## Compound Interest Questions and Answers Part-9

1. A sum of money becomes eight times in 3 years, If the rate is compounded annually. In how much time will the same amount at the same compound rate become sixteen times ?
a) 6 years
b) 4 years
c) 8 years
d) 5 years

Explanation:
\eqalign{ & {\text{Let principal = P}} \cr & {{Case (I)}} \cr & {\text{Time = 3 years,}} \cr & {\text{Amount = 8P}} \cr & 8{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & {\left( 2 \right)^3} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & {\text{Taking cube root of both sides,}} \cr & {\text{2 = }}\left( {1 + \frac{{\text{R}}}{{100}}} \right) \cr & {\text{R = 100 }}\% \cr & {{Case (II)}} \cr & {\text{Let after t years it will be 16 times}} \cr & 16{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{\text{t}}} \cr & 16 = {\left( 2 \right)^{\text{t}}} \cr & {\left( 2 \right)^4} = {\left( 2 \right)^{\text{t}}} \cr & {\text{t}} = 4 \cr & {\text{Required time}} {\text{(t) = 4 years}} \cr}

2. A sum of money placed at compound interest double itself in 4 years. In how many years will it amount to four times itself ?
a) 12 years
b) 13 years
c) 8 years
d) 16 years

Explanation:
\eqalign{ & {\text{Let}}, {\text{ Principal}} = Rs.\,100\% \cr & {\text{Amount}} = Rs.\,200 \cr & {\text{Rate}} = r\% \cr & {\text{Time}} = 4\,{\text{years}} \cr & A = P \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^n} \cr & 200 = 100 \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} \cr & 2 = {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} - - - - \left( i \right) \cr & {\text{If}}\,{\text{sum}}\,{\text{become}}\,{\text{8}}\,{\text{times}}\,{\text{in}}\,{\text{the}}\,{\text{time}}\,n\,{\text{years}} \cr & 4 = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {2^2} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} - - - - \left( {ii} \right) \cr & {\text{Using}}\,{\text{eqn}}\,\left( i \right)in\left( {ii} \right),\,{\text{we}}\,{\text{get}} \cr & {\left( {{{\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]}^4}} \right)^2} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^{8}} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & n = 8\,{\text{years}}. \cr}

3. The compound interest on Rs. 30000 at 7% per annum for a certain time is Rs. 4347. The times is = ?
a) 3 years
b) 4 years
c) 2 years
d) 2.5 years

Explanation:
\eqalign{ & {\text{Principal = Rs}}{\text{. 30000}} \cr & {\text{CI = Rs 4347}} \cr & {\text{Rate = 7}}\% \cr & {\text{By using formula, }} \cr & \left( {30000 + 4347} \right) = 30000{\left( {1 + \frac{7}{{100}}} \right)^{\text{t}}} \cr & 34347 = 30000{\left( {1 + \frac{7}{{100}}} \right)^{\text{t}}} \cr & \frac{{34347}}{{30000}} = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & \left( {\frac{{11449}}{{10000}}} \right) = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & {\left( {\frac{{107}}{{100}}} \right)^2} = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & {\text{t}} = 2\,{\text{years}} \cr}

4. A money lender borrows money at 4% per annum and pays the interest at the end of the year. He lends it at 6% per annum compound interest compounded half yearly and receives the interest at the end of the year. In this way, he gains Rs. 104.50, a year. The amount of money be borrows, is ?
a) Rs. 4500
b) Rs. 5000
c) Rs. 5500
d) Rs. 6000

Explanation:
\eqalign{ & {\text{Let the sum Rs}}{\text{. }}x{\text{ }} \cr & {\text{C}}{\text{.I}}{\text{. when compounded half yearly}} {\text{ = Rs}}{\text{.}}\left[ {x \times {{\left( {1 + \frac{3}{{100}}} \right)}^2} - x} \right] \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{10609}}{{10000}}x - x} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{609x}}{{10000}}} \right) \cr & {\text{C}}{\text{.I}}{\text{. when compounded yearly}} {\text{ = Rs}}{\text{.}}\left[ {x \times \left( {1 + \frac{4}{{100}}} \right) - x} \right] \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{26x}}{{25}} - x} \right) \cr & = {\text{Rs}}{\text{.}}\frac{x}{{25}} \cr & \therefore \frac{{609x}}{{10000}} - \frac{x}{{25}} = 104.50 \cr & \frac{{209x}}{{10000}} = 104.50 \cr & x = \left( {\frac{{104.50 \times 10000}}{{209}}} \right) \cr & x = 5000 \cr}

