As we have seen in the previous exercise, the method of computing compound interest was very lengthy and required more time to solve. In Exercise 14.2 of Chapter 14, we shall discuss some formulae for the computation of compound interest, which becomes very easy to solve and is less time-consuming. Students can access the RD Sharma Solutions, formulated by subject experts to help them prepare well for the annual exam. Students can refer to and download the PDF from the links given below.
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1. Compute the amount and the compound interest in each of the following by using the formulae when :
(i) Principal = Rs 3000, Rate = 5%, Time = 2 years
(ii) Principal = Rs 3000, Rate = 18%, Time = 2 years
(iii) Principal = Rs 5000, Rate = 10 paise per rupee per annum, Time = 2 years
(iv) Principal = Rs 2000, Rate = 4 paise per rupee per annum, Time = 3 years
(v) Principal = Rs 12800, Rate = 7 ½ %, Time = 3 years
(vi) Principal = Rs 10000, Rate = 20% per annum compounded half-yearly, Time = 2 years
(vii) Principal = Rs 160000, Rate = 10 paise per rupee per annum compounded half yearly, Time = 2 years.
Solution:
By using the formula,
A = P (1 + R/100) n
Let us solve
(i) Given, P = Rs 3000, rate = 5%, time = 2years
A = P (1 + R/100) n
= 3000 (1 + 5/100)2
= 3000 (105/100)2
= Rs 3307.5
Compound interest (CI) = A-P = Rs 3307.5 – 3000 = Rs 307.5
(ii) Given, P = Rs 3000, rate = 18%, time = 2years
A = P (1 + R/100) n
= 3000 (1 + 18/100)2
= 3000 (118/100)2
= Rs 4177.2
Compound interest (CI) = A-P = Rs 4177.2 – 3000 = Rs 1177.2
(iii) Given, P = Rs 5000, rate = 10%, time = 2years
A = P (1 + R/100) n
= 5000 (1 + 10/100)2
= 5000 (110/100)2
= Rs 6050
Compound interest (CI) = A-P = Rs 6050 – 5000 = Rs 1050
(iv) Given, P = Rs 2000, rate = 4%, time = 3years
A = P (1 + R/100) n
= 2000 (1 + 4/100)3
= 2000 (104/100)3
= Rs 2249.72
Compound interest (CI) = A-P = Rs 2249.72 – 2000 = Rs 249.72
(v) Given, P = Rs 12800, rate = 7 ½ % = 15/2% = 7.5%, time = 3years
A = P (1 + R/100) n
= 12800 (1 + 7.5/100)3
= 12800 (107.5/100)3
= Rs 15901.4
Compound interest (CI) = A-P = Rs 15901.4 – 12800 = Rs 3101.4
(vi) Given, P = Rs 10000, rate = 20 % = 20/2 = 10% (quarterly), time = 2years = 2 × 2 = 4years
A = P (1 + R/100) n
= 10000 (1 + 10/100)4
= 10000 (110/100)4
= Rs 14641
Compound interest (CI) = A-P = Rs 14641 – 10000 = Rs 4641
(vii) Given, P = Rs 160000, rate = 10% = 10/2% = 5% (half yearly), time = 2years = 2×2 = 4 quarters
A = P (1 + R/100) n
= 160000 (1 + 5/100)4
= 160000 (105/100)4
= Rs 194481
Compound interest (CI) = A-P = Rs 194481 – 160000 = Rs 34481
2. Find the amount of Rs. 2400 after 3 years, when the interest is compounded annually at the rate of 20% per annum.
Solution:
Given details are,
Principal (p) = Rs 2400
Rate (r) = 20% per annum
Time (t) = 3 years
By using the formula,
A = P (1 + R/100) n
= 2400 (1 + 20/100)3
= 2400 (120/100)3
= Rs 4147.2
∴ Amount is Rs 4147.2
3. Rahman lent Rs. 16000 to Rasheed at the rate of 12 ½ % per annum compound interest. Find the amount payable by Rasheed to Rahman after 3 years.
Solution:
Given details are,
Principal (p) = Rs 16000
Rate (r) = 12 ½ % per annum = 12.5%
Time (t) = 3 years
By using the formula,
A = P (1 + R/100) n
= 16000 (1 + 12.5/100)3
= 16000 (112.5/100)3
= Rs 22781.25
∴ Amount is Rs 22781.25
4. Meera borrowed a sum of Rs. 1000 from Sita for two years. If the rate of interest is 10% compounded annually, find the amount that Meera has to pay back.
Solution:
Given details are,
Principal (p) = Rs 1000
Rate (r) = 10 % per annum
Time (t) = 2 years
By using the formula,
A = P (1 + R/100) n
= 1000 (1 + 10/100)2
= 1000 (110/100)2
= Rs 1210
∴ Amount is Rs 1210
5. Find the difference between the compound interest and simple interest. On a sum of Rs. 50,000 at 10% per annum for 2 years.
Solution:
Given details are,
Principal (p) = Rs 50000
Rate (r) = 10 % per annum
Time (t) = 2 years
By using the formula,
A = P (1 + R/100) n
= 50000 (1 + 10/100)2
= 50000 (110/100)2
= Rs 60500
CI = Rs 60500 – 50000 = Rs 10500
We know that SI = (PTR)/100 = (50000 × 10 × 2)/100 = Rs 10000
∴ Difference amount between CI and SI = 10500 – 10000 = Rs 500
6. Amit borrowed Rs. 16000 at 17 ½ % per annum simple interest. On the same day, he lent it to Ashu at the same rate but compounded annually. What does he gain at the end of 2 years?
Solution:
Given details are,
Principal (p) = Rs 16000
Rate (r) = 17 ½ % per annum = 35/2% or 17.5%
Time (t) = 2 years
Interest paid by Amit = (PTR)/100 = (16000×17.5×2)/100 = Rs 5600
Amount gained by Amit:
By using the formula,
A = P (1 + R/100) n
= 16000 (1 + 17.5/100)2
= 16000 (117.5/100)2
= Rs 22090
CI = Rs 22090 – 16000 = Rs 6090
∴ Amit total gain is = Rs 6090 – 5600 = Rs 490
7. Find the amount of Rs. 4096 for 18 months at 12 ½ % per annum, the interest being compounded semi-annually.
Solution:
Given details are,
Principal (p) = Rs 4096
Rate (r) = 12 ½ % per annum = 25/4% or 12.5/2%
Time (t) = 18 months = (18/12) × 2 = 3 half years
By using the formula,
A = P (1 + R/100) n
= 4096 (1 + 12.5/2×100)3
= 4096 (212.5/200)3
= Rs 4913
∴ Amount is Rs 4913
8. Find the amount and the compound interest on Rs. 8000 for 1 ½ years at 10% per annum, compounded half-yearly.
Solution:
Given details are,
Principal (p) = Rs 8000
Rate (r) = 10 % per annum = 10/2% = 5% (half yearly)
Time (t) = 1 ½ years = (3/2) × 2 = 3 half years
By using the formula,
A = P (1 + R/100) n
= 8000 (1 + 5/100)3
= 8000 (105/100)3
= Rs 9261
∴ CI = Rs 9261 – 8000 = Rs 1261
9. Kamal borrowed Rs. 57600 from LIC against her policy at 12 ½ % per annum to build a house. Find the amount that she pays to the LIC after 1 ½ years if the interest is calculated half-yearly.
Solution:
Given details are,
Principal (p) = Rs 57600
Rate (r) = 12 ½ % per annum = 25/2×2% = 25/4% = 12.5/2% (half yearly)
Time (t) = 1 ½ years = (3/2) × 2 = 3 half years
By using the formula,
A = P (1 + R/100) n
= 57600 (1 + 12.5/2×100)3
= 57600 (212.5/200)3
= Rs 69089.06
∴ Amount is Rs 69089.06
10. Abha purchased a house from Avas Parishad on credit. If the cost of the house is Rs. 64000 and the rate of interest is 5% per annum compounded half-yearly, find the interest paid by Abha after one year and a half.
Solution:
Given details are,
Principal (p) = Rs 64000
Rate (r) = 5 % per annum = 5/2% (half yearly)
Time (t) = 1 ½ years = (3/2) × 2 = 3 half years
By using the formula,
A = P (1 + R/100) n
= 64000 (1 + 5/2×100)3
= 64000 (205/200)3
= Rs 68921
∴ CI = Rs 68921 – 64000 = Rs 4921
11. Rakesh lent out Rs. 10000 for 2 years at 20% per annum, compounded annually. How much more he could earn if the interest be compounded half-yearly?
Solution:
Given details are,
Principal (p) = Rs 10000
Rate (r) = 20% per annum
Time (t) = 2years
By using the formula,
A = P (1 + R/100) n
= 10000 (1 + 20/100)2
= 10000 (120/100)2
= Rs 14400
When the interest is compounded half yearly,
Rate = 20/2 % = 10%
Time = 2×2 years = 4years
By using the formula,
A = P (1 + R/100) n
= 10000 (1 + 10/100)4
= 10000 (110/100)4
= Rs 14641
∴ Rakesh could earn Rs (14641 – 14400) = Rs 241 more
12. Romesh borrowed a sum of Rs. 245760 at 12.5% per annum, compounded annually. On the same day, he lent out his money to Ramu at the same rate of interest, but compounded semi-annually. Find his gain after 2 years.
Solution:
Given details are,
Principal (p) = Rs 245760
Rate (r) = 12.5% per annum
Time (t) = 2years
By using the formula,
A = P (1 + R/100) n
= 245760 (1 + 12.5/100)2
= 245760 (112.5/100)2
= Rs 311040
When compounded semi-annually,
Rate = 12.5/2% = 6.25%
Time = 2×2 years = 4years
By using the formula,
A = P (1 + R/100) n
= 245760 (1 + 6.25/100)4
= 245760 (106.25/100)4
= Rs 313203.75
∴ Romesh gain is Rs (313203.75 – 311040) = Rs 2163.75
13. Find the amount that David would receive if he invests Rs. 8192 for 18 months at 12 ½ % per annum, the interest being compounded half-yearly.
Solution:
Given details are,
Principal (p) = Rs 8192
Rate (r) = 12 ½ % per annum = 25/2×2 = 25/4% = 12.5/2% (half yearly)
Time (t) = 18 months = 18/12 = 1 ½ years = (3/2) ×2 = 3years
By using the formula,
A = P (1 + R/100) n
= 8192 (1 + 12.5/2×100)3
= 8192 (212.5/200)3
= Rs 9826
∴ Amount is Rs 9826
14. Find the compound interest on Rs. 15625 for 9 months, at 16% per annum, compounded quarterly.
Solution:
Given details are,
Principal (p) = Rs 15625
Rate (r) = 16% per annum = 16/4 = 4% (quarterly)
Time (t) = 9 months = 9/12 ×4 = 3quarters of a year
By using the formula,
A = P (1 + R/100) n
= 15625 (1 + 4/100)3
= 15625 (104/100)3
= Rs 17576
∴ CI = Rs 17576 – 15625 = Rs 1951
15. Rekha deposited Rs. 16000 in a foreign bank which pays interest at the rate of 20% per annum compounded quarterly, find the interest received by Rekha after one year
Solution:
Given details are,
Principal (p) = Rs 16000
Rate (r) = 20% per annum = 20/4 = 5% (quarterly)
Time (t) = 1 year = 4 quarters of a year
By using the formula,
A = P (1 + R/100) n
= 16000 (1 + 5/100)4
= 16000 (105/100)4
= Rs 19448.1
∴ CI = Rs 19448.1 – 16000 = Rs 3448.1
16. Find the amount of Rs. 12500 for 2 years compounded annually, the rate of interest being 15% for the first year and 16% for the second year.
Solution:
Given details are,
Principal (p) = Rs 12500
Rate1 (r) = 15% and Rate2 = 16%
Time (t) = 2 years
By using the formula,
A = P (1 + R1/100 × 1 + R2/100)
= 12500 (1 + 15/100 × 1 + 16/100)
= 12500 (1.15 × 1.16)
= Rs 16675
∴ Amount after two years is Rs 16675
17. Ramu borrowed Rs. 15625 from a finance company to buy scooter. If the rate of interest be 16% per annum compounded annually, what payment will he have to make after 2 ¼ years?
Solution:
Given details are,
Principal (p) = Rs 15625
Rate (r) = 16%
Time (t) = 2 ¼ years
By using the formula,
A = P (1 + R/100 × 1 + R/100)
= 15625 (1 + 16/100)2 × (1 + (16/4)/100)
= 15625 (1 + 16/100)2 × (1 + 4/100)
= 15625 (1.16)2 × (1.04)
= Rs 21866
∴ Amount after 2 ¼ years is Rs 21866
18. What will Rs. 125000 amount to at the rate of 6%, if the interest is calculated after every four months?
Solution:
Given details are,
Principal (p) = Rs 125000
Rate (r) = 6% per annum
Time (t) = 1 year
Since interest is compounded after 4months, interest will be counted as 6/3 = 2% and
Time will be 12/4 = 3quarters
By using the formula,
A = P (1 + R/100) n
= 125000 (1 + 2/100)3
= 125000 (102/100)3
= Rs 132651
∴ Amount is Rs 132651
19. Find the compound interest at the rate of 5% for three years on that principal which in three years at the rate of 5% per annum gives Rs. 12000 as simple interest.
Solution:
Given details are,
Simple interest (SI) = Rs 12000
Rate (r) = 5% per annum
Time (t) = 3 years
SI = (PTR)/100
P = (SI×100) / (T×R)
= (12000×100) / (3×5)
= 1200000/15
= 80000
By using the formula,
A = P (1 + R/100) n
= 80000 (1 + 5/100)3
= 80000 (105/100)3
= Rs 92610
∴ CI = Rs 92610 – 80000 = Rs 12610
20. A sum of money was lent for 2 years at 20% compounded annually. If the interest is payable half-yearly instead of yearly, then the interest is Rs. 482 more. Find the sum.
Solution:
Given details are,
Rate (r) = 20% per annum = 20/2 = 10% (half yearly)
Time (t) = 2 years = 2 × 2 = 4 half years
Principal be = Rs P
P (1 + R/100) n – P (1 + R/100) n = 482
P (1 + 10/100)4 – P (1 + 20/100)2 = 482
P (110/100)4 – P (120/100)2 = 482
P (1.4641) – P (1.44) = 482
0.0241P = 482
P = 482/0.0241
= 20000
∴ Amount is Rs 20000
21. Simple interest on a sum of money for 2 years at 6 ½ % per annum is Rs. 5200. What will be the compound interest on the sum at the same rate for the same period?
Solution:
Given details are,
Rate = 6 ½ % per annum = 13/2%
Simple Interest (SI) = Rs 5200
Time (t) = 2 years
By using the formula,
SI = (PTR)/100
P = (SI×100) / (T×R)
= (5200×100) / (2×13/2)
= (5200×100×2) / (2×13)
= 1040000/26
= Rs 40000
Now,
P = Rs 40000
R = 13/2% = 6.5%
T = 2years
By using the formula,
A = P (1 + R/100) n
= 40000 (1 + 6.5/100)2
= 40000 (106.5/100)2
= Rs 45369
∴ CI = Rs 45369 – 40000 = Rs 5369
22. What will be the compound interest at the rate of 5% per annum for 3 years on that principal which in 3 years at the rate of 5% per annum gives Rs. 1200 as simple interest.
Solution:
Given details are,
Rate = 5 % per annum
Simple Interest (SI) = Rs 1200
Time (t) = 3 years
By using the formula,
SI = (PTR)/100
P = (SI×100) / (T×R)
= (1200×100) / (3×5)
= 120000/15
= Rs 8000
Now,
P = Rs 8000
R = 5%
T = 3years
By using the formula,
A = P (1 + R/100) n
= 8000 (1 + 5/100)3
= 8000 (105/100)3
= Rs 9261
∴ CI = Rs 9261 – 8000 = Rs 1261
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