Question

The difference between the simple interest and compound interest at the same rate on Rs 5000 for two years is 98. What is the rate of interest?
[a] 10%
[b] 12%
[c] 10.5%
[d] 14%

Answer 1

Hint: Assume that the rate of interest r% per annum. Use the fact that in the simple interest the amount on which the interest is levied is constant. However, in the compound interest, the amount on which the interest is levied is changed after each year adding the interest gained that year to the total amount. Hence form an equation in r. Solve for r and hence find the rate of interest. Verify your answer.

Complete step by step answer:
Let the rate of interest be r% per annum.
When simple interest is levied on the sum.
We have Principal(P) = Rs 5,000, rate(r) = r and time (T) = 2 years.
We know that simple interest is given by $I=\dfrac{P\times r\times T}{100}=\dfrac{5000\times r\times 2}{100}=100r$
Hence the simple interest on the amount is $100r$
For compound interest we have
The principal for the first year is 5000, rate of interest is r and time is 1 year.
Hence the interest for the first year is $I=\dfrac{5000\times r}{100}=50r$
Principal for the second year is 50r+5000, rate of interest is r and time is 1 year.
Hence the interest for the second year is $I=\dfrac{\left( 5000+50r \right)\times r}{100}=50r+0.5{{r}^{2}}$
Hence the total interest levied is $50r+0.5{{r}^{2}}+50r=100r+0.5{{r}^{2}}$
Hence the difference between the simple interest and compound interest is $100r+0.5{{r}^{2}}-100r=0.5{{r}^{2}}$
But given that the difference between the compound interest and simple interest is 98.
Hence, we have
$\begin{align}
  & 0.5{{r}^{2}}=98 \\
 & \Rightarrow {{r}^{2}}=196 \\
\end{align}$
Hence, we have
$r=\pm 14$
But since r is the rate of interest r > 0
Hence, we have
$r=14$
Hence the rate of interest levied is 14%
Hence option [d] is correct.

Note:
[1] Verification: We can verify the correctness of our solution by checking that @ 14% the interest difference between simple interest and compound interest is 98.
We know that simple interest is given by $I=\dfrac{P\times r\times T}{100}$, where P is the principal, r is the rate of interest and T is the time.
Hence, we have
$I=\dfrac{5000\times 14\times 2}{100}=1400$
Also, we know that the compound interest is given by $I=P\left\{ {{\left( 1+\dfrac{r}{100} \right)}^{n}}-1 \right\}$, where P is the principal, r is the rate of interest and n is the time.
Hence, we have
$I=5000\left\{ {{\left( 1+\dfrac{14}{100} \right)}^{2}}-1 \right\}=1498$
Hence the difference between the compound interest and simple interest is 1498 – 1400 =Rs 98.
Hence our solution is verified to be correct.