Question

Rs. 16000 lent on 10% per annum compounded half yearly amounts to Rs. 18522. Find the time period for which the sum was lent.

Answer 1

Hint: Using compound interest formula we first evaluate time period for the given amount of money. Then after putting all the values which are given in the question i.e. (principal, amount, rate) we formulate out the required quantity. With help of this, we calculate the time period for which the sum was lent.

Complete step-by-step answer:

Now, as per the statement given in the question, the principal amount of lent in rupees is:

Principal = Rs. 16000

Also, from the given statement, the amount of lent in rupees is given below:

Amount = Rs. 18522

Also, the given rate of interest on the principal amount of the lent is:

Rate = 10% annually

First, we calculate rate of interest of half yearly time by using the formula:

Rate = $\dfrac{1}{2}\times 10=5%$ half yearly.

Now, we have to find the time period for the principal amount.

Formula of compound interest can be elaborated as:

A = P$\times {{\left( 1+\dfrac{R}{100} \right)}^{t}}$

Here, R = Rate of interest

           A = Amount

           P = Principal amount

           t = time period

Putting the values as derived above in the formula:

$  18522=16000{{\left( 1+\dfrac{5}{100} \right)}^{t}} $

$ \Rightarrow \dfrac{18522}{16000}={{\left( 1+\dfrac{5}{100} \right)}^{t}} $

Reducing the left-hand side by dividing with common multiple we get,

$\Rightarrow \dfrac{9261}{8000}={{\left( \dfrac{21}{20} \right)}^{t}}$

We can also write left hand side as,

${{\left( \dfrac{21}{20} \right)}^{3}}={{\left( \dfrac{21}{20} \right)}^{t}}$

Now, when comparing both sides we get the period of loan i.e. $t=3$ half years.

So, the time period in years would be:

$t=1\dfrac{1}{2}years$.

Hence, the time for which the loan was lent was one and half years.

Note: Formula of compound interest plays an important role in the evaluation process. All the dependent variables must be utilized in correct form like in this question half time period is utilized so care must be emphasized on dividing the final answer by 2.