Compound Interest

M. Velrajan

Simple Interest 

interest for 1st year = interest for 2nd year 

                                 = interest for 3rd year = . . . . 

Principal amount P

After 1 year amount = P + interest for 1 year for P

After 2 years amount = P + 2 × interest for 1 year 

After 3 years amount = P + 3 × interest for 1 year

Compound Interest

A. Principal amount P

1st year interest = interest for 1 year for P

2nd year interest = interest for 1 year for P 

                                            + interest for interest for 1 year for P

                              = interest for 1 year for P 

                                            + interest for 1st year interest for 1 year

 3rd year interest = interest for 1 year for P 

                                            + interest for 2nd year interest for 1 year   

After 1 year amount = P + interest for 1 year for P

After 2 years amount = P + interest for 1 year for P 

                                            + interest for interest for 1 year for P

                                     = P + interest for 1 year amount for 1 year

After 3 years amount = P + interest for 2 years amount for 1 year   

Compound interest at 6 % per annum compounded annually means that interest per year per 100 is 6 and 

after 1 year every 100 becomes 100 + 6 = 106,

after 2 years every 100 becomes 100 + 6 + 6 + 6 × 6/100 

                                                    = 112 + 0.36 = 112.36,

after 3 years every 100 becomes 

100 + 6 + 6 + 6 × 6/100 + 6 + 6 × 6/100 + 6 × 6/100 + 6 × 6/100 × 6/100 

= 100 + 3 × 6 + 3 × 6 × 6/100 + 6 × 6/100 × 6/100 

= 118 + 3 × 0.36 + 0.0216, etc.

Note that there is no difference in simple and compound interest after 1 year.

After 2 years,

compound interest - simple interest 

= 0.36 = 6 × 6/100 

= interest for the one year interest.

After 3 years,

compound interest - simple interest 

= 3 × 6 × 6/100 + 6 × 6/100 × 6/100

= 3 × interest for the one year interest + interest for the interest for the one year interest

1. A certain sum grows up to Rs.4840 in 2 years and up to Rs.5324 in 3 years on

compound interest. The rate  of interest is   

(1) 5%  (2) 10%   (3) 15%      (4) 20%

Interest for 4840 for one year = 5324 - 4840 

                                                   = 484

Interest rate = interest for 100 for one year   

                      = (484/4840) × 100 = 10%

2. The simple interest on a certain principal amount for 4 years at 10% per annum

is half of the compound interest on Rs. 1000 for 2 years at 20% annum. 

Find the principal amount.

1.Rs 500      2. Rs 450     3. Rs. 650    4. Rs. 550

The compound interest on Rs. 1000 for 2 years at 20% annum : 

I year interest  = 1000 × 20/100 = 200

II year interest = 200 + (200 × 20/100) = 200 + 40 = 240

2 years compound interest = 200 + 240 = 440

The simple interest on principal amount for 4 years 

at 10% per annum = ½ × 440 = 220

The simple interest on principal amount for 1 year = 220/4 = 55

For interest 10, principal amount = 100

For interest 55, principal amount = (100/10)  × 55 = 550     Ans. 4.

3. A man deposits Rs 5000 in a bank at the rate of 12% simple interest for 2 years.

He also deposits another amount of Rs 5000 in another bank at the same rate of

12% compound interest compounded annually. Find the difference between the

simple interest and the compound interest generated on the amount after 2 years.

1.Rs 72    2. Rs 1200    3. Rs 1272    4. Rs 272

The difference between the simple interest and the compound interest 

in 2 years = interest for the one year interest  

First year Interest for 5000 = 5000 × 12/100 = 600

Difference   =  Interest for the interest 600  

                     = 600 × 12/100 = 72

Aliter:

      I year     II year

SI    600       600

CI   600       600 + 600 × 12/100 = 600 + 72

Difference = 72

4. Find the difference between simple interest and compound interest for a sum of

Rs. 8,000 lent at 10% p.a. in 2 years.            

(a)90                        (b)100                  (c)80                            (d)70

The difference between the two in two years 

                                                           = interest for the interest for the first year

First year Interest for 8000 = 800

Difference   =  Interest for the interest 800  = 80

Aliter:

      I year     II year

SI    800       800

CI   800       800 + 800 × 10/100 = 800 + 80

Difference = 80

5. A man invested an amount in two schemes in the ratio of 2 : 3 at the rate of 

20% p.a. and 10% p.a. on compound interest respectively. If the man gets a total interest of Rs. 1208 after two years from both the schemes, find the amount 

invested by the man? 

(a) 6000 Rs.       (b) 4800 Rs.    (c) 5000 Rs.     (d) 4500 Rs    (e) 4000 Rs.

Since 2 + 3 = 5, we take the total amount as 500. 

Then in I scheme 200 and in II 300

Amount          200        300

I year int          40          30

II yr int            40          30

Int for I yr int   8            3

Total int    = 40 + 48 + 30 + 33 = 151 

Amount = (500/151) × 1208 = 4000

6. The simple interest on a sum of money for 3 years at an interest rate of 6% p.a.

is Rs 6750. What will be the compound interest rounded off on the same sum at the

same rate for the same period, compounded annually is close to: 

1. 7133    2. 7633   3. 7103   4. 7163

One year simple interest = 6750/3 = 2250

Compound interest : 

I year       2250

II year      2250 + 2250 × 0.06 = 2250 + 135 = 2385

III year    2385 + 2385 × 0.06 = 2385 + 143.1 = 2528.1

3 years compound interest = 2250 + 2385 + 2528.1 = 7163.1

Ans. 4.

