From the above table, we can see that in Simple interest case, Interests earned in each year is same, but in the compound interest, except for the first year, interests are different. This is due to the fact that in the CI case interest earned in the 1st year is considered as principal and interest will be calculated on this too. In the first box 10+1 can be explained as 10 is the interest on the principal, 1 is the interest on the first year interest. For the 2nd box, 10 is the interest on principal, 1 is interest on first year interest, another 1 is interest on second year 10, and 0.1 is the interest on 1.

## Compound Interest formula:

There is no direct formula for Compound interest but $P{\left( {1 + \displaystyle\frac{R}{{100}}} \right)^n}$ gives us the Amount which includes principal as well as interest. So to find the interest we need to substract principal from the amount.

CI = A - P = \(P{\left( {1 + \dfrac{R}{{100}}} \right)^n} - P\)

###
**Compound interest through Pascal Triangle:**

Find the compound interest on Rs.6000 at the rate 10% per annum for 3 years.

Formula: A = $P{\left( {1 + \displaystyle\frac{R}{{100}}} \right)^n}$ = $6000{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)^3}$

$ \Rightarrow 6000 \times {\left( {\displaystyle\frac{{11}}{{10}}} \right)^3}$ = 7986

Compound Interest = 7986 - 6000 = 1986

**Alternate method:**

We can solve compound interest problems easily with the use of pascal triangle.

###
**To find the difference between compound interest and simple interest for two years :**

Compound interest for two years = P $\left( {{\rm{1 + r}}} \right)^{\rm{2}} $ - PSimple interest for two years = P x r x 2 = 2Pr

Here, r = $\displaystyle\frac{{{\rm\text{Rate of interest}}}}{{100}}$

= [P $\left( {{\rm{1 + r}}} \right)^{\rm{2}} $ - P] - (2Pr)

= P (1 + 2r + ${\rm{r}}^{\rm{2}} $) - P - (2Pr)

= P + 2Pr + P${\rm{r}}^{\rm{2}} $ - P - 2Pr = P${\rm{r}}^{\rm{2}} $

Hence, difference between compound interest and simple interest @ r % p.a. for two years = P(${\rm{r}}^{\rm{2}} $)

###
**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(${\rm{r}}^{\rm{2}} $) = Pr (r)But, Pr = Principal x $\displaystyle\frac{{{\rm{Rate}}}}{{{\rm{100}}}}$ = Simple Interest for one year

= $\displaystyle\frac{1}{2}$ x Total Simple Interest

Therefore, Difference between compound interest and simple interest for 2 years

= Pr (r) = $\displaystyle\frac{1}{2}$ x Simple Interest x (r)

= $\displaystyle\frac{1}{2}$ r of Simple Interest

For 2 years: $\displaystyle\frac{1}{2}$r of simple interest

Alternate Method:

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 :**

${\rm{[P(1 + r}}{{\rm{)}}^{\rm{3}}}{\rm{ - P] - P \times 3 \times r}}$$ \Rightarrow {\rm{[P(1 + 3r + 3}}{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\rm{r}}^{\rm{3}}}{\rm{) - P] - P \times 3 \times r}}$

$ \Rightarrow {\rm{3P}}{{\rm{r}}^{\rm{2}}}{\rm{ + P}}{{\rm{r}}^{\rm{3}}}$

$ \Rightarrow {\rm{Pr(3r + }}{{\rm{r}}^{\rm{2}}}{\rm{)}}$

Difference for 3 years = P${\rm{r}}^{\rm{2}} $ (3 + r)

###
**To find the difference between Compound Interest and Simple interest, when Simple Interest for 3 years is given:**

For 3 years: $\displaystyle\frac{1}{3}$r (3 + r) of simple interestNote: 'r' is $\displaystyle\frac{{{\rm\text{Rate of interest}}}}{{{\rm{100}}}}$

Difference between compound interest and simple interest for 3 years = P${\rm{r}}^{\rm{2}} $ (3 + r) = Pr (r) (3 + r)

