Compound Interest – Rate from second- and third-year interests: Rohit earns ₹ 1,440 as interest for the second year and ₹ 1,656 for the third year on the same sum at compound interest. What is the annual rate?

Introduction / Context:Under compound interest, each year’s interest equals the previous year’s interest multiplied by (1 + r). Knowing two consecutive years of interest allows direct extraction of the rate without needing the principal value.

Given Data / Assumptions:

• Second-year interest I2 = 1,440
• Third-year interest I3 = 1,656
• Annual rate r (constant)

Concept / Approach:I3 = I2 * (1 + r). Therefore (1 + r) = I3 / I2. Compute the ratio and convert to a percentage to obtain r.

Step-by-Step Solution:Compute ratio: 1,656 / 1,440 = 1.15Hence 1 + r = 1.15 ⇒ r = 0.15Convert to percent: r = 15%

Verification / Alternative check:If r = 15%, any year’s interest grows by 15% to the next year: 1,440 * 1.15 = 1,656, which matches perfectly. The principal cancels out in this ratio method.

Why Other Options Are Wrong:12%, 10%, 18%, or 16% produce 1 + r values of 1.12, 1.10, 1.18, and 1.16 respectively, none of which equal the observed 1.15 ratio.

Common Pitfalls:Trying to back out principal and then the rate is unnecessary and error-prone. Use the consecutive-interest ratio to shortcut directly to r.

Final Answer:15%