Question

A certain sum of money placed on compound interest amounts to Rs.4000 in 3 years and Rs.5000 in 4 years, Calculate the rate of interest.

Answer 1

Hint: In this question a different amount is given for certain years so by using the compound interest formula we will get two equations for amount Rs.4000 and Rs.5000 respectively, then we will compare the two obtained equations to find the rate of interest.

Complete step-by-step answer:
Given
Amount in 3 years =Rs4000
Amount in 4 years =Rs5000
We know Compound interest on a sum of money is given by the formula
\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}\]
Let the initial principal sum of money be x
When sum of money amounts to Rs.4000 in 3 years
 $ A_1 $ =Rs.4000
 $ T_1 $ =3
Hence we can write
\[\Rightarrow 4000 = x{\left( {1 + \dfrac{r}{{100}}} \right)^3} - - (i)\]
Now when sum of money amounts to Rs.5000 in 4 years
 $ A_2 $ =Rs.5000
 $ T_2 $ =4
Hence we can write
\[\Rightarrow 5000 = x{\left( {1 + \dfrac{r}{{100}}} \right)^4} - - (ii)\]
Now divide equation (ii) by equation (i), hence we can write
\[\dfrac{{5000}}{{4000}} = \dfrac{{x{{\left( {1 + \dfrac{r}{{100}}} \right)}^4}}}{{x{{\left( {1 + \dfrac{r}{{100}}} \right)}^3}}}\]
By further solving
\[
\Rightarrow \dfrac{5}{4} = \dfrac{{{{\left( {1 + \dfrac{r}{{100}}} \right)}^4}}}{{{{\left( {1 + \dfrac{r}{{100}}} \right)}^3}}} \\
\Rightarrow \dfrac{5}{4} = 1 + \dfrac{r}{{100}} \\
\Rightarrow \dfrac{r}{{100}} = \dfrac{1}{4} \\
\Rightarrow r = \dfrac{1}{4} \times 100 \\
   = 25\% \\
 \]
Therefore the rate of interest at which a certain sum amounts to Rs.4000 in 3 years and Rs.5000 in 4 years \[ = 25\% \]

Note: If the interest rates are different for every year, Compound interest can also be calculated by \[A = P\left( {1 + \dfrac{{{r_1}}}{{100}}} \right)\left( {1 + \dfrac{{{r_2}}}{{100}}} \right).......\]for shortcut methods, but interest is being calculated year by year for better understanding.
Compound interest
\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}\]
Where
P is the initial principal sum of money
R is the interest rate in percentage
T is the time period in a year.