Now, Pr = Principal x $\displaystyle\frac{{{\rm{Rate}}}}{{{\rm{100}}}}$ = Simple Interest for one year

= $\displaystyle\frac{{\rm{1}}}{{\rm{3}}}$ x Total Simple Interest

Therefore, Difference between compound interest and simple interest for 3 years

= Pr (r) (3 + r) = $\displaystyle\frac{{\rm{1}}}{{\rm{3}}}$ x Simple Interest x (r) (3 + r)

= $\displaystyle\frac{{\rm{1}}}{{\rm{3}}}$r (3 + r) of Simple Interest

###
**To find simple interest when compound interest is given:**

If difference between compound interest and simple interest is $\displaystyle\frac{{\rm{1}}}{{\rm{x}}}$ of Simple interest Then, difference between compound interest and simple interest is $\displaystyle\frac{{\rm{1}}}{{{\rm{x - 1}}}}$ of Compound Interest.Hint: Compound Interest is more than Simple Interest.

And, Compound Interest = Simple 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 =$\displaystyle\frac{{10}}{2}$= 5% (for half year)

P = 6,000

A = 6,000${\left( {1 + \displaystyle\frac{5}{{100}}} \right)^4}$ = Rs. 7,293

Similarly, R =$\displaystyle\frac{{10}}{2}$= 5% (for half year)

P = 6,000

A = 6,000${\left( {1 + \displaystyle\frac{5}{{100}}} \right)^4}$ = 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 = $\displaystyle\frac{{a \times 2 \times 10}}{{100}} = \displaystyle\frac{a}{5}$ and CI = Amount – a = a${\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)^2}$– a = $\displaystyle\frac{{21}}{{100}} \times a$

CI – SI = Rs. 372

$\displaystyle\frac{{21}}{{100}} \times a - \displaystyle\frac{a}{5}$ = Rs. 372

a = Rs. 37,200

**Alternate method:**

We know that the difference in interest comes from second year. Assume principal is Rs.100 then Interests are calculated as below.

$ \Rightarrow \displaystyle\frac{{372}}{1} \times 100 = 37200$

**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 × $\displaystyle\frac{{100}}{{10}}$ = Rs. 3,720

3. Find compound interest on Rs. 10000 at 10% p.a. for 4 years, if interest is compounded annually.

Amount = Rs. 10000 x $\left( {\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}} \right)^{\rm{4}} $ = 14641

Therefore, Compound interest = Rs. 14641 - Rs. 10000 = Rs. 4641

Note: Steps for calculation of ${\rm{11}}^{\rm{4}} {\rm{:}}$

11 x 11 = 121; 121 x 11 = 1331; 1331 x 11 = 14641

Approximate Method:

We know that,Therefore, Compound interest = Rs. 14641 - Rs. 10000 = Rs. 4641

Note: Steps for calculation of ${\rm{11}}^{\rm{4}} {\rm{:}}$

11 x 11 = 121; 121 x 11 = 1331; 1331 x 11 = 14641

Approximate Method:

Compound interest = Simple interest + Interest on simple interest

Simple interest = 10000 x $\displaystyle\frac{{\rm{1}}}{{{\rm{10}}}}$ x 4 = Rs. 4000

Therefore, Simple interest for one year = Rs. 4000 ${\rm{ \div }}$ 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, ${\rm{2}}^{\rm{2}} {\rm{ = 4}}{\rm{.}}$

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?

