

!;:.

2002 - 2003 ~ 5%

What is the overali percentage of increase in graduates from 2000 to 2003?

Sol: Reema's salary in 1999 = (24,000 + 16,000) x 25 = 10,000

Sol: Let number of graduates in 2000 be 100. Therefore number of graduates In 2001 = (1DO)+ 10% of 100 = 110 In 2002 = (110) + 25% of 110 = 137.5 In 2003 = (137.5) + 5% of 1~7.5 = 144.375

100 Reema's salary in 2000 = 16,000 Percentage increase in Reema's Salary = 16,000-10,000 x100=60% 10,000

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3.08 If the price of an item goes up by 20% by what percentage should it be reduced to bring it down to the original price?

~ ~

:c

-

..

method:

=782.

In this problem, the percenta~e reduction

20 x 100 (100+ 20) ] and can be generated as {(100x)/(100 + x)}%

[

--'? :::0 of the salaries of A and 8 is 3 : 2%. :-. .-oa;.percentage is A's salary greater -=- 0-0
---=':'- .ef"ratio 3 : 22/3=3:

~=~:~=9: 8 3 3 3 --'! ~ of salaries of A and 8 is 9 : 8. Le re- 8's salary is 8 A's salary is 1 more -z- 3's.

~ta9F by which A's salary is greater -~ os is -x100=12.5% 8 8r

Now, to bring this down to the original price, we have to effect a reduction of 20 out of 120. Hence, percentage reduction 20 2 =. -x100 = 1613% . 120

850 =782

~ squal to 714; 714 92% =92 x 84

--e ~ight of a triangle is increased by -: c and the base by 10%. What is the =--sequent increase in area?

Sol: Note the words "MORE THAN"and "LESS THAN"given in the problem. When we say Deshpande's salary is 25% MORE THAN,if we take Ram Prakash's salary as 100, the working out of the problem becomes easier.

. In general

2

=15% =~xh=0.15h 100

~.. ~ight =h + 0.15h =1.15h -~e inbase =~xb=0.1b 100 ~ base = b + 0.1b= 1.1b 1 'e«Area = -(1.15h)(1.1b) 2 :: -:-:m(1.265) '"'Uease in Area =(0.1325) bh =-crcentage increase in area :: :::1325)bh x100=26.5% rO.5)bh ~

the object or quantity that follows

MORE THAN or LESS THAN should be made as 100. In this case, if Ramprakash's salary = 100, then Deshpande's salary = 125. Ram Prakash's salary is less than Deshpande's by 25. .

-ta' area = ..!.bh in height

can be writtenas

3.09 If Deshpande's. salary is 25% more than Ram Prakash's salary, then by what percentage is Ram Prakash's salary, less than Deshpande's salary?

.. -=- ...tialheight be 'h' and the initialbase be -zsase

NewPrice=120,due to 20%increase.

0

-~

,~

.

. =3.5C

::

~

Sol: Let originalprice =100

,="714

As a Percentage it is 25 x 100 = 20%

-125

n this problem,' the percentage calculated in

the

last

step

can

be

written

as

]

25 x100. (100 + 25)

~

3.10 If the price of coffee goes up by 25% what should be the percentage reduction in the quantity consumed so that the total expenditure on coffee remains the same? ,

Sol: Let p be the price and q the quantity initially. Since the price went up by 25%, the new price is 1.25p. If the new quantity is z, what

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{

/1

Profit = 250 - 200 = Rs.50 Profit 50 Profit%=-x100=x 100 = 25% CP 200

is asked is'z as a percentage of the original quantity q. This will be equal to q - z x 100 q Since the total expenditure is .the same, q. p.q. = 1.25p. z ~ z=1.25

3.12 A merchant gains 25% by selling a book for Rs.20. Whatwould be his percentage gain or loss if he sold at (1) Rs.12 (2) Rs.10 (3) Rs.18 (4) Rs.24

:. Percentage reduction in quantity is: q q-1.25 x100 = 0.25 x100= 25 =20% q 1.25 125 n this problem, the percentage

the

quantity 25

~(100 + 25)

can

be

x100.

reduction i

written

as

~

{In this problem also, the percentage reduction in quantity can be written as

Sol: S.P = Rs.20, Profit = 25% 20 - CP Profit%= x100=25 ~ C.P= RS.16 C.p' . (1) S.P = 12, q.P = 16 16-12 . Loss = x100=25; loss Is25% 16

(2)S.P= 10,C.P= 16

10 100} t100 + 10) x In the above' three examples, if the percentage given initially is x, what is asked to

.

be found .

IS

Loss%=

100x

16

x100=37.5:

.

loss IS 37.5% .

(3) S.P = 18, C.P = 16

(100 + x) .

Wecan generalize each of the three cases as below:

.

If the value of an item goes up/down by x%, the percentage reductionlincrement to be now made to bring it back to the original level is 100x % t100 :!: x) 0 If A .is x% morelless than B, then B is 100x I. M . .. \ % less/more than A. If the price of an item goes up/down by x%, then the quantity consumed should be

.

16-10

.

100x

reducedltncreased by (100:!:. x) % so that the total expenditure remains the same.

18-16 Profit%=-x100=12.5;Profit 16

.

. IS 12.5%.

(4)S.P = 24, C.P = 16 24-16 .. Profit%= x100=50; Profit IS 50% 16 . 3.13 By selling a table lamp at Rs.260, Soman makes a profit of 30%. Find the cost price of the table lamp.

Sol: CostPrice=260x

100 -Rs.200 (100+30)

3.14 The selling price of 10 pencils is the cost price of 14 pencils. Find the profit percentage.

Sol: Such problems where there is no amount

3.11 A shopkeeper bought a watch for Rs.200 and sold it for RS.250. What is his profit percentage? Sol: S.P = Rs.250. C.P = Rs.200

specified for the Cost Price (C.P) or the Sales Price (S.P), the best approach is to assume cost of each unit to be Re. 1 and proceed. Let cost of each pencil be Re.1. Then cost of 10 pencils is Rs.10 and cost of 14 pencils is Rs.14. Since S.P. of 10 pencils is equal to the cost of 14 pencils, which we know is Rs.14, the S.P of 10 pencils RS.14.