7. The compound interest on Rs. 16,000 for 9 months at 20% per annum compounded quarterly is

a. Rs. 2,599   b. Rs. 2,572   c. 2, 522  d. Rs. 2,502  e. Answer not known

Interest per quarter = 20/4 = 5%

For 16000,  I quarter interest  = 16000 × 5/100 = 800

II quarter interest = 800 + 800 × 5/100 = 800 + 40 = 840

 III quarter interest = 840 + 840 × 5/100 = 840 + 42  = 882

9 months compound interest = 800 + 840 + 882 = 2522

8. What is the compound interest on a sum of Rs.12,000 for 2 5/8 years at 8%p.a.,

when the interest is compounded annually?(nearest to a rupee)  

(a)2712     (b)2642     (c)2697    (d)2654

Interest rate for the last 5/8 year = 8 × 58 = 5%

I year interest  = 120 × 8 = 960

II year interest = 960 + 960 × 0.08 = 960 + 76.8 = 1036.8

5/8 year interest = (12000 + 960 + 1036.8) × 0.05 

                           = 13996.8 × 0.05 = 699.84

Total interest = 2696.64 = 2697

9. Find the principal that yields a compound interest of Rs. 1,632 in 2 years at 4%

rate of interest per annum. 

(a)Rs.10,000            (b)Rs.20,000           (c)Rs. 30,000            (d)Rs. 40,000

Int. for 100 = 8+, for 10000 = 800+, for 20000 = 1600+, for 30000 = 2400+.  

Since int is 1632, principal is  20000

Aliter :  Take Principal  100

I year int.     4

II year int.    4 and Int. for I int. 4 × 0.04 = 0.16

Total int.  8.16

Ans. = (100/8.16) × 1632 = (10000/816) × 1632 = 20000

 

The amount P becomes after n years at r% compounded annually  

                                                                                               = P(1 + r/100)n  

And r = ((A/P)1/n - 1)100, where A is the amount after n years

10. At what rate of compound interest per annum will a sum of Rs. 1,200 become

Rs. 1348.32 in 2 years     

(a)6%           (b)6.5 %         (c)7%        (d)7.5%

Interest for 2 years = 1348.32 - 1200 = 148.32.  

Interest for 1 year = 148.322 = 74.16

6 × 12 = 72, 6.5 × 12 = 78 > 74.16. Ans. 6%

(or) At 6% rate interest = 72 + 72 + 4.32 

(or) rate of interest = (74.16/1200) × 100 = 6 

Aliter: 1200(1 + r/100)2 = 1348.32

                       (1 + r/100)2 = 1348.32/1200 = 1.1236

                        1 + r/100 = (1.1236)½ = 1.06

                               r/100 = 0.06

                                 r = 6

(or) r = ((A/P)1/n - 1)100

        r = ((1348.32/1200)½ - 1)100 = (1.06 - 1)100 = 6

11. At what rate of compound interest does a sum of money become nine fold in 2

years?      (a)100%                (b)200%                  (c)300%                  (d)150%

In 2 years 100 becomes 900

Compound Interest for 100 in 2 years = 900 - 100 = 800

2 years compound interest for 100 at 100% = 100 + 100 + 100 = 300

2 years compound interest for 100 at 200% = 200 + 200 + 400 = 800

( or)

r = ((A/P)1/n - 1)100

   = ((9P/P)½ - 1)100 = 2 × 100 = 200

(or)

P(1 + r/100 )2 = 9P

  (1 + r/100)2 = 9 = 32 

    1 + r/100 = 3

     r/100 = 3 - 1 = 2

     r = 200%  

12. A certain sum of money doubles in 4 years at a compound rate of interest per

annum. The rate of interest is [Given : 21/2 = 1.414, 21/4 = 1.1892]

1. ~ 14.9 %          2. ~ 16.8 %        3. ~ 18.9 %       4. ~ 12.9 %

In 4 years 100 becomes 200

4 years compound interest for 100 = 200  - 100 = 100

4 years simple interest for 100 at highest rate 18.9% = 76 approx. 

Hence for other rates interest is less than 76. So, Ans. 3.

( or)

r = ((A/P)1/n - 1)100

   = ((2P/P)¼  - 1)100 = (1.1892 - 1) × 100 = 18.92

(or)

P(1 + r/100 )4 = 2P

(1 + r/100 )4 = 2 

r/100 = 21/4 - 1 = 1.1892 - 1 = 0.1892

 r = 18.92       Ans. 3. 

13. A sum of money placed at C.I. doubles itself in 5 years. It will amount to eight

times itself at the same rate of interest in  ------ years.

(a)5            (b)10               (c)15                    (d)20

In 5 years the amount doubles. So every 5 years 2 times.

In 10 years 4 times

In 15 years 8 times

(or)

P(1 + r/100)5 = 2P

 (1 + r/100)5  = 2

8 = 23 = ((1 + r/100)5)3

8P = P((1 + r/100)15 

Eight times in 15 years

14. In how many years will a sum of money triple itself at 5% CI Payable

annually?

(a)22            (b)22 1 2                  (c)23                  (d)23 1 2

P(1 + 5/100 )n = 3P

(105/100)n = 3

1.05n = 3

n log 1.05 = log 3

n = log 3/log 1.05 = 22.517    Ans.  (b)