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To find the difference between compound interest and simple interest for two years :
Compound interest for two years = P (1 + r) 2 - P Simple interest for two years = P x r x 2 = 2Pr Rate of interest Here, r = 100 = [P (1 + r) 2 - P] - (2Pr) = P (1 + 2r + r 2) - P - (2Pr) = P + 2Pr + P r 2 - P - 2Pr = Pr 2 Hence, difference between compound interest and simple interest @ r % p.a. for two years = P( r 2) Difference = Principal x
( )
Rate 2 100
To find the difference between Compound Interest and Simple interest, when Simple Interest for 2 years is given:
Difference between compound interest and simple interest for 2 years = P( r 2) = Pr (r) Rate But, Pr = Principal x = Simple Interest for one year 100 1 = x Total Simple Interest 2 Therefore, Difference between compound interest and simple interest for 2 years 1 = Pr (r) = x Simple Interest x (r) 2 1 = r of Simple Interest 2 1 For 2 years: r of simple interest 2 Alternate Method: Without the above derivation, we can calculate the difference by using the reasoning given below. We know that Compound interest for first year = Simple Interest for one year Compound Interest for second year = Simple Interest for one year + Interest on Simple Interest for the first year (have a look at the table) Therefore Difference = Interest on Simple interest of first year
To find the difference between compound interest and simple interest for three years : [P(1 + r) 3−P] − P × 3 × r ⇒
[P(1 + 3r + 3r 2 + r 3) − P] − P × 3 × r
⇒
3Pr 2+ Pr 3
⇒
Pr(3r + r 2)
Difference for 3 years = P r 2 (3 + r)
To find the difference between Compound Interest and Simple interest, when Simple Interest for 3 years is given:
For 3 years: Note: 'r' is
1
r (3 + r) of simple interest 3 Rate of interest
100 Difference between compound interest and simple interest for 3 years = P r 2 (3 + r) = Pr (r) (3 + r) Rate Now, Pr = Principal x = Simple Interest for one year 100 1 = x Total Simple Interest 3 Therefore, Difference between compound interest and simple interest for 3 years 1 = Pr (r) (3 + r) = x Simple Interest x (r) (3 + r) 3 1 = r (3 + r) of Simple Interest 3
To find simple interest when compound interest is given? If difference between compound interest and simple 1 1 interest is of Simple interest Then, difference between compound interest and simple interest is of x x− 1 Compound Interest.
Hint: Compound Interest is more than Simple Interest. And, Compound Interest = S imple Interest + Difference Therefore, If we shift from Simple Interest to Compound Interest, smaller fraction is required and vice-versa.
To find Compound Interest for two years, when Simple interest is given:
Compound Interest = Total Simple interest + (Simple interest for one year x Rate of Interest)
Practice problems
1. Find the amount for Rs. 6000 at 10% per annum, compounded semi-annually for 2 years.Here n = 2 years x 2 = 4 periods Similarly, R =
10 2
= 5% (for half year)
P = 6,000
(
A = 6,000 1 +
5 100
)
4
= Rs. 7,293
Interest = Rs.7293 - 6000 = 1293
Alternate method: Using pascal triangle: the coefficients are 4, 6, 4, 1 for 4 years.
= 4 X (300) + 6 X (15) + 4 X ( .75) + () = 1293 We can stop after using 3 coefficients as the 4th coefficient multiple is too small and may not cause any big difference in our answer.
2. The difference between the CI and SI on a certain amount at 10% per annum for 2 years, compounded annually is Rs. 372. Find the principal. Let the principal be a. SI =
a × 2 × 10
100
=
a
5
(
and CI = Amount – a = a 1 +
CI – SI = Rs. 372 a 21 × a − = Rs. 372 100 5 a = Rs. 37,200
10 100
)
2
– a =
21 100
× a
Alternate method: We know that the difference in interest comes from second year. Assume principal is Rs.100 then Interests are calculated as below.
Now the difference is Rs.1 but actually it was given as Rs.372. If principal is Rs.100 difference is Rs.1, what is the principal if difference has to be Rs.372 372 × 100 = 37200 ⇒ 1 Alternate method:
The above problem has an alternate method. You need to understand the fact that for 1st period, SI = CI.
