Question : A sum was lent for one year at the rate of 16 percent per annum on compound interest (compounding annually). If the compounding had been done half yearly, then the interest would have increased by Rs. 64. What was the sum lent?
Option 1: Rs. 10000
Option 2: Rs. 6000
Option 3: Rs. 14000
Option 4: Rs. 20000
Correct Answer: Rs. 10000
Solution :
We know, Total Amount = Principal × (1 + $\frac{\text{Rate}}{100})^{\text{Time}}$
Let the lending amount be $x$.
⇒ Interest earned with annually compounding
= $x (1 + \frac{16}{100}) - x$
= $x \times\frac{116}{100} - x$
= $\frac{116x}{100}-x$
⇒ Interest earned with half-yearly compounding
= $x (1 + \frac{16}{200})^2- x$
= $x(1+\frac{2}{25})^2-x$
= $x(\frac{27}{25})^2- x$
Given,
The difference in interest earned is Rs. 64.
⇒ $[x(\frac{27}{25})^2- x] - [\frac{116x}{100}-x] = 64$
⇒ $x[\frac{729}{625}-\frac{116}{100}]=64$
⇒ $x[\frac{729}{625}-\frac{29}{25}]=64$
⇒ $x(\frac{729-725}{625}) = 64$
⇒ $x\times\frac{4}{625}=64$
⇒ $x = \frac{64}{4}\times625= 10000$
Hence, the correct answer is Rs. 10,000.
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