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, $\left( {{\rm{1 + }}\displaystyle\frac{{{\rm{Rate}}}}{{{\rm{100}}}}} \right)^{{\rm{Time}}} {\rm{ = }}\displaystyle\frac{{{\rm{Amount}}}}{{{\rm{Principal}}}}$

Therefore, $\left( {{\rm{1 + r}}} \right)^{\rm{3}} {\rm{ = }}\displaystyle\frac{{{\rm{2662}}}}{{{\rm{2000}}}}{\rm{ = }}\displaystyle\frac{{{\rm{1331}}}}{{{\rm{1000}}}}{\rm{ = }}\left( {\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}} \right)^{\rm{3}} $

Therefore, 1 + r = $\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}$

Therefore, r = $\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}{\rm{ - 1 = }}\displaystyle\frac{{\rm{1}}}{{{\rm{10}}}}$ = 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, $\left( {{\rm{1 + r}}} \right)^{\rm{3}} {\rm{ = }}\displaystyle\frac{{{\rm{18522}}}}{{{\rm{16000}}}}{\rm{ = }}\displaystyle\frac{{{\rm{9261}}}}{{{\rm{8000}}}}{\rm{ = }}\left( {\displaystyle\frac{{{\rm{21}}}}{{{\rm{20}}}}} \right)^{\rm{3}} {\rm{ = }}\left( {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{20}}}}} \right)^{\rm{3}} $

Therefore, Rate of interest = $\displaystyle\frac{{\rm{1}}}{{{\rm{20}}}}$ = 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

Therefore, Rate of interest = $\displaystyle\frac{{{\rm{2522}}}}{{{\rm{480}}}}$ = 5 + (Remainder is Rs. 122)

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 $ \ge $ 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.

The sum amounts to $\displaystyle\frac{{3456}}{{2880}} = \displaystyle\frac{6}{5}$ times of itself

Therefore, Principal = $2880 \div \left( {\displaystyle\frac{6}{5}} \right)^2 = 2880 \times \displaystyle\frac{5}{6} \times \displaystyle\frac{5}{6}$ = Rs. 2000

8. A man borrows Rs. 2100 and undertakes to pay back with compound interest @ 10% p.a. in 2 equal yearly installments at the end of first and second year. What is the amount of each installment?Here, (1 + r) = 1 + $\displaystyle\frac{1}{{10}} = \displaystyle\frac{{11}}{{10}}$

Ratio of principals of two instalments = 1 : $\displaystyle\frac{{10}}{{11}}$ = 11 : 10

Sum of ratios = 11 + 10 = 21

Therefore, Principal of first instalment = 2100 x $\displaystyle\frac{{11}}{{21}}$ = Rs. 1100

Therefore, Instalment = Principal of first instalment x (1 + r)

= 1100 x $\displaystyle\frac{{11}}{{10}}$ = Rs. 1210

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?

Here, (1 + r) = 1 + $\displaystyle\frac{1}{{20}} = \displaystyle\frac{{21}}{{20}}$

Ratio of principals of two instalments = 1 : $\displaystyle\frac{{20}}{{21}}$ = 21 : 20

Sum of ratios = 21 + 20 = 41

Therefore, Principal of first instalment = $\displaystyle\frac{{21}}{{41}}$ x 820 = Rs. 420

Therefore, Instalment = Principal of first instalment x (1 + r) = 420 x $\displaystyle\frac{{21}}{{20}}$ = Rs. 441

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?

Here, (1 + r) = 1 + $\displaystyle\frac{1}{5} = \displaystyle\frac{6}{5}$

Ratio of principals for three years = 1 : $\displaystyle\frac{5}{6}:\left( {\displaystyle\frac{5}{6}} \right)^2 $

= $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 = $\displaystyle\frac{{36}}{{91}}$ x 1820 = Rs. 720

Therefore, Installment = Principal of first installment x (1 + r) = 720 x $\displaystyle\frac{6}{5}$ = 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 + r) = 1 + $\displaystyle\frac{1}{{10}} = \displaystyle\frac{{11}}{{10}}$

Then, $\displaystyle\frac{x}{y} = \left( {\displaystyle\frac{{11}}{{10}}} \right)^{7 - 5} = \left( {\displaystyle\frac{{11}}{{10}}} \right)^2 = \displaystyle\frac{{121}}{{100}}$

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 + r) = 1 + $\displaystyle\frac{1}{{25}} = \displaystyle\frac{{26}}{{25}}$

Then, $\displaystyle\frac{{\rm{x}}}{{\rm{y}}}$ = $\left( {\displaystyle\frac{{26}}{{25}}} \right)^{2 - 3} = \left( {\displaystyle\frac{{26}}{{25}}} \right)^{ - 1} = \displaystyle\frac{{25}}{{26}}$

Therefore, Mohan's share = $\displaystyle\frac{{25}}{{51}}$ 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, Rate = $\displaystyle\frac{{10}}{2}$% = 5% = $\displaystyle\frac{1}{{20}}$

And, Time = 2 x 1 = 2 half-years.

