The difference between the compound interest and the simple interest on a sum for one and a half years at 12.5% per annum is Rs 2450, the compound interest being compounded half-yearly. Find the principal.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Rs 204800
With half-yearly compounding, the rate per half-year is half the yearly rate and the number of periods is twice the number of years. Simple interest is worked out at the yearly rate for the same time.
Rate per half-year \(\displaystyle =\frac{12.5}{2}=6.25\%=\frac{1}{16}\); periods \(=1.5\times2=3\)
Compound interest \(\displaystyle =P\left[\left(\frac{17}{16}\right)^{3}-1\right]=P\left(\frac{4913}{4096}-1\right)=\frac{817}{4096}P\)
Simple interest \(\displaystyle =\frac{P\times12.5\times1.5}{100}=\frac{3}{16}P=\frac{768}{4096}P\)
Difference \(\displaystyle =\frac{817-768}{4096}P=\frac{49}{4096}P\)
\(\displaystyle \frac{49}{4096}P=2450\), so \(\displaystyle P=\frac{2450\times4096}{49}=50\times4096=204800\)
The principal is Rs 204800.