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',ow S.P and C.P of 10 pencils is RS.14 and D,S10 respectively. Ne get profit percentage as follows.

14-10

The computation is done as follows:

P~ofit%=-'x100=40% 10 ~'ffiC£NTAGE

Percentage

POINTS

-'0: :c"ICept of "percentage points" is important in ~ ...sageof percentages. Percentage points is - ~ ::~erence of two percentage figures.

-

-=' ...s understand this with an example. : -~ that rice forms 20% of total food grain :--r:..dion in Year I and 30% of total food grain :-. ::...ction in Year II.

- -e are asked to find out the percentage - ,93Se in the production of rice, calculating :"" :e""~ge

increase

.':..2C x 100 and

from

20

to

30

as

saying it is 50% increase is

'

-

Cost Price = S.P. - C.P. SP.- C.P. Profit = x 100 = CP.

Profit = Sale Price

',

Pr ofit x 100 C.P. Loss = C.P. - S.P. Loss Percentage Loss = x 100 C.P. It is customary to express Profit/Loss as percentage of Cost Price. However,' in some problems it may specifically be given that profitlloss percentage has been calculated on the selling price or the student may be ask~d to calculate the profit/loss percentage on the selling price. Unless such specific directions are given, the profit/loss percentage is always to be calculated 01')the cost price.

I

":- ::o
-

'=~ see by taking the following figures that the JE~ge increase in rice production need not ::;o:~. Year I Year II, 960 1000 5000 3200 - -a 'oodgrains 20% 30% ::.~ as percent of -=.0.'oodgrains -== .:;I,ilerice is 20% of total food grains in Year :-:: 30% of total food grains in Year II, we find =-e actual production of rice has not even ::. -.E

-

...

it decreased

from 1000 in Year I to

fear II.

- ,,-

business/commercial

environment

given by S.P. = C.P. x

100

Given the cost price (C.P.) and loss percentage p%. the selling price will be

(100 - P) 100

the most

~ '-.ant concern is about the profitlloss, of the :-S3Ction conducted. .

-~ SEI-UNGPRICE (S.P) and the COST PRICE

: = Of an article determine the profit or loss -.a:o:~ the particular transaction. -"t.-"""'C'"a1t

Given the cost price (C.P.) and profit percentage p%, the selling price will be . (100+p )

given by S.P. = CP. x

... "tm AND LOSS

-«

The following simple rules can be remembered for this purpose.

'

-:-~

E

. Given ProfitlLoss percentage along with S.P., C.P. can be found out and similarly, given Profit/Loss percentage along with C.P., S.P. can be found out by using the concepts discussed at the beginning of this chapter (where, if percentage increase or decrease is given, we can find out the new value from the old value or the old value from the new value).

Given the selling price (S.P.) and profit percentage p%, the cost price will be given 100 by C.P. = S.P. X I \

100 + p

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Given the selling price (S.P.) and loss percentage p%, the cost price will be given 100 by C.P. = S.P.x , \ 100 - P When two arti~les are SOLD at the; same price (Le., their S.P. is the same) such that there is a PROFIT ofp% on one article and a LOSS of p% on the other (Le., common profit or loss percentage), then, irrespective of what the S.P. actually is, the net result of the transaction is LOSS. This percentage loss is given by

Examples 3.15 The cost of 4 apples is equal to the selling price of 6 apples. Find the profit or loss percentage.

Sol: Let the cost of each apple be Re.1 :. Cost of 6 apples = RS.6. S.P. of 6 apples = Cost of 4 apples = Rs.4 Since cost price is greater than the selling price, it will be a loss in the transaction and hence 6-4 Loss Percentage

Loss percentage = (Common profit or lossY 100

=-p2

100

=-x100=331/3% . 6

3.16 A trader cheats his customers to make a profit by announcing that he sells at cost price but gives his customers only 900 ml for every litre. What is his profit percentage?

MARKED PRICE or LiSt PRICI= is the price that is indicated or marked on the product or it is the price which is given in the price list. This is the . Sol: The trader delivers only 900 ml to his price at which the product is intended to be sold. customer, while declaring it as 1 litre. However, there can be some DISCOUNT given on Hence, his cost price, in this transaction, is this price and consequently, the actual SE;LLlNG of 900 ml. If cost price is assumed as Re.1 PRICE of the product may be .Iess than the for ml, cost price of 900 ml is Rs.900 MARKED PRICE. He declares to the customer that he is delivering 1 litre and collects the sale price of SELLING PRICE = MARKED PRICE - DISCOUNT 1 litre. Because his sale price and cost price are The amount of discount given can also be same, he collects 1000 x Re.1 expressed as a percentage. DISCOUNT is always = RS.1000as sale price expressed as a percentage of the MARKED Hence, percentage profit PRICE or the LIST PRICE.

= (1000- 900) x 1~0 900

DISCOUNT percent = Marked Pr ice - SellingPrice Marked Pr ice

x 100

Certain discount is given on an article whose selling price is S.P. If further discounts are given on this discounted price, such discounts ,are referred to as successive discounts. If the successive discounts are p%, q% and r%, on a product whose selling price is S.P., then the effective price after all the discounts is given by Discounted price = S.P. x (100 - p)(1 00 - q)(100 - r)

100x100x100

-

100

-9-

-

111,

o/c go

3.17 A sells to B a desk at 10% profit, B sells it to C for 15% profit. If C pays RS.506for it what is the price at which A bought the desk?

Sol: Let the price at which A bought the desk = p 110

( ) C's C.P= (Up) (100)=1.1 B'sC.P = P =1.1p 100 .

115

x 1.15p .

'C bought the desk at RS.506. :.1.1 x 1.15 p = 506 ::p = Rs.400. Hence, A bought the desk at Rs.400.

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: -! 'zra~ sells a chair at a loss of 10%. If he -ac sold it at a profit of 5%, he would have ~ RS.75. What is the cost price of the

3.21 If goods are purchased for RS.1200 and onefourth of them sold at a loss of 10%, then at what profit percentage should the rest be sold to obtain a profit percentage of 1O%?