5. The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half yearly is = ?
a) 6.06%
b) 6.07%
c) 6.08%
d) 6.09%

Explanation: Amount of Rs. 100 for 1 year when compounded half yearly
\eqalign{ & {\text{ = Rs}}{\text{.}}\left[ {100 \times {{\left( {1 + \frac{3}{{100}}} \right)}^2}} \right] \cr & = {\text{Rs}}.106.09 \cr & {\text{Effective rate}} \cr & {\text{ = }}\left( {106.09 - 100} \right)\% \cr & = 6.09\,\% \cr}

6. A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:
a) Rs. 120
b) Rs. 121
c) Rs. 122
d) Rs. 123

Explanation:
\eqalign{ & {\text{Amount}} = {1600 \times {{\left( {1 + \frac{5}{{2 \times 100}}} \right)}^2} + 1600 \times \left( {1 + \frac{5}{{2 \times 100}}} \right)} \cr & = {1600 \times \frac{{41}}{{40}} \times \frac{{41}}{{40}} + 1600 \times \frac{{41}}{{40}}} \cr & = {1600 \times \frac{{41}}{{40}}\left( {\frac{{41}}{{40}} + 1} \right)} \cr & = {\frac{{1600 \times 41 \times 81}}{{40 \times 40}}} \cr & = Rs.\,3321 \cr & C.I. = Rs.\,\left( {3321 - 3200} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,121 \cr}

7. The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is Rs. 1. The sum (in Rs.) is:
a) 625
b) 630
c) 640
d) 650

Explanation:
\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{sum}}\,{\text{be}}\,Rs.\,x.\,{\text{Then}}, \cr & {\text{C}}{\text{.I}}{\text{.}} = {x{{\left( {1 + \frac{4}{{100}}} \right)}^2} - x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\frac{{676}}{{625}}x - x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{51}}{{625}}x \cr & {\text{S}}{\text{.I}}{\text{.}} = {\frac{{x \times 4 \times 2}}{{100}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2x}}{{25}} \cr & \therefore \frac{{51x}}{{625}} - \frac{{2x}}{{25}} = 1 \cr & x = 625 \cr}

8. There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
a) Rs. 2160
b) Rs. 3120
c) Rs. 3972
d) Rs. 6240

Explanation:
\eqalign{ & {\text{Let}}\,{\text{P = Rs}}{\text{.}}\,{\text{100}}\,{\text{Then}},\, \cr & \,\,\,\,\,{\text{S}}{\text{.I}}{\text{. = }}\,{\text{Rs}}{\text{.}}\,{\text{60}}\,{\text{and}} \cr & \,\,\,\,\,\,\,\,{\text{T = 6}}\,{\text{years}} \cr & R = {\frac{{100 \times 60}}{{100 \times 6}}} = 10\% \,p.a. \cr & {\text{Now}},\,P = Rs.\,12000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,T = 3\,{\text{year}}\,{\text{and}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,R = \,10\% \,p.a. \cr & {\text{C}}{\text{.I}}{\text{.}} = Rs.\,\left[ {12000 \times \left\{ {{{\left( {1 + \frac{{10}}{{100}}} \right)}^3} - 1} \right\}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {12000 \times \frac{{331}}{{1000}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,3972 \cr}

9. What is the difference between the compound interests on Rs. 5000 for $$1\frac{1}{2}$$ years at 4% per annum compounded yearly and half-yearly?
a) Rs. 2.04
b) Rs. 3.06
c) Rs. 4.80
d) Rs. 8.30

\eqalign{ & {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{compounded}}\,{\text{yearly}} = Rs.\left[ {5000 \times \left( {1 + \frac{4}{{100}}} \right) \times \left( {1 + \frac{{\frac{1}{2} \times 4}}{{100}}} \right)} \right] \cr & = Rs.\left( {5000 \times \frac{{26}}{{25}} \times \frac{{51}}{{50}}} \right) \cr & = Rs.5304 \cr & {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{in}}\,{\text{compounded}}\,{\text{half - yearly}} = Rs.\,\left[ {5000 \times {{\left( {1 + \frac{2}{{100}}} \right)}^3}} \right] \cr & = Rs.\,\left( {5000 \times \frac{{51}}{{50}} \times \frac{{51}}{{50}} \times \frac{{51}}{{50}}} \right) \cr & = Rs.\,5306.04 \cr & {\text{Difference}} = Rs.\,\left( {5306.04 - 5304} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,2.04 \cr}
b) $$2\frac{1}{2}$$
\eqalign{ & {\text{Amount}} = Rs.\,\left( {30000 + 4347} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,34347 \cr & {\text{Let}}\,{\text{the}}\,{\text{time}}\,{\text{be}}\,n\,{\text{years}} \cr & 30000\,{\left( {1 + \frac{7}{{100}}} \right)^n} = 34347 \cr & \Rightarrow {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{34347}}{{30000}} = \frac{{11449}}{{10000}} = {\left( {\frac{{107}}{{100}}} \right)^2} \cr & n = 2\,{\text{years}} \cr}