Ratio and Proportion RATIO AND PROPORTION We use numbers in everyday life quite frequently.One of the uses of numbers is for comparison.When two things of same same kind are attributed numerical values,we values,we are able to compare them.This comparison is expressed in phreases like ‘is greater than’,’is multiple of’ etc.
Let us take an example exa mple , a familiar situation in which Vignesh scored 17 runs while Vinay amassed 51 runs in an inning of cricket.Then we say, 1) Vinay scored 34 runs more than Vignesh or Vignesh scored 34 runs less than Vinay or 2) Vinay scored three times as many runs as Vignesh or we say that Vignesh scored only one third of the runs scored by b y Vinay. When we compare in the way as (2), we are finding the ratio between the two numbers.In short,the ratio between two quantities ‘a’ and ‘b’( where b>0 is the value of fraction a/b in its lowest terms) Let us revise revise a few things about ratio. To find the ratio of the first number to the second one ,we find ‘what multiple of second number is the first number?’ and this is done by dividing the first number by the second one. For example, the ratio ratio of 17 to 51 =17/51=1/3 =17/51=1/3 The ratio of 50 to 30=50/30=5/3 The phrase, ‘the ratio of 17 to 51’ is written as ’17:51’ and read as ’17 is to 51’ While comparing two quantities in terms of ratio, we must bear in mind the following: 1) The two two quanti quantitie tiess must must be of same same kind kind.. 2) The units units of measure measurement ment of the two quantities quantities must must be the the same. same. 3) As the ratio ratio denotes denotes how many many times times is one quantity quantity of of the other, other, it is a pure pure number( without any unit of measurement) For example 4m : 80 cm=400cm: 80cm=5:1 1hr 30 min : 2 hrs 15 min=90 min:135min=2:3 The numbers involved in a ratio are called its ‘terms’
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Ratio and Proportion That expression of a ratio,both of whose terms do not have common factor other than one, is called the ratio in its lowest terms.Thus by cancelling the common factor of the two terms 105 and 135, we obtain the lowest form f the ratio 105:135 1 05:135 as 7:9. Percentage is a special kind of a ratio . it is a ratio having its second term 100.
Please note down certain important aspects of ratios. 1) when we consider consider the the ratio of two two numbers numbers as a:b ,then ,then the first first number number need not be ‘a’ and the second number need not be ‘b’.They can be ka and kb, where k is any non zero multiple of a and b. 2) Two ratios a : b and c: d (a/b and c/d) are said said to be equal if a x d = b x c. 2) If two ratios ratios are a:b and b:c, b:c, they are briefly briefly written written as a:b:c.For a:b:c.For example example ,when ,when we say that the ratio ratio of measures of angle A angle B and angle C of triangle ABC are three is to four is to five( 3 :4 :5) We really mean to say that m< A : m
Proportion is a very familiar and an important mathematical concept.Let us try to understand this. Suppose a fruitseller fruitseller tells you that the the price of oranges is Rs.20 a dozen,that is,Rs 20 for 12 oranges If you want to buy six oranges,how do you determine their cost? As the number of oranges is half of one dozen,their cost also has to be half.Therefore,the cost of 6 oranges is half of rs.20 i.e. Rs 10. In other words,you think that the cost of oranges in proportion to their number. Mathematically,proportion is defined as follows: When a/b=c/d, the numbers a,b,c,d are in proportion. When a,b,c,d are in proportion ,they are respectively called the first, the second,the third and the fourth propotionals. a and d are called the extremes, while b and c are called the means.
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Ratio and Proportion
For example,14/21=18/27 Therefore 14,21,18,27 are in proportion. 14 and 27 are the extremes while 21 and 18 are the means. You know that when a/b=c/d,then the products a x d and b x c are equal. So when four numbers are in proportion, the product of extremes is equal to the product of means. The concept of proportion need not be restricted to only two equal ratios. It may be extended thus. If a/b=c/d=e/f…..,then a,b,c,d,e… are said to be in proportion. For example,5/9=20/36=15/27. So 5,9,20,36,15,27 are in proportion. Continued proportion Consider the ratios 25:20 and 20:16. These ratios are equal . So, the numbers 25,20,20,16 are in proportion. The means of this proportion are equal. So, we say, the numbers25,20,16 are in continued proportion. Generally ,when a / b = b / c then a,b,c are in continued proportion,also,when a/b=b/c, we get b2 = a x c When a, b, c are in continued proportion ,b is called the ‘geometric mean’ or ‘ mean proportional’ between a and c. From the above discussion, we get four equivalent equ ivalent statements. (1) (1) a/b= a/b=b/ b/cc 2 (2) b =a x c (3) a,b,c are are in continued continued proportion proportion and and (4) b is the the geometric geometric mean mean of a and c.
The concept of continued proportion is extended thus. If a/b=b/c=c/d=……then a,b,c,d….. are said to be in continued proportion. For example, the ratios 8:12,12:18,18:27 are equal. So the numbers 8,12,18,27 are in continued proportion.