The difference between the values of CI and SI is because of accumulated interest building on interest which is reinvested. Therefore, for period 2, the difference between CI and SI is the interest for 1 period on the interest of period 1.
In the above example, the difference being 372 is the interest generated on interest for period 1 on the principal.
Interest for period 1 = Rs. 372 ×
100
= Rs. 3,720 10 100 Therefore, principal = Rs. 3,720 × = Rs. 37,200. 10 3. Find compound interest on Rs. 10000 at 10% p.a. for 4 years, if interest is compounded annually. Amount = Rs. 10000 x
( )
11 4 10
= 14641
Therefore, Compound interest = Rs. 14641 - Rs. 10000 = Rs. 4641
Note: Steps for calculation of 114 : 11 x 11 = 121; 121 x 11 = 1331; 1331 x 11 = 14641
Approximate Method:
We know that, Compound interest = Simple interest + Interest on simple interest 1 Simple interest = 10000 x x 4 = Rs. 4000 10 Therefore, Simple interest for one year = Rs. 4000 ÷ 4 = Rs. 1000. Therefore, Interest on interest = 0 + 10% of Rs. 1000 + 10% of Rs. 2000 + 10% of Rs. 3000 + interest on interest = 0 + 100 + 200 + 300 + interest on this amount = 600 + interest on this amount Therefore, Compound interest = Rs. 4000 + Rs. 600 + interest on Rs. 600 = Rs. 4600 + Interest on Rs. 600 Out of given options, amount nearest to it is Rs. 4641.
4. If a certain sum of money invested at a certain rate of compound interest doubles in 5 years. In how many years will it become 4 times?. Since, 2 2 = 4. Therefore, The amount will become 4 times in 2 x 5 = 10 years.
5. At what rate per cent of compound interest, a sum of Rs. 2000 will amount to Rs. 2662 in 3 years?
(
We know that, 1 +
100
Therefore, (1 + r) 3 = Therefore, 1 + r = Therefore, r =
11 10
)
Rate Time
2662 2000
=
=
Amount Principal
1331 1000
=
( )
11 3 10
11 10
−1 =
1 10
= 10%
6. A man invested Rs. 16000 at compound interest for 3 years, interest compounded annually. If he got Rs. 18522 at the end of 3 years, what is rate of interest?
Here, (1 + r) 3 =
18522 16000
=
9261 8000
Therefore, Rate of interest =
( ) ( 21 3
= 1
20
20
=
1+
)
3
1 20
= 5%
'Approximate Method': Compound interest = Rs. 18522 - Rs. 16000 = Rs. 2522 Let, the amount is invested at 1% p.a. simple interest. Then, simple interest of 3 years = 16000 x 1% x 3 = Rs. 480 2522 Therefore, Rate of interest = = 5 + (Remainder is Rs. 122) 480 We know that compound interest is more than simple interest. Note: If Rate is 6%, then simple interest = 480 x 6 = 2880, which is more than the given compound interest which is not possible. Therefore, Rate of interest ≥ 6% is not possible. Therefore, Rate of interest is 5% p.a.
