Question

If the difference between the C. I compounded half-yearly and simple interest on a sum at $10\%$ per annum for one year is Rs.25, the sum is

Answer 1

Hint: We start solving the problem by finding the interest rate compounded half-yearly by taking half the value of the interest rate annually. We then assign the variable for the sum and find the compound interest of sum for year using the formula $C.I=P\left( {{\left( 1+\dfrac{r}{100} \right)}^{n}}-1 \right)$. We then find the simple interest per annum using the formula $S.I=\dfrac{npr}{100}$. We then take the difference between compound interest and simple interest and equate it to Rs. 25 to find the required value of a sum.

Complete step-by-step solution:
According to the problem, we are given that the difference between the C. I compounded half-yearly and simple interest on a sum at 10% per annum for one year is Rs.25. We need to find the value of the sum.
Let us assume the sum is ‘x’. We have given that the interest rate per annum is 10%.
Let us find the interest rate for half-yearly = $\dfrac{10\%}{2}=5\%$.
We know that the compound interest is calculated as $C.I=P\left( {{\left( 1+\dfrac{r}{100} \right)}^{n}}-1 \right)$.
Where P = principal sum,
r = rate of interest in %,
n = time in years.
Now, let us find the compound interest that is laid on the sum ‘x’ for a year at an interest rate of 5% half-yearly.
So, we get $C.I=x\left( {{\left( 1+\dfrac{5}{100} \right)}^{2}}-1 \right)$.
$\Rightarrow C.I=x\left( {{\left( 1+0.05 \right)}^{2}}-1 \right)$.
$\Rightarrow C.I=x\left( {{\left( 1.05 \right)}^{2}}-1 \right)$.
$\Rightarrow C.I=x\left( 1.1025-1 \right)$.
$\Rightarrow C.I=0.1025x$ ---(1).
We know that the simple interest is calculated as $S.I=\dfrac{npr}{100}$.
Where P = principal sum,
r = rate of interest in %,
n = time in years.
Now, let us find the simple interest that is laid on the sum ‘x’ for a year at an interest rate of 10% per annum.
So, we get $S.I=\dfrac{1\times x\times 10}{100}$.
$\Rightarrow S.I=\dfrac{x}{10}$.
$\Rightarrow S.I=0.1x$ ---(2).
Let us find the difference between Compound interest and simple interest obtained from equations (1) and (2).
So, we get $C.I-S.I=0.1025x-0.1x$.
$\Rightarrow C.I-S.I=0.0025x$.
According to the problem, this difference is equal to Rs. 25.
So, we get $25=0.0025x$.
\[\Rightarrow x=\dfrac{25}{0.0025}\].
\[\Rightarrow x=Rs.10000\].
So, we have found the value of the sum as Rs. 10,000. The correct option for the given problem is (c).

Note: We should not confuse the formulas of simple interest and compound interest while solving this problem. We should not make calculation mistakes as there is a good amount of calculation present in the problem. We can also find the values of compound interest and simple interest after finding the value of the sum. Similarly, we can expect problems to find the rate of interest for compound interest at which the compound interest is equal to simple interest.