Question

Assertion: A sum of money amounts to ₹ 8400 after 1 year and ₹ 9261 after 3 years compounded annually. Then the rate of interest is 5%.

Reason: To find the rate when 2 non-successive years amounts are given, they are divided.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Given:

• Amount after 1 year, A₁ = ₹ 8400
• Amount after 3 years, A₃ = ₹ 9261 
• Interest: compounded annually

Step 1: Use the formula for compound interest

`A = P(1 + r/100)^n`

Let P = principal.

• After 1 year: `A_1 = P(1 + r/100) = 8400`
• After 3 years: `A_3 = P(1 + r/100)^3 = 9261`

Step 2: Divide the amounts for non-successive years

To eliminate P, divide A₃ by A₁:

`A_3/A_1 = (P(1 + r/100)^3)/(P(1 + r/100)^1) = (1 + r/100)^2`

`9261/8400 = 1.1025`

`(1 + r/100)^2 = 1.1025`

Step 3: Take square root to find the rate

`1 + r/100 = sqrt(1.1025) = 1.05`

`r/100 = 0.05`

`r = 5%`

Assertion is true.

Step 4: Check the reason

• Reason: “To find the rate when 2 non-successive years amounts are given, they are divided.”
• This is exactly what we did: divided A₃ by A₁ to eliminate P and solve for r. 

Reason is true and it correctly explains the assertion.