7. A sum of money amounts to Rs. 2880 in 2 years and 3456 in 3 years at compound interest. Find the sum. Rs. 2880 amounts to Rs. 3456 in one year. 3456 6 The sum amounts to = times of itself 2880 5 Therefore, Principal = 2880 ÷
()
6 2 5
= 2880 ×
5 6
×
5 6
= Rs. 2000
8. A man borrows Rs. 2100 and undertakes to pay back with compound interest @ 10% p.a. in 2 equal yearly 1 11 installments at the end of first and second year. What is the amount of each installment?Here, (1 + r) = 1 + = 10 10 10 Ratio of principals of two instalments = 1 : = 11 : 10 11 Sum of ratios = 11 + 10 = 21 11 Therefore, Principal of first instalment = 2100 x = Rs. 1100 21 Therefore, Instalment = Principal of first instalment x (1 + r) 11 = 1100 x = Rs. 1210 10
9. A man borrows Rs. 820 and undertakes to pay back with compound interest @ 5% p.a. in 2 equal yearly instalments at the end of first and second year. What is the amount of each installment? 1 21 Here, (1 + r) = 1 + = 20 20 20 Ratio of principals of two instalments = 1 : = 21 : 20 21 Sum of ratios = 21 + 20 = 41 21 Therefore, Principal of first instalment = x 820 = Rs. 420 41 21 Therefore, Instalment = Principal of first instalment x (1 + r) = 420 x = Rs. 441 20 10. A man borrows Rs. 1820 and undertakes to pay back with compound interest @ 20% p.a. in 3 equal yearly installments at the end of first, second and third years. What is the amount of each installment? 1 6 Here, (1 + r) = 1 + = 5 5 Ratio of principals for three years = 1 :
5 6
:
()
5 2 6
= 6 2 : 6 x 5 : 5 2 (On multiplying each ratio by 6 2) = 36 : 30 : 25 Sum of the ratios = 36 + 30 + 25 = 91 Therefore, Principal of first installment =
36 91
x 1820 = Rs. 720
Therefore, Installment = Principal of first installment x (1 + r) = 720 x
6 5
= Rs. 864
11. A certain sum is to be divided between A and B so that after 5 years the amount received by A is equal to the amount received by B after 7 years. The rate of interest is 10%, interest compounded annually. Find the ratio of amounts invested by them. Let the sum (principal) received by A and B are x and y. 1 11 (1 + r) = 1 + = 10 10 Then,
x y
=
( )
11 7 −5 10
=
( )
11 2 10
=
121 100
Hence, the ratio in which the sum is divided = 121 : 100.
12. A father wants to divide Rs. 5100 between his two sons, Mohan and Sohan who are 23 and 24 at present.
Divide the amount in such a way that if their shares are invested at compound interest @ 4% p.a. they will receive equal amount on attaining the age of 26 years. Find Mohan's share. Let, Mohan and Sohan receives Rs. x and Rs. y respectively at present. 1 26 (1 + r) = 1 + = 25 25 Then,
x y
=
( )
26 2 −3 25
=
( )
26 −1 25
Therefore, Mohan's share =
25 51
=
25 26
x Rs. 5100 = Rs. 2500
13. Find the difference between Compound Interest and Simple Interest on Rs. 4000 for 1 year at 10% p.a., if the interest is compounded half-yearly. Since, interest is compounded half-yearly. Therefore, Rate of interest is halved and time is doubled. Therefore, 10 1 Rate = % = 5% = 2 20 And, T ime = 2 x 1 = 2 half-years. 1 1 Therefore, Difference between Compound Interest and Simple Interest = Rs. 4000 x x = Rs. 10 20 20 14. Find the difference between Compounded Interest and Simple Interest on Rs. 1000 for 3 years at 10% p.a., if interest is compounded annually. Difference between Compound Interest and Simple Interest for 3 years = P r 2 (3 + r) = Rs. 1000 x
(
3+
1 10
1 10
x
1 10
x
)
= Rs. 31
15. Find the difference between Compound Interest and Simple Interest on Rs. 10000 for 4 years at 10% p.a., if interest is compounded annually. Difference between Compound Interest and Simple Interest for 4 years = P r 2 (6 + 4r + r 2) = 10000 x
(
6+
4 10
+
= 10000 x
1 100 1 100
1 10
x
) ×
641 100
= Rs. 641
16. If Compound Interest on a certain sum for 2 years @ 5% p.a. is Rs. 328, the Simple interest will be ?
1 10
x
Suppose, Compound Interest for first year = Rs. 100 Then, Compound Interest for second year = Rs. 105 Total Compound Interest for two years = (Rs. 100 + Rs. 105) = Rs. 205 And S imple Interest for two years = 2 x Rs. 100 = Rs. 200 If Compound Interest is Rs. 205, Simple Interest = Rs. 200 If Compound Interest is Rs. 328, Simple Interest = Rs. 328 x
200 205
= Rs. 320
Alternative Method: 1 Rate = 5% = 20 Difference between Compound interest and Simple interest 1 1 1 = = × of simple interest 2 20 40 1 1 = of the compound interest = x Rs. 328 = Rs. 8 41 41 Therefore, Simple interest = Compounded interest - Difference = Rs. 328 - Rs. 8 = Rs. 320
17. If a certain sum of money invested at a certain rate of compound interest doubles in 6 years. In how many years will it become 8 times? Solution: Since, 2 3 = 8. Therefore, The amount will become 8 times in 3 x 6 = 18 years.