Therefore, Difference between Compound Interest and Simple Interest = Rs. 4000 x $\displaystyle\frac{1}{{20}}$ x $\dfrac{1}{{20}}$ = Rs. 10

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${\rm{r}}^{\rm{2}} $ (3 + r) = Rs. 1000 x $\displaystyle\frac{1}{{10}}$ x $\frac{1}{{10}}$ x $\left( {3 + \dfrac{1}{{10}}} \right)$ = 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${\rm{r}}^{\rm{2}} $ (6 + 4r + ${\rm{r}}^{\rm{2}} $) = 10000 x $\displaystyle\frac{1}{{10}}$ x $\displaystyle\frac{1}{{10}}$ x $\left( {6 + \displaystyle\frac{4}{{10}} + \displaystyle\frac{1}{{100}}} \right)$

= 10000 x $\displaystyle\frac{1}{{100}} \times \displaystyle\frac{{641}}{{100}}$ = Rs. 641

16. If Compound Interest on a certain sum for 2 years @ 5% p.a. is Rs. 328, the Simple interest will be ?

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 Simple 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 $\displaystyle\frac{{200}}{{205}}$

= Rs. 320

Alternative Method:

Rate = 5% = $\displaystyle\frac{1}{{20}}$

Difference between Compound interest and Simple interest

= $\displaystyle\frac{1}{2} \times \frac{1}{{20}} = \displaystyle\frac{1}{{40}}$ of simple interest

= $\displaystyle\frac{1}{{41}}$ of the compound interest = $\displaystyle\frac{1}{{41}}$ x Rs. 328 = Rs. 8

Therefore, Simple interest = Compounded interest - Difference

= Rs. 328 - Rs. 8 = Rs. 320

Therefore, $\left( {{\rm{1 + r}}} \right)^{\rm{3}} {\rm{ = }}\displaystyle\frac{{{\rm{2662}}}}{{{\rm{2000}}}}{\rm{ = }}\displaystyle\frac{{{\rm{1331}}}}{{{\rm{1000}}}}{\rm{ = }}\left( {\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}} \right)^{\rm{3}} $

Therefore, 1 + r = $\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}$

Therefore, r = $\displaystyle\frac{{{\rm{11}}}}{{{\rm{10}}}}{\rm{ - 1 = }}\displaystyle\frac{{\rm{1}}}{{{\rm{10}}}}$ = 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, $\left( {{\rm{1 + r}}} \right)^{\rm{3}} {\rm{ = }}\displaystyle\frac{{{\rm{18522}}}}{{{\rm{16000}}}}{\rm{ = }}\displaystyle\frac{{{\rm{9261}}}}{{{\rm{8000}}}}{\rm{ = }}\left( {\displaystyle\frac{{{\rm{21}}}}{{{\rm{20}}}}} \right)^{\rm{3}} {\rm{ = }}\left( {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{20}}}}} \right)^{\rm{3}} $

Therefore, Rate of interest = $\displaystyle\frac{{\rm{1}}}{{{\rm{20}}}}$ = 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

Therefore, Rate of interest = $\displaystyle\frac{{{\rm{2522}}}}{{{\rm{480}}}}$ = 5 + (Remainder is Rs. 122)

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 $ \ge $ 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.

