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

3150

3096

3180

3221

✅ Correct Option: 4

Amount after 3 years $= Rs\ 13,920$

Amount after 6.5 years $= Rs\ 18,960$

Since interest is simple, the difference between the two amounts gives the interest earned over the gap in time.

SI for $3.5$ years $= 18,960 - 13,920 = Rs\ 5,040$

SI for $1$ year $= \dfrac{5040}{3.5} = Rs\ 1,440$

SI for $3$ years $= 3 \times 1440 = Rs\ 4,320$

$P = 13,920 - 4,320 = Rs\ 9,600$

$R = \dfrac{1440 \times 100}{9600 \times 1} = 15\%$ per annum

Now the same sum is invested for 2 years at the same rate, but compounded every 6 months.

When compounded half-yearly:

Rate per half-year $= \dfrac{15}{2} = 7.5\%$

Number of half-year periods in 2 years $= 2 \times 2 = 4$

$A = P\left(1 + \dfrac{r}{100}\right)^n$

$A = 9600 \times (1.075)^4$

$(1.075)^2 = 1.155625$

$(1.075)^4 = (1.155625)^2 \approx 1.3355$

$A = 9600 \times 1.3355$

$A \approx Rs\ 12,820.50$

$CI = A - P$

$CI = 12,820.50 - 9,600$

$CI \approx Rs\ 3,220.50$

Therefore, the total interest earned is nearest to $Rs\ 3,220$ .

The answer is Option 4.

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