MCQ's
1. A sum of money becomes Rs.6690 after three years and Rs.10,035 after 6 years on compound interest. The sum is : a. Rs.4400 b. Rs.4445 c. Rs.4460 d. Rs.4520
Correct Option: C Explanation: Let the sum be P.
[
R
Then, P 1 +
[
and P 1 +
100
]
R
100
]
3
= 6690.........(i)
6
= 10, 035 ..... (ii)
Dividing (ii) by (i), we get
(
1+
R
100
)
3
10035
=
(
P= 6690 ×
2 3
6690
=
3 2
)
=Rs.4460
2. Rs.1600 at 10% per annum compound interest compound half-yearly amount to Rs.1944.81 in a. 2 years b. 3 years 1 c..1 years 2 1 d. 2 years 2 Correct Option: A Explanation:
(
5
1600 1 +
T
= 1944.81
( ) ( ) ( )
⇒
=
100
)
21 20
441 2 400
r
=
=
1944.81 1600.00
=
194481 160000
21 4 20
T = 4 (Half - years) or T = 2 years
3. The difference between simple interest and compound interest on a sum for 2 years at 8%, when the interest is compounded annually Rs.16. If the interest was compounded half-yearly, the difference in two interests would be nearly : a. Rs.16
b. Rs.16.80 c. Rs.21.85 d. Rs.24.64 Correct Option: D Explanation: For Ist year, S.I = C.I Thus, Rs.16 is the S.I on S.I for 1 year, which at 8% is thus Rs.200 i.e.S.I on the principal for 1 year is Rs.200 Principal = Rs.
(
100 × 200 8× 1
)
= Rs.2500
Amount for 2 years, compounded half-yearly
[
(
=Rs. 2500 × 1 +
4 100
) ] 4
=Rs.2924.64
C.I = Rs.424.64 Also, S.I = Rs.
(
2500 × 8 × 2 100
)
=Rs.400
Hence, [(C.I)-(S.I)] = Rs. (424.64-400)=Rs.24.64
4. The difference in C.I and S.I for 2 years on a sum of money is Rs.160. If the S.I for 2 years be Rs.2880, the rate percent is : 5 a. 5 % 9 1 b. 12 % 2 1 c. 11 % 9 d. 9% Correct Option: C Explanation: S.I for 1 year = Rs.1440 S.I on Rs.1440 for 1 year = Rs.160 Hence, rate percent =
(
100 × 160 1440 × 1
)
1 = 11 % 9
5. The value k of a machine depreciates every year at the rate of 10% on its value at the beginning of that year. If the present value of the machine is Rs.729, its worth 3 years ago was : a. Rs.947.10 b. Rs.800 c. Rs.1000 d. Rs.750.87 Correct Option: C Explanation:
(
10
P= 1 −
P=Rs.
100
(
)
3
= 729
729 × 10 × 10 × 10 9×9×9
)
=Rs.1000
6. The least number of complete years in which a sum of money put out at 20% C.I. will be more than doubled is : a. 3 b. 4 c. 5 d. 6 Correct Option: B Explanation:
(
x 1 +
Now,
20 100
(
6 5
)
×
n
> 2 x or 6 5
×
6 5
×
() ) 6
n
5
6 5
>2
>2
n = 4 years
7. A sum of Rs.550 was taken a loan. This is to be repaid in two equal annual instalments. If the rate of interest be 20% compounded annually, then the value of each instalment is : a. Rs.421 b. Rs.396 c. Rs.360 d. Rs.350
Correct Option: C Explanation: Let the value of each instalment be Rs.x. Then, x
(
1+
x
+ 20
) (
20
= 550
)
2
1+ 100 100 5 x 25 x or + = 550 or x = 360 6 36
8. A loan was repaid in two annual instalments of Rs.112 each. If the rate of interest be 10% per annum compounded annually, the sum borrowed was : a. Rs.200 b. Rs.210 c. Rs.217.80 d. Rs.280 Correct Option: B Explanation: Principal = (Present value of Rs.121 due 1 year hence ) + (Present value of Rs.121 due 2 years hence ) 121 121 = Rs. =Rs.210 + 10 10 2 1+ 1+ 100 100
(
) (
)
9. A sum amounts to Rs.2916 in 2 years and to Rs.3149.28 in 3 years at compound interest. The sum is : a. Rs.1500 b. Rs.2000 c. Rs.2500 d. Rs.3000 Correct Option: C Explanation: Let P be the principal and R% per annum be rate.