=-aor?

io

~ difference in profit and loss percentages :s5 - (-10) = 15% .E CP of chair = C . 75 'Stltc ofC = Rs.75 :. C=-=Rs.500 0.15

1200 =Rs.300 4

Sol: C.P of one fourth of goods = -

S.P of these goods at 10% loss = 300 x 0.9 = Rs.270 Let S.P. of the rest of the goods be X. 10% profit on RS.1200 gives S.P = Rs.1320 = X + 270 ~X = 1050

: .:: ~d bought 12 kg of cashewnuts for Qs.360. He was forced to' sell them at a loss if as much money as the S.P of 3 kg. At ,'.'18tprice did he sell the cashewnuts? 30

ass = C.P - S.P ~et S.P per kg = 5 -055 = 3 x S.P. per kg = 35 3s = 360 - 125

(

a certain number of oranges at 12

Sol: Let's assume S.P = Rs.100. Since profit = 162/3% of S.P, his profit is Rs.162f3. :. C.P = 100 - 16213= 831/3; Profit = 162/3

::>erRs.9 and the same number at 18 per ~s.9. If she sells them at 18 per Rs.15, does she gain or lose and by what percentage? :e

~et the number of oranges that Asha bought be 2X Le. X oranges at each of the 2 prices. ~e C.P of X oranges at 12 per RS.9

3:

= x.( 19J=

.

Similarly, C.P of X oranges at 18 per RS.9

=X.C~)=

16%

:. ProfitPercentage=

3 5X 4 x 100 (because 4

5X >5X ) 3 4 5X 5X --=

-'U"'lphant

3

-5X 4

4 x100=

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R

-5X 4

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x100=

33113%

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x 100 = 20%

Sol: When 2 items are sold at the SAME selling price, one at p% profit and the other at p% loss, irrespective of what the selling price is, the net result is always a loss and the loss 2 percentage = L .Here, the common profit 100 or loss = 15% 152 :. Net loss = = 2.25% 100 So, Prasad makes 21/4%loss overall.

3X X 5X C.P of the oranges = + - =424 . 15 10X 5X S.P. of the oranges = 2X.==18 6 3 5X 5X ---

=

t;'

83/3

3.23 Prasad sells two TV sets, one at a loss of 15% and the other at a profit of 15%. Find the loss or gain percentage in the overall transactions.

~

Profit%

)

3.22 Pavan calculates his profit on selling price and finds it to be 16%%. What is his actual profit percentage?

= 5 = Rs.24 :..:: :'shabuys

.

1050 2 :. Profit % = x 100 - 100 = 16/3% 900

3.24 If Alok sells an article at three-fifth of its selling price he makes a loss of 25%. What will be the profit or loss percentage, if he sells it at the actual selling price? . Sol: Let the C.P be 100. When sold at 3/5th of S.P. the loss is 25%. This means S.P. in this case = 75. This 75 is 3/5 times the actual S.P.

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PARTNERSHIPS Hence 75 = (~) S.P. =>Actual S.P = 125 If he sells at the actual S.P., then Alok makes a profit of 25 against a cost price of 100 Le. 25% profit. 3.25 A trader' marks his pr~ducts 30% over his

cost and then gives a discount of 5%. What

,

is his profit percenta,ge in the transaction?

Sol: Let C.P. be 100, then marked price is 130. On this, a discount of 5% is given. 5% of 130 = 6.5, Hence the product is are sold at

123.5 (130 - 6.5)

,

Two or more people can get together to do business by pooling their resources. The money put in by each of the partners ,is called his "INVESTMENT" or "CAPITAL." All the people who have invested money in the partnership are called PARTNERS. While two or more partners would have invested money, it is not necessary that all of them should be involved in the day-to-day running of the business. The partners involved in the day-to day activities of the business are called "working partners" and the others are called "sleeping partners" ,Or"dormant partners.".

:. Profit percentage = 23%%

3.26 ~anjay gives a discount of 15% on the marked price and in the process makes a profit of 19%. By what perrpntage did he mark the product over the cost price? Sol: Let C.P. be 100. Then the S.P is 119 (as profit percentage is 19) This S.P. of 119 has been obtained after a discount of 15% on the marked price. In other words, the S.P. is 85% of the marked price. 119 :. Marked Price = -=140 0.85 Since C.P. = 100 and the marked price is 140, we can conclude that Sanjay has marked his product 40% over the cost price. ,

3.27 Ajay sold his cycle ata loss of 9%. If he sold ,

it for RS.75more, he would have made a profit of 16%. Find the cost of the cycle.

Sol: Let cost price be 100. Since he sold at a loss of 9% his S.P would be 91. But, ,if he sold at a profit of 16%, his S.P would be 116 Le. the difference betweelJ the two selling prices is 25. But, we are given that the difference in the two selling prices is Rs.75. If difference between selling prices is 25, the C.P is 100.

:. If differenceis 75,then

'

75 C.P. = -x100=300 25 Hence, the cost price of Ajay's cycle is RS.300.

The profits left after paying the working partners' remuneration/commission are shared amongst all the partners. Sometimes, the partners also take interest on their investments and only the remaining profits are shared by the partners.

Sharing of profits among the partners also depends ,on the understanding between the partners. However, if no special scheme of sharing the profits is specified (in a problem), then the profits are shared based on the investments of the partners. There are three different possibilities that exist here. If the partners invest DIFFERENT amounts each for the SAME period of time, then the profits at the end of the year are shared in the ratio of their investments. If the partners invest the SAME amounts for DIFFERENT periods of time, then the profits at the 'end of the year are shared in the ratio of the time periods for which their respective investments have been in business. If the partners invest DIFFERENT amounts and' the time periods for which their investments are in the business are also DIFFERENT, then the profits at the end of the year are shared in the ratio of the product (investment x time period) for each partner.

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., 5 0.1 P + -(0.9P)=58,OOO 13

--.;:-=, :"':". De problems that are' modelled along 'II! ;;-~g of profits in partnerships. An example :r -.. -.pe is where a particular facility (like .-:-; : ;rador for ploughing their fields by three ~J"'"' :~Ie) is used by more than one party has to be shared by all the concerned ar= -e~

=> 5.8P =58000 13 ' =>P = 1,30,000:. Waqar'ssharewill be 8 =-x (0.9)(1,30,000) = Rs.72,OOO (or) 13 .

