An amount of Rs 8,000 is deposited into an account paying 7% annual interest compounded quarterly. After how many years (correct to one decimal place) will the balance in the account first reach Rs 12,400?

Introduction / Context:
Here the principal, target amount, interest rate, and compounding frequency are known, and we are asked to determine the time required for the principal to grow to the target amount. This is a classic compound interest problem where we solve for the time variable. Because the interest is compounded quarterly, we must work with a quarterly rate and convert the total number of quarters back into years. This question also reinforces the use of logarithms in financial mathematics.

Given Data / Assumptions:

• Principal (P) = Rs 8,000.
• Required future amount (A) = Rs 12,400.
• Nominal annual interest rate (R) = 7% per annum.
• Interest is compounded quarterly, so m = 4 compounding periods per year.
• We want the time T in years such that the amount reaches Rs 12,400.

Concept / Approach:
The general compound interest formula with compounding m times per year is:
A = P * (1 + (R / 100) / m)^(m * T)Here, A, P, and R are given. We must solve for T. Rearranging involves dividing both sides by P, then taking logarithms to isolate T. The periodic rate is R / 100 / m and the total number of compounding periods is m * T.

Step-by-Step Solution:
Step 1: Write the formula with the values: 12400 = 8000 * (1 + 0.07 / 4)^(4 * T).Step 2: Compute the quarterly rate: 0.07 / 4 = 0.0175. So 12400 = 8000 * (1.0175)^(4T).Step 3: Divide both sides by 8000 to get (1.0175)^(4T) = 12400 / 8000.Step 4: Simplify the ratio: 12400 / 8000 = 1.55.Step 5: Thus (1.0175)^(4T) = 1.55.Step 6: Take natural logarithms: ln(1.55) = 4T * ln(1.0175).Step 7: Solve for T: T = ln(1.55) / (4 * ln(1.0175)). Numerically this is approximately equal to 6.3 years.

Verification / Alternative check:
To check reasonableness, a rough comparison is useful. At about 7% interest with annual compounding, the rule of 72 suggests doubling time is about 72 / 7 ≈ 10.3 years. We are not doubling from 8000 to 16000, but only going to 12400, which is a growth factor of 1.55. That should take noticeably less than 10.3 years. Quarterly compounding is slightly more favorable than annual compounding, giving an effective rate a bit higher than 7%, which further reduces the time. A value around 6 to 7 years is therefore quite reasonable, and 6.3 years matches the precise calculation.

Why Other Options Are Wrong:
2.3, 3.3 and 4.3 years are all too short to achieve a 55% increase at only 7% per annum, even with quarterly compounding. An investment cannot grow from 8,000 to 12,400 in such a small time at this modest rate. 5.3 years is closer but still does not satisfy the exact compound interest equation. Only about 6.3 years matches the solution obtained by properly solving the compound growth equation.

Common Pitfalls:
Common mistakes include using the simple interest formula to estimate time, forgetting to divide the annual rate by 4 for quarterly compounding, or failing to multiply the time by 4 in the exponent. Some students also try to guess the answer rather than using logarithms, which can lead to large errors. Working directly from the compound interest formula and handling the logarithms carefully leads to the correct time period.

Final Answer:
The deposit of Rs 8,000 will grow to Rs 12,400 at 7% interest compounded quarterly in approximately 6.3 years.