(
Then P 1 +
R
100
)
3
=3149.28 ........ (i)
(
R
and P 1 +
100
)
2
=2916 ..........(ii)
On dividing (i) and (ii) we get
(
1+
R
100
)
3149.28
=
2916 233.28 233.28 or 100 = or R = × 100 = 8% 2916 2916
(
8
Now P 1 +
100 27
27
or P ×
)
2
= 2916
× = 2916 25 25 2916 × 25 × 25 or P = =Rs. 2500 27 × 27
10.A sum of money amounts to Rs.10648 in 3 years and Rs.9680 in 2 years. The rate of interest is : a. 5% b. 10% c. 15% d. 20% Correct Option: B Explanation: Let P be the principal and R% annum be the rate. Then.
(
P 1+
100
)
(
100
R
and P 1 +
3
= 10648...(i)
R
)
2
= 9680 ....(ii)
On dividing (i) by (ii), we have
(
)
R 10648 = 100 9680 R 968 1 or = = 100 9680 10 1 or R = × 100 = 10% 10
1+
11. The difference between simple interest and compound interest at the same rate for Rs.5000 for 2 years is
Rs.72. The rate of interest is : a. 10% b. 12% c. 6% d. 8% Correct Option: B Explanation:
[ ( [(
5000 × 1 +
⇒
50 00
⇒
1+
⇒
1+
R 2
100
R 2 =
(
R 2
100
+
) ] ) ]
R 2
100 2 R
100
72 5000
− 5000 −
−1 −
−1 −
R
50
5000 × 2 × R
R
50 =
100
= 72
= 72
72 5000
)
× 10000 = 144 or R = 12%
12. The compound interest on a certain sum of money for 2 years at 10% per annum is Rs.420. The simple interest on the same sum at the same rate and for the same time will be : a. Rs.350 b. Rs.375 c. Rs.380 d. Rs.400 Correct Option: D Explanation:
(
Let principal be P. Then, P 1 + S.I = Rs.
2000 × 2 × 10 100
P 2
100
)
− P = 420 ⇒ P =Rs.2000
= Rs.400
13.The difference between the compound interest and simple interest on a certain sum at 5% per annum for 2 years is Rs.1.50. The sum is : a. Rs.600
b. Rs.500 c. Rs.400 d. Rs.300 Correct Option: A Explanation: Let the sum be Rs. 100. Then. S.I = Rs.
(
C.I = Rs.
[{
100 × 5 × 2 100
(
)
100 × 1 +
= Rs.10
)} 2
5 100
− 100
]
Difference between C.I and S.I. = Rs.
= Rs.
(
41 4
41 4
)
− 10 =Rs.0.25
0.25 : 1.50 : : 100 : x 1.50 × 100 x= = Rs.600 0.25 14. A sum of money placed at C.I doubles itself in 5 years. It will amount to eight times itself in : a. 15 years b. 20 years c. 12 years d. 10 years Correct Option: A Explanation:
(
Let the principal P and rate be r% . Then, 2P = P 1 +
(
1+
r
100
)
5
= 2
( )) (
Let it be 8 times in t years . Then, 8p = p
(
or 1 +
r
100
)
t = 15 years
t
=8=(2) 3 =
((
1+
r
100
1+
5 3
=
1+
r
100 r
100
) )
t
15
r
100
)
5
or
15. The simple interest on a certain sum for 2 years at 10% per annum is Rs.90. The corresponding compound interest is : a. Rs.99 b. Rs.95.60 c. Rs.94.50 d. Rs.108 Correct Option: C Explanation: Sum = Rs.