-

=--~

- similar to sharing of profits in a

;:;a--~-'P

Total Profit - Payment to Wasim = 1,30,000 - 58,000 = Rs.72,OOO :...:!! 5a-.a and Banta invest RS.21,000 and =-$ . "',500 respectively in a business and at -e end of the year they make a profit of =526,400. Find their individual shares in the ~~. io

S""'Ce both their investments

are there in the

: si"less for the same duration (1 year), :"'J~ will be shared in the ratio of their -.estments i.e. 21,000: 17,500 = 6: 5. Santa's Share 6 = - x 26,400 = Rs.14,400 11 3.a'lta's Share = ~x 11

3.31 A. started a business with Rs.40,OOO. After 2 months B joined him with Rs.60,OOO. C joined them after some more time with Rs.1,20,000. At the end of the year, out of a total profit of RS.3,75,OOO, C gets Rs.1,50,000 as his share. How many months after B joined.the business did C join? shares of profits is (40,000 x 12) : (60,OOOx 10) : (1,20,000 x P)

Sol: The ratio of the

[Here 'P' is the number of months that C was with the business] = 24 : 30 : 6P = 4: 5 : P C's Share = --.e..- of total profit p+q But this is given to be equal to RS.1,50,000 out of a total of Rs.3,75,000.

26,400 = RS.12,OOO .

Hence --.e..-= 1,50,000=> --.e..- = 3:..:s °:erana starts a business with Rs.45,OOO. ~"'fee months later Sanjana joins her with' °05.30,000. At the end of the year in what -aoo should they share the profits? ~

S~aring of. profits will be in the ratio of nvestments multiplied by the time period. '1encethe ratio is 45,000 x 12) : (30,OOOx 9) 2: 1

.

3.32 Peter started a business with Rs.20,OOO

.

=

:.:: .'Iasim started a business with

X.

Ratio of shares of profits is :25,000 x 12) : (60,000 x 8) 5.: 8 ...et the total profit be P, as Wasim receives

=

10% of this as commission, the remaining 90% of P is shared in the ratio of 5 : 8. I-ience Wasim's receipts will be 5/131h of 90% of total profit plus his commission. -"V'"'phant

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John

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later

with

Rs.30,OOO.After 2 months Peter withdr~w Rs.5000 of his capital and 2 more months later John brought in Rs.20,OOOmore. At the end of the year what should be the ratio in which they should share ttie profits?

Rs.25,OOO

a"d after 4 months Waquar joined him with Qs.60,OOO. Wasim received RS.58,OOO r-cluding 1O%.of the profits as commission 'or managing the business. What amount did Naquar receive?

'p+q 3,75,000 p+q 5 =>p=6 So, C was with the business for 6 months. Hence C joined the business 4 months after, B joined the business.

Sol: ,Here, even for each individua1, the capital was not the same for the entire period his money was in the business. So, the term of the ratio for a person will be sum of products of investment multiplied by time period for different parts of the year. Peter has 20,000 for 6 months and then since he withdrew 5000 he had onry 15,000

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for the rest of the 6 months. His term of the

ratiowill be

.

(20,000 x 6) + (15,000 x 6) = 210,000 John joined with 30,000 which remained unchanged for 4 months and then he brought in 20,000 more. So he had 50,000 for 4 months only as he joined 4 months after the business was began. His term of the ratio will be (30,000 x 4) + 50,000 x 4) = 3,20,000 :. The ratio of shares of the profit will be 210,000 : 320,000 = 21 : 32 3.33 The working partner of a business gets as his commission 10% of the profits left after his commission is paid. If the working partner's commission is Rs.30,000 then, find the total profit.

Sol: Let total profit be P. The profit left after the working partner's commission of Rs.30,OOOis (P - 30,000). .10% of this is the working partner's commission. So we have (0.1)(P - 30,000) = 30,000 :=) (0.1)P = 33,000 :. P = RS.3,30,000

STOCKS AND SHARES A limited company raises capital by floating shares. It is also referred to as stock. The capital required is divided into small units called shares. In India, the generally accepted value for such a unit is RS.10 or RS.100. This is called the Face Value or Par Value. The shares of a public limited company are traded in the market place and depending on the demand for the share, the price fluctuates. The rate at wllich a share is bought or sold in the market is the Market Value of the share. This fluctuates. If the market value is more than the face value of the share, then we say that such a share is quoting at a "premium." If the market value is less than the .face value of the share, then we say that such

ashare

is quoting

at a "discount."

The people who are holding the shares are called shareholders. The company distributes a part of its profits from its operations as dividend to the

shareholders. The dividend is expressed as a perce7ntage of the Par Value. Whenever any company quotes a dividend percentage figure, it goes without saying that it is a percentage of the

facevalue.

.. 10 0 f d IVId en d =

.

.

Dividend Amount

x 100 Par Value Dividend is always calculated only on the 'FACE VALUE' or the 'PAR VALUE' only irrespective of what price the share was purchased at. 01

The government also deals with stock where it issues bonds or other form of stock with a certain face value and a certain assured rate of interest. This stock is then traded in the market as per the regulations of the government. Since the government stock comes with fixed rate of return, the stock is normally referred to by the percentage of the return. For example, if 5% is the rate of return (of stock whose face value is Rs.100), then such stock is referred to as 5% stock. The face value of the government bond is normally RS.100. Supposing this stock yielding 5% return (on face value) is purchased by somebody at Rs.95, then we say that person has purchased "5% stock at 95". Instead, if he purchases it at R&.108, then we say that he has purchased "5% stock at 108". In the case where he purchased 5% stock at 95, to buy one unit of that stock, he pays Rs.95. But since the face value is Rs.100, the return or income he gets at the end of the year will be 5% of 100, Le., RS.5. . In this case, since he receives an income of Rs.5 per year by investin~ Rs.95, his rate of return is 5/95 x 100 which is 5 /19% . To compare two investments (Le., investments in two different stocks), we compare the rate of

. return

for both investments and whichever gives a higher rate of return is a better investment.

