Discussion
A sum of ₹ 12,000 amounts to ₹ 15,972 at a certain rate percent per annum in \(1\dfrac{1}{2}\) years, when the interest is compounded half-yearly. What will be the amount of the same sum in same time and with the same rate, if interest is compounded annually ?
Answer: A
Compounded annually, the amount is ₹ 15,840.
Half-yearly for 1½ years means 3 periods:
\[ \begin{aligned} \left(1 + \frac{r}{200}\right)^{3} &= \frac{15972}{12000} = 1{\cdot}331 = (1{\cdot}1)^{3} \\ \frac{r}{200} &= 0{\cdot}1 \Rightarrow r = 20\% \end{aligned} \]
Compounded annually, one full year at 20% is followed by half a year at simple interest of 10%:
\[ 12000 \times 1{\cdot}2 \times 1{\cdot}1 = 15840 \]
The amount is a little less than ₹ 15,972 because interest is added less often. ₹ 15,950 and ₹ 15,420 do not follow from a 20% rate, and ₹ 14,520 is the amount after only two half-years at 10%.
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