The sum amounts to $\displaystyle\frac{{3456}}{{2880}} = \displaystyle\frac{6}{5}$ times of itself

Therefore, Principal = $2880 \div \left( {\displaystyle\frac{6}{5}} \right)^2 = 2880 \times \displaystyle\frac{5}{6} \times \displaystyle\frac{5}{6}$ = Rs. 2000

8. A man borrows Rs. 2100 and undertakes to pay back with compound interest @ 10% p.a. in 2 equal yearly installments at the end of first and second year. What is the amount of each installment?Here, (1 + r) = 1 + $\displaystyle\frac{1}{{10}} = \displaystyle\frac{{11}}{{10}}$

Ratio of principals of two instalments = 1 : $\displaystyle\frac{{10}}{{11}}$ = 11 : 10

Sum of ratios = 11 + 10 = 21

Therefore, Principal of first instalment = 2100 x $\displaystyle\frac{{11}}{{21}}$ = Rs. 1100

Therefore, Instalment = Principal of first instalment x (1 + r)

= 1100 x $\displaystyle\frac{{11}}{{10}}$ = Rs. 1210

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?

Here, (1 + r) = 1 + $\displaystyle\frac{1}{{20}} = \displaystyle\frac{{21}}{{20}}$

Ratio of principals of two instalments = 1 : $\displaystyle\frac{{20}}{{21}}$ = 21 : 20

Sum of ratios = 21 + 20 = 41

Therefore, Principal of first instalment = $\displaystyle\frac{{21}}{{41}}$ x 820 = Rs. 420

Therefore, Instalment = Principal of first instalment x (1 + r) = 420 x $\displaystyle\frac{{21}}{{20}}$ = Rs. 441

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?

Here, (1 + r) = 1 + $\displaystyle\frac{1}{5} = \displaystyle\frac{6}{5}$

Ratio of principals for three years = 1 : $\displaystyle\frac{5}{6}:\left( {\displaystyle\frac{5}{6}} \right)^2 $

= $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 = $\displaystyle\frac{{36}}{{91}}$ x 1820 = Rs. 720

Therefore, Installment = Principal of first installment x (1 + r) = 720 x $\displaystyle\frac{6}{5}$ = 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 + r) = 1 + $\displaystyle\frac{1}{{10}} = \displaystyle\frac{{11}}{{10}}$

Then, $\displaystyle\frac{x}{y} = \left( {\displaystyle\frac{{11}}{{10}}} \right)^{7 - 5} = \left( {\displaystyle\frac{{11}}{{10}}} \right)^2 = \displaystyle\frac{{121}}{{100}}$

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 + r) = 1 + $\displaystyle\frac{1}{{25}} = \displaystyle\frac{{26}}{{25}}$

Then, $\displaystyle\frac{{\rm{x}}}{{\rm{y}}}$ = $\left( {\displaystyle\frac{{26}}{{25}}} \right)^{2 - 3} = \left( {\displaystyle\frac{{26}}{{25}}} \right)^{ - 1} = \displaystyle\frac{{25}}{{26}}$

Therefore, Mohan's share = $\displaystyle\frac{{25}}{{51}}$ 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, Rate = $\displaystyle\frac{{10}}{2}$% = 5% = $\displaystyle\frac{1}{{20}}$

And, Time = 2 x 1 = 2 half-years.

Therefore, Difference between Compound Interest and Simple Interest = Rs. 4000 x $\displaystyle\frac{1}{{20}}$ x $\dfrac{1}{{20}}$ = Rs. 10

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${\rm{r}}^{\rm{2}} $ (3 + r) = Rs. 1000 x $\displaystyle\frac{1}{{10}}$ x $\frac{1}{{10}}$ x $\left( {3 + \dfrac{1}{{10}}} \right)$ = 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${\rm{r}}^{\rm{2}} $ (6 + 4r + ${\rm{r}}^{\rm{2}} $) = 10000 x $\displaystyle\frac{1}{{10}}$ x $\displaystyle\frac{1}{{10}}$ x $\left( {6 + \displaystyle\frac{4}{{10}} + \displaystyle\frac{1}{{100}}} \right)$