If somebody means that Rs.1000. If that person

is the the will

holding Rs.1 000' "worth of stock", it face value of stock he is holding is face value of the stock is Rs.100, be holding 10 units of such stock.

Typical problems in Shares and Stocks may include finding as to which out of given investment is a better one or finding the annual income or change in income from a certain investment or change in portfolio, etc.

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-~ =-~

::.x eMS are very similar to problems in .;.-:: -oss Percentages except for involving ~- ~o Ot;;fas given above.

=-=-:: -e examples we are going to look at,the ~ ~ -E o~the stock is to be taken as Rs.100 ~~ . se specified.

~..

::.;...

'"Z is the annual income from Rs.21,SOO ... ested in 3% stock at 7.S% premium?

ID

- 5'to premium means the market value is =5 .D7.S0. =:'"'CeRs.21,SOOis invested in this stock, the . 21S00 -.rOOr of Units = = 200 107.S . -- 's is 3% stock, so each unit of this stock == face value of Rs.100) will give an income ~ Rs.3 at the end of the year. ZOOunits will give an annual income of .

2::0 x : ~

3

= Rs.600

3.36 A man owned Rs.2S,000 worth of 6% stock. When it was quoting RS.228 he sold it and invested the proceeds in 7.S% stock quoting at Rs.13S, so that his annual income doubled. How much money was he left with or how much more money was he required to bring in? Sol: Rs.2S,000 worth .of stock means it refers to face value, which is Rs.100. Hence he owned 2S0 units. When he sells at Rs.228, his sales proceeds will be 2S0 x 228 = Rs.S7,OOO. Since each unit sold gave him Rs.6 as income per annum, his annual incomewas' 2S0 x 6 = 1SOO.With the new. instrument, income doubled; new income = 2 x 1S00 = Rs.3,OOO. To get annual income of Rs.3,OOO,he must have bought 3000n.S units = 400 units. Market Price of 7.S% stock was 13S. To buy 400 units he will needed to pay 400 x 13S = Rs.54,Ooo.

The sale proceeds from 6% stock was Rs.S7,OOO.Hence, the difference between

'.~ of the following is a better investment - 4% stock at 84 or 8% stock at 128? '.'e can calculate the rate of return for each :~ these investments and decide which is ~tter. Another approach is to take a certain amount as invested in each of these two stocks and calculate the income from each stock. For this purpose, instead of taking any arbitrary amount, if we take the amount "wested as the product of the market value of both the stocks, calculations' become simple. In this case, let amount invested be 84 x 128. 84x128 In 4% stock we get x4=S12 84 The same amount in 8% stock we get

Rs.S4,OOOand Rs.S7,OOO Le. Rs.3,OOO is the amount he is left with after the transaction.

3.37 A person invests Rs.19,400 in S% stock at 97. He then sells it when it is quoting RS.104. He then reinvests this money in 4% stock at 100, which he sells when the stock is quoting 10S. Find the overall profit of the transaction.

,

Sol: At RS.97 per unit. Rs.19,400

19400/97 = 200 units.

These whe'n sold at RS.104 each, he realises 104 x 200 = Rs.20,800. This is invested in 4% stock at 100. This gets

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208Units. h 1m -= 100 This stock is then sold when it is quoting 10S giving him. 10S x 208 RS.21,840 So, on the whole he made a profit of RS.2440.(= 21840 -19400)

=

84x128 x8=672 128 Since, the annual income from 8% stock is higher, it is a better investment.

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. Concept Revisited Questions 1; The price of an article is first increased by 10% and then decreased by 10%. What is the change in the price? (1) It is increases by 1%. (2) It is decreases by 1%. (3) No change in the price. (4) None of these 2. The price-of a product is decreased by 10% and then increased by 10%. What is the change in the price? (1) It is increases by 1%. (2) It is decreases by 1%. (3) No change in the price. (4) None of these 3. The price of a product becomes RS.54 after its price is increased by 35%. What was the original price of the product? 4.

The salaries of two persons are equal. If the salary of one person is increased by 10% and that of the other is decreased by 10%, then what is the change in the total salary of the two persons? (1) It is increases by 1%. (2) It is decreases by 1%. (3) No change in the total salary. (4) None ofthese

5. The price of a product is reduced to Rs.72 after it is decreased by 10%. What is the original price ofthe product? 6. The price of a commodity A is double that of commodity B. If the price of A is increased by 10% and that of B is increased by 25%, then what is the percentage increase in the total price of the products A and B together?

7. The price of a fan P is thricethat of a fan Q. The price of P is increased by 10% and that of Q is decreased by 30%. What is the change in the total price of fans P and Q put together? (1) It is increase d by 2% (2) It is decreased by 2% (3) No change in the total price (4) None of these

8. The price of a refrigerator is increased by 25%. By what percent should the price of the refrigerator be decreased to bring it back to the original price? . (1) 25% (2) 20% (3) 16213%

(4)

13113%

9. This year, the number of children in a colony is four times that in the previous year, What is the percentage increase in the number of children in the colony? (1) 4% (2) 400% (3) 300% (4) 200% 10. In an examination, Pradeep got 10% more marks than Suresh. By what percentage are the marks of Suresh less than that of Pradeep? (1) 1~% (3) 9/11%

(2) .13~13% (4) 11 19%

11. The cost price of a product is Rs.80 and its selling price is Rs.104. What is the percentage of profit? (1) 24% (2) 36% (3) 30% (4) 16% 12. The cos.t price of a chair is Rs.240.The profit made by the shopkeeper, after selling it by giving of 10% discount on marked price, is RS.30.What is the marked price? 13. The salaries of 30 employees of a company are increased by 10% and that of the remaining 20 employees of. the same company are increased by 20%. What is the percentage increase in the total salary of the 50 employees? (1) 15% (2) 26% (3) 14% (4) Cannot be determined 14. A shopkeeper sells an item at Rs.36 and incurs a loss of 10%..At what price should the shopkeeper sell it to gain 30%? 15. The cost price of a washing machine is 80% of the 'selling price. What is the percentage of the profit? 16. The cost prices of two products are equal. One product is sold at 16% profit and the other product is sold at 8% loss. What is the overall percentage of profit?

