Question

At a certain simple rate of interest, a given sum amounts to Rs 13920 in 3 years, and to Rs 18960 in 6 years and 6 months. If the same given sum had been invested for 2 years at the same rate as before but with interest compounded every 6 months, then the total interest earned, in rupees, would have been nearest to:

• A. \(3096\)
• B. \(3221\)
• C. \(3180\)
• D. \(3150\)

Hint:

When you know the amounts at two different times under simple interest, the difference of amounts directly gives you the interest for the extra period. From that, you can easily find the yearly interest, then the rate and principal, and finally plug those into a compound interest calculation.

Answer 1

The Correct Option is B

Approach: With simple interest, the amount grows by a fixed sum every year. So the jump between two given amounts, divided by the time gap, instantly gives the yearly interest — from there the principal and rate fall out. Then switch to half-yearly compounding.

Step 1: Yearly simple interest.
Amount rises from \(13920\) (3 yrs) to \(18960\) (6.5 yrs).
\[ \Delta I = 18960 - 13920 = 5040 \ \text{over}\ (6.5 - 3) = 3.5\ \text{years}. \]
\[ \text{SI per year} = \frac{5040}{3.5} = 1440. \]

Step 2: Principal and rate.
Interest in first 3 years \(= 1440 \times 3 = 4320\), so
\[ P = 13920 - 4320 = 9600. \]
\[ R = \frac{1440}{9600} \times 100 = 15\%\ \text{p.a.} \]

Step 3: Compound, half-yearly, for 2 years.
Rate per half-year \(= \frac{15}{2} = 7.5\%\); number of periods \(= 2 \times 2 = 4\).
\[ A = 9600 \,(1.075)^4. \]
\((1.075)^2 = 1.155625\), then \((1.155625)^2 \approx 1.33547\).
\[ A \approx 9600 \times 1.33547 \approx 12820.5. \]

Step 4: Interest earned.
\[ \text{CI} = 12820.5 - 9600 \approx 3220.5 \approx 3221. \]

Total interest \(\approx\) Rs 3221 — option (2).

Answer 2

Approach: Rather than raising \((1.075)\) to the fourth power directly, first convert the half-yearly compounding into a single effective ANNUAL rate, then apply that rate for 2 years. This reduces the calculation from a 4th-power problem to a squaring problem.

Step 1: Recover principal and rate.
Yearly simple interest \(= \dfrac{18960 - 13920}{6.5 - 3} = \dfrac{5040}{3.5} = 1440\). Principal \(= 13920 - 3(1440) = 9600\). Rate \(= \dfrac{1440}{9600} \times 100 = 15\%\) per annum, so \(7.5\%\) every half-year.

Step 2: Convert to an effective annual rate.
Compounding \(7.5\%\) twice a year, the effective annual growth factor is
\[ (1.075)^2 = 1.155625, \]
i.e. an effective annual rate of \(15.5625\%\).

Step 3: Apply this rate for 2 years and find the interest.
\[ A = 9600(1.155625)^2 = 9600(1.335470\ldots) \approx 12820.5. \]
\[ \text{Interest} = 12820.5 - 9600 \approx 3220.5 \approx 3221. \]

Total interest \(\approx\) Rs 3221, option (2).