Question

A sum of money becomes Rs.6500 after 3 years and Rs.10562.50 after 6 years on compound interest. The sum is:
A. Rs.4000
B. Rs.4500
C.Rs.4800
D. None of these.

Answer 1

Hint: we will assume that the sum is Rs.x and rate of interest be R%. Using these data given in question, we will first calculate the amount after three years at compound interest and then calculate the amount after 6 years at compound interest. Finally solve the two equations to get the sum and rate.

Complete step by step answer:
Let the sum be Rs.x and rate of interest be R%.
We know that amount(A) at compound interest is calculate as:
 A= $P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$ (1)
Where,
 P is the principal or sum.
R is the rate of interest.
‘n’ is time for which interest is calculated.
 Putting the values in equation 1, we get:
$6500 = x{\left( {1 + \dfrac{R}{{100}}} \right)^3}$ (2)
Again using the formula to calculate the amount we have:
A = $P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$ . where, ‘n’ is time duration of interest.

On putting the values, we get:
$10562.50 = x{\left( {1 + \dfrac{R}{{100}}} \right)^6}$ (3)
On dividing equation 2 by equation 1, we get:
$\dfrac{{10562.5}}{{6500}} = \dfrac{{{{\left( {1 + \dfrac{R}{{100}}} \right)}^6}}}{{{{\left( {1 + \dfrac{R}{{100}}} \right)}^3}}} = {\left( {1 + \dfrac{R}{{100}}} \right)^3}$
On solving the RHS, we get:
${\left( {1 + \dfrac{R}{{100}}} \right)^3} = 1.625$
On taking cube root on both sides, we get:
$\left( {1 + \dfrac{R}{{100}}} \right) = {\left( {1.625} \right)^{1/3}} = 1.175$
$ \Rightarrow $ R= (1.175 – 1)$ \times $100 = 17.5%
Putting the value of rate in equation 2, we get
$6500 = x{\left( {1 + \dfrac{{17.5}}{{100}}} \right)^3}$
$ \Rightarrow $ x = $\dfrac{{6500}}{{{{\left( {1 + \dfrac{{17.5}}{{100}}} \right)}^3}}} \simeq 4000$
Therefore, the sum of money = Rs.4000.

So, the correct answer is “Option A”.

Note: In this question, the important thing is to assume the unknown and then apply the formula. You should remember the formula to calculate amount in case of compound interest. The simple interest is given as:
$SI = \dfrac{{P \times R \times T}}{{100}}$ and amount = SI + Principal. Principal is also called sum.