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- S'C 8 started a business. A invested ~ : ~ and 8 invested Rs.20,OOQ.If they 1= =.$8 400 profit, at the end of one year, ~ ""'21 is 8's profit? --'= ::=:st price of 60 articles is RS.10 each. -= :r:cles were sold at Rs.12 each. At what ;r .:..=s-'QiJldthe remaining article be sold, so -= ~

A

a an overallprofitof RS.5'perarticle?

--: ~ o,g price of a product is such that, the ~ ~ade on selling 3 units is equal to the

~

~

of 2 units. What is the percentage

::r~

4-=r5

salary before he got an increment E;; ~ of the total income of the family. If ~ "cement is 1/6thof his salary,then what ::e--:s-;age of his family's total income is his ~~

:::::- a 'ar'1i/y, the savings per month are 30%

= -e :otal income. If the expenditure on is increased' from 30% to 40% of

E:.-CZ':(Y\

'lIE :::-.::. '1come, then find the of savings, as a :P:'e"'~ of the expenditure on education, 7:". ::eO there is no change in the savings.

. :

. 23 '3% . x,"'o

(2) 75% (4) Cannotbe determined

"- z,- -= 8r'1increment in

Ramu's salary, the ~ =.,'pEmditure of the family was 60% of the === "come. Ramu's salary was equal to the ::a sa-."lgS of the family. If Ramu got 50% ~e<'\t in his salary, then what are the new ~ sa.~ngsof the family as a percentage of 1[ ~ ""come,keeping expenditureconstant?

-

i

~ :-~

~

price for 'a company to manufacture is Rs.60. The company sold the

7:-;:..e to

a dealer. for Rs.70. The dealer sells

"""! :-::CJct to a shopkeeper for Rs.85 and the sr::::...:.;per sells it to a customer for Rs.102. ~ S ::'.Ie percentage of profit for the company? .

..

:

~3%

.:.'~c,%

..

26. The cost price of a product is RS.60. It is increasedby 25% and then again by 331/3%and sold.What is the selling price of the product? 27. Anil and Nikhil invested Rs.10,000 each to start a business. Anil invested his money for eight months and Nikhil invested his money for the whole year. At end of the year, if the profit was RS.1,OOO,then what was Anil's share? .

per sale of one unit?

=-a ES"" got an increment, such that his =--c...moil to his family's total income is -==zsedfrom 30% to 40%. What is the "'- =-n to his family's total income?

-

25. In problem 24, who got the highest profit on selling the product?

(2) 20%' (4)

28. Rajath and Manoj started a business. Rajath invested Rs.10,OOOfor eight months of the year and Manoj invested RS.6,OOOfor the entire year. If the profit at the end of the year was Rs.5,700, what was the share of Manoj? 29. What is the annual income for a person, who invested Rs.20,OOOin 5% stock? 30. What is the annual income of a person, who invested Rs.10,450 in 6% stock at 4.5% premium? 31. What is the annual income of a person, who invested Rs.19,500 in 7% stock at 2.5% discount? 32. A person invested Rs.18,400 in 7% stock at Rs.92. If the person sells it when it is quoting Rs.107, then what is profit for the person? 33. A man invested Rs.17,400 in 6% stock at 13% discount. What is the yield percent at the end of the year, approximately? 34. Ajay invested RS.3, 120 in 8% stock at Rs.104, Sujan invested RS.3,800 in 5% stock at Rs.95. Who will get more income at the end of the year?

35. To manufacture a product X a company needs raw materials A, Band C and others. The cost price of A is 10% of the total cost price of X. The cost price of each of Band C is 15% of the total cost price of X. The costs of A, Band C are increased by 10% each and the total cost has increased by 10%. By what percentage has the cost of others increased?

70%

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Exercise

- 3(a)

Directions for questions 1 to 28: Select the

sales were the same, what were the total sales this month? (1) Rs.57,5DO (2) Rs.52,500 (3) Rs.67,500 (4) Cannot be determined.

correct alternative from the given choices.

1. The strength of St. Paul's Model High School increased by 10% this year. By what percent was the strength less last year, when .compared to this year? (2) 11% (1) 10% (3) 11.! % 9

-

(4) 9 ~% 11

2. Water in a cylindrical vessel is poured into another cylindrical vessel whose radius is 10% m'ore than that of the first vessel. By what percent is the height of the water level in the second vessel more or less than that iA the first vessel? 1 (1) 17.4% less (2) 9 -11 %less (2) 17-4%more (4) 21% less 3. A's weight is 10% less than that of Band C's weight is 10% more than that of B. By what percent is A's weight less than that of C? (2 ) 18~%

(1) 20%

11

.

(4) None of these

(3) 1~% 11

4. On St. Vincent's birthday, chocolates were distributed to the children of the primary classes. If the total strength of the primary classes increases by .12%this year compared to that of the last year and the total number of chocolates distributed increases by 34.4%, the number of chocolates given to each child increases by what percent? (1) 22.4% (2) 24.1% (3) 18% (4) 20% 5. A salesperson used to get a commission of 9% of his total' sales for a month. With a change in the terms of employment, he now gets a fixed amount of RS.3000 per month plus a commission of only 5% on the excess . of his sales over Rs.10,OOO.If the difference in the amounts that he received last month and this month is Rs.400, though the total Triumphant

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6. The price of groundnuts increased by x% per week over two successive weeks. If at the beginning, two kilograms were. availabie for RS.80 and after the two weeks they were available for the 105.80, what is the value ofx? (2) 115 (3) 15 (4) 11.5 (1) 1.5 7. The volume of a sphere A is 701%7% less than that of B. If the area of B is x% greater than that of A, x is equal to 2

.

