Rajiv invested a sum of  ₹ 7,000 at 20% p.a. for 1$\frac{1}{2}$ years, compounded half yearly, and his friend Amit invested a sum of ₹8,000 at 10% p.a. for 2 years, compounded annually. What is the difference between the interest received by Rajiv and Amit, at maturity ?

The correct option is ₹637.

Step 1: Calculate the compound interest received by Rajiv

Principal invested by Rajiv (P₁) = ₹ 7,000

Annual interest rate (R₁) = 20% p.a.

Time period = 1$\frac{1}{2}$ years = 3 half-years

Since the interest is compounded half-yearly, the rate per half-year (r₁) is half of the annual rate:

r₁ = $\frac{\mathrm{20\%}}{2}$ = 10% per half-year

Number of compounding periods (n₁) = 3

Using the compound interest amount formula:

A₁ = P₁ × (1 + $\frac{{\mathrm{r}}_1}{100}$)^n₁

A₁ = 7000 × (1 + $\frac{10}{100}$)³

A₁ = 7000 × ($\frac{11}{10}$)³

A₁ = 7000 × $\frac{1331}{1000}$

A₁ = 7 × 1331 = ₹ 9,317

Interest received by Rajiv (CI₁):

CI₁ = A₁ - P₁ = ₹ 9,317 - ₹ 7,000 = ₹ 2,317

Step 2: Calculate the compound interest received by Amit

Principal invested by Amit (P₂) = ₹ 8,000

Annual interest rate (R₂) = 10% p.a.

Time period (n₂) = 2 years (compounded annually)

Using the compound interest amount formula:

A₂ = P₂ × (1 + $\frac{{\mathrm{R}}_2}{100}$)^n₂

A₂ = 8000 × (1 + $\frac{10}{100}$)²

A₂ = 8000 × ($\frac{11}{10}$)²

A₂ = 8000 × $\frac{121}{100}$

A₂ = 80 × 121 = ₹ 9,680

Interest received by Amit (CI₂):

CI₂ = A₂ - P₂ = ₹ 9,680 - ₹ 8,000 = ₹ 1,680

Step 3: Calculate the difference between the interest received by Rajiv and Amit

Difference = CI₁ - CI₂

Difference = ₹ 2,317 - ₹ 1,680 = ₹ 637

Thus, the difference between the interest received by Rajiv and Amit at maturity is ₹637.