= 10000 x $\displaystyle\frac{1}{{100}} \times \displaystyle\frac{{641}}{{100}}$ = Rs. 641

16. If Compound Interest on a certain sum for 2 years @ 5% p.a. is Rs. 328, the Simple interest will be ?

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 Simple 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 $\displaystyle\frac{{200}}{{205}}$

= Rs. 320

Alternative Method:

Rate = 5% = $\displaystyle\frac{1}{{20}}$

Difference between Compound interest and Simple interest

= $\displaystyle\frac{1}{2} \times \frac{1}{{20}} = \displaystyle\frac{1}{{40}}$ of simple interest

= $\displaystyle\frac{1}{{41}}$ of the compound interest = $\displaystyle\frac{1}{{41}}$ x Rs. 328 = Rs. 8

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, ${\rm{2}}^{\rm{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.

Then, P ${\left[ {1 + \displaystyle\frac{R}{{100}}} \right]^3} = 6690$.........(i)

and P ${\left[ {1 + \displaystyle\frac{R}{{100}}} \right]^6} = 10,035$ ..... (ii)

Dividing (ii) by (i), we get

${\left( {1 + \displaystyle\frac{R}{{100}}} \right)^3} = \displaystyle\frac{{10035}}{{6690}} = \displaystyle\frac{3}{2}$

P=$\left( {6690 \times \displaystyle\frac{2}{3}} \right)$=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

c..$1\displaystyle\frac{1}{2}$ years

d. $2\displaystyle\frac{1}{2}$ years

Correct Option: A

Explanation:

1600${\left( {1 + \displaystyle\frac{5}{{100}}} \right)^T} = 1944.81$

$ \Rightarrow {\left( {\displaystyle\frac{{21}}{{20}}} \right)^r} = \displaystyle\frac{{1944.81}}{{1600.00}} = \displaystyle\frac{{194481}}{{160000}}$

=${\left( {\displaystyle\frac{{441}}{{400}}} \right)^2} = {\left( {\displaystyle\frac{{21}}{{20}}} \right)^4}$

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. $\left( {\displaystyle\frac{{100 \times 200}}{{8 \times 1}}} \right) = $Rs.2500

Amount for 2 years, compounded half-yearly

=Rs.$\left[ {2500 \times {{\left( {1 + \displaystyle\frac{4}{{100}}} \right)}^4}} \right]$=Rs.2924.64

C.I = Rs.424.64

Also, S.I = Rs.${\left( {\displaystyle\frac{{2500 \times 8 \times 2}}{{100}}} \right)}$=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 :

a. $5\displaystyle\frac{5}{9}$%

b. $12\displaystyle\frac{1}{2}$%

c. $11\displaystyle\frac{1}{9}$%

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 = $\left( {\displaystyle\frac{{100 \times 160}}{{1440 \times 1}}} \right)$ = $11\displaystyle\frac{1}{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:

P=${\left( {1 - \displaystyle\frac{{10}}{{100}}} \right)^3} = 729$

P=Rs.$\left( {\displaystyle\frac{{729 \times 10 \times 10 \times 10}}{{9 \times 9 \times 9}}} \right)$=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:

$x{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)^n} > 2x$ or ${\left( {\displaystyle\frac{6}{5}} \right)^n} > 2$

Now, $\left( {\displaystyle\frac{6}{5} \times \displaystyle\frac{6}{5} \times \displaystyle\frac{6}{5} \times \displaystyle\frac{6}{5}} \right) > 2$

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,

$\displaystyle\frac{x}{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)}} + \displaystyle\frac{x}{{{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)}^2}}} = 550$

or $\displaystyle\frac{{5x}}{6} + \displaystyle\frac{{25x}}{{36}} = 550$ or x = 360

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 )

= Rs. $\displaystyle\frac{{121}}{{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)}} + \displaystyle\frac{{121}}{{{{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)}^2}}}$ =Rs.210