(2) 125

(1) (1900)3 9 (3) 225

(4) None ofthese

8. All the water in a big tank A is emptied into two smaller empty tanks Band C. The volume of water in C is 331/3%ofthat in A. If 300 liters of water, which had gone into B had instead gone into C, B would have 50% more water than C. What was the volume of water in A? (1) 2000 liters (2) 4500 liters (3) 1200 liters (4) 1800 liters 9. Four teachers, five students' and two members of the non-teaching staff went as a group on a visit to a temple. They contributed the followingamounts towards the expenses. The teachers contributed RS.100 each, the students RS.30 each and the non-teaching staff Rs.25 each. What percent of the total expenditure did each student pay? (1) 16~% 3

(2) 4% (3) 4.!% 6

(4) 5%

10. By what. percentage does the annual income from RS.21400 invested in a 3% stock at 7% premium exceed that from RS.15600 invested in a 2% stock at 4% premium? (1) 80% (2) 150% (3) 100% (4) 200%

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.

,,- - :J!:-::-=ought ,

Rs.7200 worth of 4% stock 90. When the stock is quoting at

~

;

~- ~s

~

~

-i: se,ls it. He invests this money in

... -:.:.. q.;oting at RS.70 and sells it when ~ :-:::. .vas quoting RS.80. What is the :2:"..=-;a;e 'Ofprofithe makesfromthe overall =r::=-':'""

0""'"

-

-

:

-

(2) 33 ..! % 3

I.

(4) 37..! % 2

-. '1!- ~:::: . for Rs. 2250 more, he would have - -.=- so~ his computer at a loss of 5%. Had

17. The tetal. expenditure (E) of a mess is given by E = F + cn, where F is the fixed cost, n is the number 'Ofpeeple eating in the mess and c is the cost per person. One menth F, nand c increased by,. 50%, 25% an,d 20% respectively over the previous month. What is the percentage increase in the total expenditure?

~:~ a ;rofit of 10%. What is the cost price of ~ : :-o...ter? . =,5 15,000 (2) RS,16,000 - =~ 20,000 (4) Rs. 24,000 r 'S

~) 50%

. -~-:.a; ...atbought 2 quintals of dal. Due to the ::ii:t: condition of his shop, he had to sell it

.. a ~urry. He incurred a loss equal to the ~'= p"ce of 30 kg of dal. If he bought the ~.;" "or Rs.4600, find the selling price each

"7

:E-

=s20 ::'>518

:

(2) Rs.22 (4) None of these

- - :-q>keeper

normally makes a prefit of 20%.

- ~ certain transaction, he weighed 900 gm ~!Ead

'Of1 kg withouthis knowledgedue to

.

en-orin the weighing scale. If he changes less than what he nermally changes, orZ' s his actual profit or loss percentage?

= .

5 3. % prefit 3 :

63. % loss 3

:

'0% loss

-

'::J% profit

.

16. Hari has just eneugh money to buy 50 papayas 'Or30 watermelens. He decides to spend 'Only 90% of his money and buy 9 watermelons. How many papayas can he buy with the remaining meney that he wants te spend? (1) 45 (2) 30 (3) 15 (4) None ofthese.

(2) 31 3. % 3 (4) Cannot be determined

~~'

18. A sports dealer bought 2000 hockey sticks fer RS.350 each. 10% of them were found te be defective and unfit to be sold. At what price should he sell the remaining sticks to get an 'Overallprofit of 8%? (1) Rs.420 . (2) Rs.400 (3) Rs.450 (4) RS.380 19. By selling 160 calculators, an electronics dealer makes a profit equal to the selling price 'Of30 calculaters. Find his profit percentage.

.

(1) 18!% 4

(2) 13~%

(3) 23J..-% 13

(4) 15~% 19

.

13

20. In any month Harish deposits m% and withdraws n% of the closing balance of the previous month. If his balance at the end of

- s~udent appears for 4 papers-English, .o::.~. Physics and Chemistry. Maximum -E.""Sfor which are in the ratio of 1 : 1 : 2 : 2. -~ ""larks are in the ratio of 4 : 8 : 13 : 15. If

March (after the withdrawal) is the same as his. balance at ~he b~ginning of Ja~ua~ .(before the deposit), which of the following IS true?

-~ got 80% of the total maximum marks, in many papers did he get more than 80%?

(1) ~ < m < n 2

'7',

. (3)

-

1

(2) 2

=

3

(4) Cannotbe determined

~

Institute of Management

--11898194/95

Education

Fax: 040-27847334

m>n

(4) m < ~

2

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(2) m = n

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21. A trader cheats both his supplier and customer by using faulty weights. When he buys from the supplier, he takes 10% more than .the indicated weight. When he sells to his customer, he gives 10% less than the indicated weight. If he sells at 'cost-price' (i.e., charges the cost price of the indicated weight), what is his profit percent? (1) 20% (2) 10% . (3) 11~% . 9

(4) 22!% 9

22. A trader cheats both his supplier and customer by using false weights. While ~uying from his suppliers, he takes 10% more than the indicated weight. When he sells to his customer, he gives the customer a weight such that if 10% of that is added to the weight, the weight claimed by the trader is obtained. If he charges the cost price of the weight that he claims, find his profit percentage (1) 20%

(2) 21%

(3) 21~% 3

(4) 22~% 9

25. A person sold two washing machines, each for Rs.8000. On one he gained 111/9%and on the other he lost 91/11%.Find his overall loss or profit. 2 (1) 2-%profit (2) 2% profit 99 (3) 2% loss (4) None of these 26. A salesman makes a commission of a% on the first Rs.3000 worth of sales and b% commission on all further sales during the month if any. If he makes RS.960 from a total sales of Rs.7000 in Jan and Rs.1110 from a total sales of Rs.8000 in Feb, what is the value of b? (2) 15 (1) 12 (4) None of these (3) 18 27. Two TVs were sold both for the same price of Rs.9000. The first was sold at a profit of 25% and the other at a profit of 20%. What was the respective cost price of the first and the

secondTV?

.

(1) Rs.6400 and RS.7000 (2) RS.7200 and Rs.7500

23. The cost of production of a 50 kg bag of cement is Rs.90 and it is sold for RS.120.The gove~nmentimposes a tax of 10% of'the cost of production. If the govemm~nt changes its policy and levies the 10% tax on the selling price instead of the cost price, by what percent does the profit de,crease? (Assume ~that the selling p'riceis unchanged.) . (1) 14~% 7

.

