On a certain rate of interest a sum of Rs. 5000 becomes Rs. 16,200 in certain years at compound interest. In half of the time given, this sum will become?

Hint: We are provided with the two amounts one is the sum of beginning and the other is the compound interest at the end of a particular time period. Since we have to find the sum at half the time therefore both the time periods are equal. Taking the ratio of sum and the time we can find the sum at half time period.

Complete step-by-step answer:
Let the sum at the start is a, at half time period is x and sum at end of full time period is c.
As we have to calculate sum for half time, both the time period is same
Hence, a : x = x : c
a = 5000
c= 16,200
∴ 5000 : x = x : 16,200
\[ \Rightarrow \]\[\dfrac{{5000}}{x} = \dfrac{x}{{16,200}}\]
\[ \Rightarrow \]\[{x^2} = 5000 \times 16,200\]
\[ \Rightarrow \]\[x = \sqrt {5000 \times 16,200} \]
\[ \Rightarrow \]\[x = 9000\]
∴ The amount at the end of the half time period is Rs. 9000.

Note: Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan. Simple interest is based on the principal amount of a loan or deposit. Compound interest is based on the principal amount and the interest that accumulates on it in every time period.