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 ${{{\left( {1 + \displaystyle\frac{R}{{100}}} \right)}^3}}$=3149.28 ........ (i)

and P ${{{\left( {1 + \displaystyle\frac{R}{{100}}} \right)}^2}}$=2916 ..........(ii)

On dividing (i) and (ii) we get

${\left( {1 + \displaystyle\frac{R}{{100}}} \right)}$ = $\displaystyle\frac{{3149.28}}{{2916}}$

or ${100 = \displaystyle\frac{{233.28}}{{2916}}}$ or R = $\displaystyle\frac{{233.28}}{{2916}} \times 100 = 8\% $

Now P ${\left( {1 + \displaystyle\frac{8}{{100}}} \right)^2} = 2916$

or P $ \times \displaystyle\frac{{27}}{{25}} \times \displaystyle\frac{{27}}{{25}} = 2916$

or P = $\displaystyle\frac{{2916 \times 25 \times 25}}{{27 \times 27}}$=Rs. 2500

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 ${\left( {1 + \displaystyle\frac{R}{{100}}} \right)^3} = 10648$...(i)

and P ${\left( {1 + \displaystyle\frac{R}{{100}}} \right)^2} = 9680$ ....(ii)

On dividing (i) by (ii), we have

$\left( {1 + \displaystyle\frac{R}{{100}}} \right) = \displaystyle\frac{{10648}}{{9680}}$

or $\displaystyle\frac{R}{{100}} = \displaystyle\frac{{968}}{{9680}} = \displaystyle\frac{1}{{10}}$

or R = $\displaystyle\frac{1}{{10}} \times 100 = 10\% $

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:

$\left[ {5000 \times \left( {1 + {{\displaystyle\frac{R}{{100}}}^2}} \right) - 5000} \right] - \displaystyle\frac{{5000 \times 2 \times R}}{{100}} = 72$

$ \Rightarrow 5000\left[ {\left( {1 + {{\displaystyle\frac{R}{{100}}}^2}} \right) - 1 - \displaystyle\frac{R}{{50}}} \right] = 72$

$ \Rightarrow 1 + \displaystyle\frac{{{R^2}}}{{100}} + \displaystyle\frac{{2R}}{{100}} - 1 - \displaystyle\frac{R}{{50}} = \displaystyle\frac{{72}}{{5000}}$

$ \Rightarrow {R^2} = \left( {\displaystyle\frac{{72}}{{5000}} \times 10000} \right) = 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\left( {1 + {{\displaystyle\frac{P}{{100}}}^2}} \right) - P = 420 \Rightarrow P $=Rs.2000

S.I = Rs.$\displaystyle\frac{{2000 \times 2 \times 10}}{{100}}$= 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. $\left( {\displaystyle\frac{{100 \times 5 \times 2}}{{100}}} \right)$ = Rs.10

C.I = Rs.$\left[ {\left\{ {100 \times {{\left( {1 + \displaystyle\frac{5}{{100}}} \right)}^2}} \right\} - 100} \right]$ = Rs. $\displaystyle\frac{{41}}{4}$

Difference between C.I and S.I. = Rs. $\left( {\displaystyle\frac{{41}}{4} - 10} \right)$=Rs.0.25

0.25 : 1.50 : : 100 : x

x = $\displaystyle\frac{{1.50 \times 100}}{{0.25}}$= Rs.600

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 ${\left( {1 + \displaystyle\frac{r}{{100}}} \right)^5}$ or

${\left( {1 + \displaystyle\frac{r}{{100}}} \right)^5}$ = 2

Let it be 8 times in t years . Then, 8p = p ${\left( {1 + \displaystyle\frac{r}{{100}}} \right)^t}$

or ${\left( {1 + \displaystyle\frac{r}{{100}}} \right)^t}$=8=${(2)^3}$ = ${\left( {{{\left( {1 + \displaystyle\frac{r}{{100}}} \right)}^5}} \right)^3} = {\left( {1 + \displaystyle\frac{r}{{100}}} \right)^{15}}$

t = 15 years

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. $\left( {\displaystyle\frac{{100 \times 190}}{{2 \times 10}}} \right)$ = Rs.450

C.I = Rs. $\left[ {450 \times {{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)}^2} - 450} \right]$ = Rs.94.50

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.

