A person invested a certain amount of money at 10% annual interest, compounded half-yearly. After one and a half years, the interest and principal together became Rs 18522. The amount, in rupees, that the person had invested is

The correct answer is 16000.

We are given that a person invested a certain principal amount at 10% per annum, compounded half-yearly, and after one and a half years, the total amount (principal + interest) became Rs 18522. We need to find the original principal invested.

Step 1: Identify the compound interest formula.

The standard formula for compound interest is:

$A=P{(1+\frac{r}{100})}^n$

where A = total amount, P = principal, r = rate of interest per compounding period, and n = number of compounding periods.

Step 2: Determine the rate per compounding period and the number of periods.

Since the interest is compounded half-yearly (i.e., every 6 months):

Rate per half-year = $\frac{10}{2}=5\%$

Number of half-year periods in 1.5 years = $1.5\times 2=3$

Step 3: Substitute the known values into the formula.

$18522=P\times {(1+\frac{5}{100})}^3$

$18522=P\times {\left(\frac{21}{20}\right)}^3$

Step 4: Compute (21/20)³.

${\left(\frac{21}{20}\right)}^3=\frac{21\times 21\times 21}{20\times 20\times 20}=\frac{9261}{8000}$

Step 5: Solve for the principal P.

$18522=P\times \frac{9261}{8000}$

$P=\frac{18522\times 8000}{9261}$

$P=\frac{148176000}{9261}$

$P=16000$

Verification: If P = 16000, then:
After 1^st half-year: 16000 × 1.05 = 16800
After 2^nd half-year: 16800 × 1.05 = 17640
After 3^rd half-year: 17640 × 1.05 = 18522 ✓

This confirms that the amount the person originally invested is Rs 16000.