(2) 500 % . 297

.(3) Rs.6500 and Rs.7500 (4) Rs.6500 and Rs.7200

28. If the discount given is equal to 25% of the selling price, and the sale gives the trader a profit of 1EfI3%when calculated on his selling price, by what percent did he mark up the cost price before offering the discount? (1) 50% (2) 30% (3) 40% (4) 60%

Directions for the questions 29 to 31: These (3) ~ % 7

(4) ~% 297

24. By selling 160 calculators, an electronic dealer makes a profit equal to the cost price of 30 calculators. Find his profit percentage. (1) 18~% 4

(2) 13~% 13

(3) 23~% 13

(4) 15~% 19

Triumphant

Institute of Management

Tel: 040-27898194/95

questions are based on the data given below.

The cost of printing a book has the following components. Paper Printing Ink . Labour. Power

Payment to autho~ 40%

Educ:aoon Pvt. Ltd. (T.LM.E.) HO: 958.

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10% 5% 20% 25%

email: [email protected]

2.-.1 Floor,

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003.

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,

I

i "::0::' .s sold at a profit of 25%. The paper cost .811!! ..c Df 10%,the cost of printingink goes up . .:!:.~~and labour charges go up by SO%, while 8IIIIe a.""e"10changes in the cost of power, or the 8IIr'cr" ~ oayment

.

..-e selling price remains unchanged, what is -e profit percent? '3%

.

(2) 11

~28%

7 (4) 12~% 12 - -e manufacturer increases the selling price

:,.

~C,o and wantsa

profit of 182/11%,by what

:E!"CeOtshould he reduce his expenditure on a:our (at the increased cost)?

-

6~% 3

(2) 6.!.% 3

3.!.%

(4 ) 3~%

33

::t . ';e payment to the author increases by ::: 5%, by what percent should he increase -e selling price to get a profit of 20%? .

~s

2S%

(2) 16~% 3

(3) 33.!.% (4) 20% 3

for questions 32 to 40: Select the

:::r-;::- alternative from the given choices.

:0: - ""1angoes to an electronic goods shop with -r::ney just enough to buy DVD player. He :;e"S

an unexpecteddiscountof 20% .on the

:: .:> player which enables him to buy DVDs ~ RS.2400 in addition to the DVD player. ',"1at price does the man pay for the DVD

:'c-.er? RS.11,200 :

Rs.9,600

(2) RS.12,OOO (4) Rs.10,800

::. - Band C start a partnership. The capitals of - ~ B are in the ratio of 4 : S. The periods of ~ and C are in the ratio S : 3. The profit Sdtes of A and C are in the ratio of 4: 3. The ::a&.als of Band C are in the ratio of

. 5: 1

(2) S: 2

:; 'S: 9

(4) Cannot be determined

34. A spectacle frame was marked up by 40% and after that a discount of 20% is offered on it. If the frame cost Rs.SOO,what is the profit earned by selling it? (2) RS.24 (1) Rs.36 (3) RS.48 ,(4) RS.60 35. A wholesale vegetable vendor sold potatoes, marked at Rs.1000. Four successive discounts of 10% each, instead of the promised 40%. By what amount did the vendor defraud the customer? (1) RS.S8.10 (3) RS.6S.10

(~) Rs.S6.10 (4) Rs.66.10

36. I bought a fridge three years back for Rs.9000. The current price of a similar fridge is Rs.16,OOO. If I sell my fridge, in the condition it is in, I would get Rs.10,SOO. Instead, if I get it painted for Rs.3000, I can' sell it for the price of a new one. What discount can I offer, if I expect to make the same percentage profit as I would have made, if I sell without painting? (1) 12.S% (2). 8% (3) 10% (4) 9.6%

37. Anand and Bhargava start a. business with RS.7700 and RS.11,000 respectively. Anand stayed for the entire year. At the end. of the year they shared their profits in the ratio of 14: 1S. For how many months was Bhargava's capital there in the business? (4) 9 (1) 3 (2) 6 (3) 8 38. Sudhir, Tushar and Uday start a business with Rs.30,000, Rs.40,000 and Rs.SO,OOO respectively Sudhir stays for the entire year. Tushar leaves the business after two months but rejoins after another 4 months but only with 3/4 of his initial capital. Uday'leaves after 3 months and rejoin:~ after another S months but with only 4/5 of his capital. If the year end. profit is Rs.27,900, how much more. than Tushar did Uday get? (1) RS.1S00 (2) Rs.9300 (3) RS.3100. (4) Rs.12,400

-,.,'"'D'"at 'nstituteof ManagementEducationPvt. Ud. (TJ.M.E.)HO: 958, 2ndFloor,SiddamsettyComPi,,:x,Secunderabad- 500 003~ -~

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SMI00806/43

r

-..

39. A and 8. start a business with different capitals. A was to get 15% of the profit as salary and the rest was to be divided in the ratio of their investments. Had the entire profit been distributed in the ratio of their investments, 8 would have got Rs.1350 more that what he actually got. What is ,8's actual share of the profit? (1) Rs.7,650 (2) Rs.9,000' (3) Rs.16,500 (4) Rs.11,000

40. A, 8, C start a partnership. The capitals of A, 8 and C are in the ratio of 10 : 9 : 6 and the time period of A and 8 is in the ratio of 2: 3. 8 gets Rs.10,800 as his share out of a total profit of Rs.26,000. If Ns capital was there in the business for 8 months, for how many months was C's capital there? (1) 8 (2) 9 (3) 10 (4) 12

Key 1. 2. 3.. 4. 5.

4 1 2 4 4

6. 3 7. 2 8. 2 9. 4 10. 3

11. 2 12. 1 13. 1 14. 1 15. 2

16. 2 17. 1 18. 1 19. 3 20.2

21. 4 22. 2 23. 1 24. I 25.4

26.2 27.2 28. 1 29,3 30. I

31. 4 32.3 33.4 34. 4 35. 2

36. 37. 38. 39. 40.

1 4 1 1 4

.'

..

Triumphant Institute of Management Education Pvt. Ltd. (T.LM.i)HO: Tel: 040-27898194/95

Fax: 040-27847334

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website: www.time4education,com

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SM I 00806/44