(
100 × 190 2 × 10
[ (
)
C.I = Rs. 450 × 1 +
= Rs.450
10 100
)
2
]
− 450 = Rs.94.50
16. What is the principal amount which earns Rs.132 as compound interest for the second year at 10% per annum ? a. Rs.1000 b. Rs.1200 c. Rs.1320 d. None of these Correct Option: B Explanation: Let x be the principal at the end of first year. x × 10 × 1 Then = 132 ⇒ x = 1320 100 Let y be the original principal y × 10 × 1 = 1320 ⇒ y = 1200 Then, y + 100 17. A sum amounts to Rs.1352 in 2 years at 4% compound interest. The sum is : a. Rs.1300 b. Rs.1250
c. Rs.1260 d. Rs.1200 Correct Option: B Explanation: Let the sum be P . Then, 1352= P ⇒
⇒
26 × 25 25 1352 × 25 × 25
1+
4 100
)
2
26
1352 = P × P =
(
26 × 26
= 1250
18. The compound interest on Rs.30000 at 7% per annum for a certain time is Rs.4347. The time is : a. 2 years 1 b. 2 years 2 c. 3 years d. 4 years Correct Option: A Explanation:
( ( )
30000 × 1 +
or
107
t
100
=
7 100
)
t
34347 30000
= 30000 + 4347
=
11449 10000
=
( )
107 2 100
Time = 2 years
19. Rs.800 at 5% per annum compound interest will amount to Rs.882 in : a. 1 year b. 2 years c. 3 years d. 4 years Correct Option: B Explanation: Let time be t years
(
882 = 800 1 +
=
5 100
( ) ( ) 21 2 20
21
=
)
t
=
882 800
=
( ) 21
t
20
t
20
⇒
t = 2
time = 2 years
20. Simple interest on a sum at 4% per annum for 2 years is Rs.80.The compound interest on the same sum for the same period is : a. Rs.81.60 b. Rs.160 c. Rs.1081.60 d. None of these Correct Option: A Explanation: Principal = Rs.
C.I = Rs.
(
100 × 80 4× 2
[{ (
1000 × 1 +
)
= Rs.1000
4 100
)
2
− 1000
}]
= Rs.81.60
21. The difference of compound interest on Rs.800 for 1 year at 20% per annum when compounded half-yearly and quarterly is : a. Nil b. Rs.2.50 c. Rs.4.40 d. Rs.6.60 Correct Option: C Explanation: C.I when reckoned half-yearly
[ (
= Rs. 800 × 1 +
10 100
)
4
− 800
]
= Rs. 172.40
Difference = Rs.(172.40-168) Rs.4.40
22. The difference between simple interest and the compound interest on Rs.600 for 1 year at 10% per annum, reckoned half-yearly is : a. Nil b. Rs.6.60 c. Rs.4.40 d. Rs.1.50 Correct Option: D Explanation: S.I = Rs.
(
600 × 10 × 1
[
(
100
C.I = Rs. 600 × 1 +
)
= Rs.60
5 100
)
2
− 600
]
= Rs.61.50
Difference = Rs.(61.50-60) = Rs.1.50 1 23. The compound interest of Rs.20480 at 6 % per annum for 2 years 73 days is : 4 a. Rs.3000
b. Rs.3131 c. Rs.2929 d. Rs.3636 Correct Option: C Explanation: 73 days is 1/5 th of an year.
[
(
C.I = Rs. 20480 × 1 +
= Rs.
[(
20480 ×
17 16
×
25
)( )]
4 × 100
17 16
×
81 80
2
1+
1 5
×
25 4 × 100
)]
− 20480
− 20480 = Rs.2929
1 24. The compound interest on Rs.2800 for 1 years at 10% per annum is : 2 a. Rs.441.35
b. Rs.436.75
c. Rs.434 d. Rs.420 Correct Option: C Explanation: Amount = Rs.
[
= Rs. 2800 ×
[[
( ]
2800 × 1 +
11 100
×
21 20
10 100
)](
1+
5 100
)]
= Rs.3234
C.I = Rs.(3234-2800)= Rs.434
25. If Rs.7500 are borrowed at C.I at the rate of 4% per annum, then after 2 years the amount to be paid is : a. Rs.8082 b. Rs.7800 c. Rs.8100 d. Rs.8112 Correct Option: D Explanation:
[ (
Amount = Rs. 7500 1 +
4 100
) ] 2
[
= Rs. 7500 ×
26 25
×
26 25
]
=Rs.8112 Next>>
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