So interest on first year ending amount = $\displaystyle\frac{{x \times 10 \times 1}}{{100}} = 132 \Rightarrow x = 1320$

Let y be the original principal. Then one year interest + principal = 1320

So, y + $\displaystyle\frac{{y \times 10 \times 1}}{{100}} = 1320 \Rightarrow y = 1200$

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 ${\left( {1 + \displaystyle\frac{4}{{100}}} \right)^2}$

$ \Rightarrow 1352 = P \times \displaystyle\frac{{26}}{{25}} \times \displaystyle\frac{{26}}{{25}}$

$ \Rightarrow P = \displaystyle\frac{{1352 \times 25 \times 25}}{{26 \times 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

b. $2\displaystyle\frac{1}{2}$ years

c. 3 years

d. 4 years

Correct Option: A

Explanation:

$30000 \times {\left( {1 + \displaystyle\frac{7}{{100}}} \right)^t} = 30000 + 4347$

or ${\left( {\displaystyle\frac{{107}}{{100}}} \right)^t} = \displaystyle\frac{{34347}}{{30000}} = \displaystyle\frac{{11449}}{{10000}} = {\left( {\displaystyle\frac{{107}}{{100}}} \right)^2}$

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

$882 = 800{\left( {1 + \displaystyle\frac{5}{{100}}} \right)^t} = \displaystyle\frac{{882}}{{800}} = {\left( {\displaystyle\frac{{21}}{{20}}} \right)^t}$

= ${\left( {\displaystyle\frac{{21}}{{20}}} \right)^2} = {\left( {\displaystyle\frac{{21}}{{20}}} \right)^t} \Rightarrow t = 2$

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. $\left( {\displaystyle\frac{{100 \times 80}}{{4 \times 2}}} \right)$ = Rs.1000

C.I = Rs. $\left[ {\left\{ {1000 \times {{\left( {1 + \displaystyle\frac{4}{{100}}} \right)}^2} - 1000} \right\}} \right]$ = 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. $\left[ {800 \times {{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)}^4} - 800} \right]$ = 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. $\left( {\displaystyle\frac{{600 \times 10 \times 1}}{{100}}} \right)$ = Rs.60

C.I = Rs. $\left[ {600 \times {{\left( {1 + \displaystyle\frac{5}{{100}}} \right)}^2} - 600} \right]$ = Rs.61.50

Difference = Rs.(61.50-60) = Rs.1.50

23. The compound interest of Rs.20480 at $6\displaystyle\frac{1}{4}$% per annum for 2 years 73 days is :

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.$\left[ {20480 \times {{\left( {1 + \displaystyle\frac{{25}}{{4 \times 100}}} \right)}^2}\left( {1 + \displaystyle\frac{1}{5} \times \displaystyle\frac{{25}}{{4 \times 100}}} \right)} \right] - 20480$

= Rs. $\left[ {\left( {20480 \times \displaystyle\frac{{17}}{{16}} \times \displaystyle\frac{{17}}{{16}} \times \displaystyle\frac{{81}}{{80}}} \right)} \right] - 20480$ = Rs.2929

24. The compound interest on Rs.2800 for $1\displaystyle\frac{1}{2}$years at 10% per annum is :

a. Rs.441.35

b. Rs.436.75

c. Rs.434

d. Rs.420

Correct Option: C

Explanation:

Amount = Rs.$\left[ {\left[ {2800 \times \left( {1 + \displaystyle\frac{{10}}{{100}}} \right)} \right]\left( {1 + \displaystyle\frac{5}{{100}}} \right)} \right]$

= Rs.$\left[ {2800 \times \displaystyle\frac{{11}}{{100}} \times \displaystyle\frac{{21}}{{20}}} \right]$ = 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: