# Compound Interest

- Amount = Principal + Interest
- Simple Interest = (Principal x Rate x Time)/100
- A = P Here A = amount, P = principal, r = rate percent yearly (or every fixed period) and n is the number of years (or terms of the fixed period).
- C.I. = P , where C.I. = compound interest
- If the interest rates for the successive fixed periods are r
_{1}%, r_{2}%, r_{3}% ..., then A (amount) is given by

A=P... - S.I. (simple interest) and C.I. (compound interest) are equal for the first year (or the first term of the fixed period) on the same sum and at the same rate.
- C.I. of 2nd year (or the second term of the fixed period) is more than the C.I. of 1st year or the first term of the fixed period), and C.I. of 2nd Year -C.I. of 1st year = S.I. on the interest of the first year.
*Equal installments (with compound interest):*

Loan amount = P

where P = each equal installment

R = rate of interest per annum (or per specified period)

T = time, say 3 years (or 3 specified terms).

**Note.**If T = n years (or specified terms), then there will be n brackets.

## Exercise

- You invest Rs 5000 at 12% interest compounded annually. How much is in the account after 2 years, assuming that you make no subsequent withdrawal or deposit?
- Find the amount and the compound interest on Rs 4000 at 10% p.a. for 2½ years.
- A man invests Rs 5000 for three years at a certain rate of interest,
compounded annually. At the end of one year it amounts to Rs 5600. Calculate

(i) the rate of interest per annum,

(ii) the interest accrued in the second year,

(iii) the amount at the end of the third year. - A sum of Rs 9600 is invested for 3 years at 10% per annum at compound interest.

(i) What is the sum due at the end of the first year?

(ii) What is the sum due at the end of the second year?

(iii) Find the compound interest earned in two years.

(iv) Find the difference between the answers (ii) and (i) and find the interest on this sum for one year.

(v) Hence write down the compound interest for the third year. - Find the difference between the S.I. and C.I. on Rs 2500 for 2 years at 4% p.a., compound interest reckoned semi-annually.
- Find the amount and the compound interest on Rs 8000 in 2 years if the rate is 10% for the first year and 12% for the second year.
- A man invests Rs 6500 for 3 years at 4·5% p.a. compound interest reckoned yearly. Income tax at the rate of 20% is deducted at the end of each year. Find the amount at the end of the third year.
- Calculate the compound interest for the
*second*year on Rs 8000 invested for 3 years at 10% p.a. - Find the sum which amounts to Rs 9261 at 10% p.a. compound interest for 18 months, interest payable half-yearly.
- On what sum will the compound interest for 2 years at 5% p.a. be Rs 246?
- On what sum will the compound interest (reckoned yearly) for 3 years at 6¼% per annum be Rs 408·50?
- A man invests Rs 1200 for two years at compound interest. After one year his money amounts to Rs 1275. Find the rate of compound interest. Also find the amount which the man will get after 2 years correct to the nearest paise.
- At what rate percent per annum compound interest will Rs 2000 amount to Rs 2315·25 in 3 years?
- If Rs 50000 amounts to Rs 73205 in 4 years, find the rate of compound interest payable yearly.
- In what time will Rs 15625 amount to Rs 17576 at 4% per annum compound interest?
- In what time will a sum of Rs 2500 produce Rs 309 at 6% per annum compound interest?
- In what time will a sum of Rs 800 at 10% per annum compounded half-yearly produce Rs 126·10?
- The simple interest on a sum of money for 2 years at 4% p.a. is Rs 450.
Find the compound interest on this sum of money at the same rate

(i) for 1 year if the interest is reckoned semi-annually.

(ii) for 2 years if the interest is reckoned annually. - At what rate of compound interest will Rs 625 amount to Rs 729 after 2 years? Also find the maturity value of Rs 625 after 2 years at the above rate of simple interest.
- Ram and Bhola each borrow equal sums for 3 years at 10% p.a. simple interest and compound interest respectively. At the time of repayment, Bhola has to pay Rs 372 more than Ram. Find the sum borrowed and interest paid by each.
- The difference between the compound interest for a year payable half-yearly and the simple interest on a certain sum of money lent out at 10% p.a. for a year is Rs 15. Find the sum of money lent out.
- The difference between compound interest and simple interest in 3 years at 10% p.a. reckoned yearly is Rs 18·60. Find the sum and the compound interest.
- The amount at compound interest which is calculated yearly on a certain sum of money is Rs 1250 in one year and Rs 1375 in two years. Calculate the rate of interest.
- A certain sum of money amounts to Rs 10584 in two years and to Rs 11113·20 in three years, interest being compounded annually. Find the interest rate percent and the original sum.
- The compound interest and the simple interest on the same sum of money at the same rate percent per annum are Rs 410 and Rs 400 respectively. Find the rate of interest and the sum of money.
- The compound interest calculated yearly on a certain sum of money for the second year is Rs 880 and for the third year it is Rs 968. Find the rate of interest and the original money.
- The simple interest on a certain sum for 3 years is Rs 150 and the compound interest on the same sum at the same rate for 2 years is Rs 110. Find the rate of interest and the principal.
- A sum of money lent at C.I. on 1st April 96 amounts to Rs 2420 on 1st April 98 and to Rs 2662 on 1st April 99. Find the rate of interest and the sum.
- The simple interest in 3 years and the compound interest in 2 years on a
certain sum at the same rate are Rs 1200 and Rs 832 respectively. Find

(i) the rate of interest, (ii) the principal,

(iii) the difference between C.I. and S.I. for three years. - The value of a machine depreciates every year at the rate of 10% of its value. The machine was purchased for Rs 40000 when new and it was sold for Rs 29160. Find the number of years that the machine was used.
- A man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end of the first year and Rs 4410 at the end of the second year. If the rate of compound interest is 5% p.a., find the sum borrowed.
- A man borrowed a certain sum of money and paid it back in 2 years in two equal instalments. If the rate of compound interest was 4% p.a. and if he paid Rs 676 annually, what sum did he borrow?
- A sum of Rs 16400 is borrowed to be paid back in 2 years by two equal annual instalments allowing 5% compound interest. Find the annual payment.
- A loan of Rs 4641 is to be paid back by 4 equal annual instalments. The interest is compounded yearly at 10%. Find the value of each instalment.
- A man borrows Rs 6000 at 5% p.a. compound interest. If he repays Rs 1200 at the end of each year, find the amount outstanding at the beginning of the third year.

## Answers

**1.**Rs 6272

**2.**Rs 5082; Rs 1082

**3.**(i) 12% (ii) Rs 672 (iii) Rs 6952·64

**4.**(i) Rs 10560 (ii) Rs 11616 (iii) Rs 2016

(iv) Rs 1056, Rs 105·60 (v) Rs 1161·60

**5.**Rs 6·08

**6.**Rs 9856; Rs 1856

**7.**Rs 7227·56

**8.**Rs 880

**9.**Rs 8000

**10.**Rs 2400

**11.**Rs 2048

**12.**6¼%; Rs 1354·69

**13.**5%

**14.**10%

**15.**3 years

**16.**2 years

**17.**1½ years

**18.**(i) Rs 227·25 (ii) Rs 459

**19.**8%; Rs 725

**20.**Rs 12000; Rs 3600, Rs 3972

**21.**Rs 6000

**22.**Rs 600; Rs 196·60

**23.**10%

**24.**5% p.a., Rs 9600

**25.**5% p.a., Rs 4000

**26.**10%, Rs 8000

**27.**20%; Rs 250

**28.**10%, Rs 2000

**29.**(i) 8% (ii) Rs 5000 (iii) Rs 98·56

**30.**3 years

**31.**Rs 7000

**32.**Rs 1275

**33.**Rs 8820

**34.**Rs 1464·10

**35.**Rs 4155