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Ratio and Proportion Exercise
1. Rs 900 is to be distributed amonst A,B and C in the proportion 2:3:4. How much would C get? (1) Rs 400 (2) Rs 450 (3) Rs 500 (4) Rs 540 (5) None of these. 2. A purse contai contains, ns, in all, all, 378 coins consist consisting ing of one rupee,50 rupee,50 paise paise and 25 paise. paise. The total values of these coins are in the ratio of 13:11:7. How many coins of one rupee does the purse contain? (1) 126 (2) 78 (3) 168 (4) 136 (5) can not be be determi determined ned
3. Ramesh has two-third of the money as that with Umesh .Umesh has three fifth of the money as that with Mahesh. If Mahesh has Rs. 1,200, how much does Ramesh have? (1) Rs 720 (2) Rs 320 (3) Rs 200 (4) Rs 480 (5) none of these
4. A sum of of Rs 45 is is made up of 100 coins coins of 50 paise paise and 25 paise paise denomina denominations. tions. What What is the ratio of number of 50 paisa coins to those of 25 paisa coins? (1) 1:3 (2) 1:4 (3) 4:1 (4) 1:2 (5) None of these
5. A gave 1/5th of the money he had to B and B in turn gave 1/3rd of the money he received from A to C . if C received Rs 20,how much much money did A have with him initially? initially? (1) Rs 300
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Ratio and Proportion (2) (3) (4) (5)
Rs 320 Rs 200 Rs 480 None of these
6. If the total total attendance attendance at a party party is 252 252 ,which ,which of the followin following g cannot be the the ratio ratio of ladies and gents at the party? (1) 1:5 (2) 1:6 (3) 1:7 (4) 1:8 (5) None of these
7. Last year year ,the prices prices of tea and and coffee were were in the ratio ratio of 2:3 2:3 . Between Between last year year and this year, the price of tea has risen in the ratio of 5:6 and that of coffee, in the ratio of 7:8. If this year, a kg of coffee and a kg of tea together cost Rs 51,how much does a kg of coffee cost? (1) Rs 17.50 (2) Rs. 21 (3) Rs. 26.25 (4) Rs. 30 (5) None of these 8 The ratio of numbers of rum rum scored by a cricketer cricketer in domestic and international matches matches last year was 2:1 .This year, his performance in domestic matches improved in the ratio of 1:2 and international ones , in the ratio of 2:3 . If the total score this year in domestic and international matches was 2200 runs, how many runs did he score in international matches last year? (1) 400 (2) 600 (3) 800 (4) 1600 (5) none of these
9. The ratio of number of boys and girls in a school is 2:1. Of the girls,3/4 girls,3/4 are day scholars rd rd and 2/3 of them travel to school by bus. If 2/3 of the boys are day scholars and 3/4th of them travel by bus, what part of the student community travel to school by bus? (1)2/3 (2)1/2
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Ratio and Proportion (3)1/3 (4)2/5 (5)none of these 10. Three numbers are in the ratio ratio 1:3:5. If the sum sum of the largest and the the smallest equals the sum of the middle and 60, what is the average of these numbers? (1) 120 (2) 100 (3) 80 (4) 60 (5) none of these 11. Rs 2,400 is to be distributed distributed among A,B and C in the proportion 3:4:5 3:4:5 respectively. If the share of “A” is increased by Rs 100 1 00 at the cost of “C” , what would be the new ne w proportion of distribution? (!) 3:4:9 (2) 1:1:3 (3) 1:1:4 (4) 1:!:1 (5) 7:8:9
12.If x:y is 9:7, then then ( x+y) (x-y) is ……………… ……………… (1) 8:1 (2) 1:8 (3) 4:1 (4) 1:4 (5) None None of the the above above
13.Three numbers bear a ratio of 1:2:3 and their sum is 1102. what is the third number? (1)17 (2)34 (3)48 (4)51 (5)None of these 14. Rs 675 is to be distributed among A,B andC in such a way that Rs 10 more more than onethird of A’s share and Rs 20 more than one third of B’s share and Rs 5 more than one third of C’s share are equal . what is A’s share of money ( in Rs)? (1) 120
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Ratio and Proportion (2) (3) (4) (5)
270 285 225 None None of these these
15. If A:B= 2:3, B:C= 3:5 , C:D= 4:5,then A:B:C:D is? (1) 8:12:2 8:12:20:2 0:25 5 (2) (2) 4:3: 4:3:5: 5:25 25 (3) (3) 8:12: 8:12:4: 4:5 5 (4) (4) 2:4: 2:4:6: 6:25 25 (5) None None of these these 16.A,B and C hire a video for Rs Rs 1200. They use it for10,12 for10,12 and 18 hours respectively. respectively. If the hiring charges of the video are to be contributed by them in the proportion of hours of their use, what is the share of B? (1)Rs 360 (2)Rs 300 (3)Rs 540 (4)Rs 400 (5)None of these
17.The incomes of A and B are in the ratio 3:2, while their expenditures are in the ratio 3:1. If each of them saves Rs 1500 1 500 , what is A’s income? (1)Rs 2400 (2)Rs 1500 \ (3)Rs 1800 (4)Rs 5000 (5)None of these
18.The monthly incomes of A,B and C are are in the ratio 3: 4:5 4:5 while their savings are in in the ratio of 1:2:3 . If the the savings of A are Rs 600 per month, equivalent to 10% of his his income,how much does C spend per month (in rupees)? (1)8200 (2)6600 (3)9000 (4)7500 (5)None of these
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Ratio and Proportion 19.Rs 336 is divided between A and B so that A gets 5/16th of what B gets. What amount does A get? (1)Rs 75 (2)Rs 80 (3)Rs 100 (4)Rs 105 (5)None of these 20.Rs 150 is divided so that A gets Rs 30 more than B and B gets twice as much as C . What amount does C get? (1)Rs 20 (2)Rs 24 (3)Rs 30 (4)Rs 25 (5)None of these
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