Question

A sum of Rs. 40,000 was lent for one year at 16% per annum. If the same sum is lent for the same time and at the same rate percent but compounded half-yearly. How much more interest will he earn?

Answer 1

Hint: Start by using the formula that $A=P{{\left( 1+\dfrac{r}{100n} \right)}^{nt}}$ for the finding the amount when the interest is compounded half yearly. Also, use the formula $A=\dfrac{Ptr}{100}+P$ to get the amount in case of simple interest. Find the difference between the amounts in each case to get the amount he has earned more.

Complete step by step solution:
Before starting with the question, let us know about interest.
Interest in the financial term is the amount that a borrower pays to the lender along with the repayment of the actual principal amount.
Broadly, there are two kinds of interest, first is the simple interest, and the other is the compound interest.
Let us start by finding the amount when a simple interest of 16% per annum was given on the principal, i.e., Rs. 40,000. We know that the amount in case of principal interest is given by $A=\dfrac{Ptr}{100}+P$ , t in our case is 1 and r is 16%.
$A=\dfrac{Ptr}{100}+P=\dfrac{40000\times 1\times 16}{100}+40000=Rs.\text{ 4}6400$
Now let us move to the case in which the interest is compounded half-yearly. The rate of interest and principal is still the same, but the formula that is to be used is $A=P{{\left( 1+\dfrac{r}{100n} \right)}^{nt}}$ , where n is number of times the interest is compounded yearly, which is 2 in our case.
$A=40000{{\left( 1+\dfrac{16}{100\times 2} \right)}^{2}}=40000\times {{\left( \dfrac{108}{100} \right)}^{2}}=Rs.\text{ }46656$
Now the extra interest he got in the compound interest case is equal to the difference in amounts in two cases.
$46400-46656=Rs.\text{ 256}$
Hence, the answer to the above question is Rs. 256.

Note: Be careful with the calculations and solving part as there is a possibility of making a mistake in the calculations. It is recommended to learn all the basic formulas related to simple as well as compound interests as they are very much useful in the problems related to money exchange. The other thing you must note that if you are finding the difference in the amounts make sure that you are considering amount in both case, because it is generally seen that in case of simple interest student take $\dfrac{Ptr}{100}$ to be the amount but in actually it is the interest only.