A money-lender borrows money at 3% per annum and pays the interest at the end of the year. He lends it to another person at 6% per annum compound interest compounded half yearly and receives the interest at the end of the year. In this way, he gains Rs. 618 a year. The amount of money he borrows, is
Solution
Correct Answer: Option C
Let the amount borrows = P
We know the formula for simple interest-
\(S.I. = \frac{{P \times R \times T}}{{100}}\)
Where:
S.I. = simple interest
P = principal
R = rate = 3%
T = time = 1 yr
\(S.I.\; = \frac{{\;P\; \times \;3\; \times \;1\;}}{{100}} = \frac{{3{\rm{P}}}}{{100}}\)
He lends it to another person at 6% per annum compound interest compounded half yearly
We know the formula for compound interest-
\(A = P{\left( {\;1\; + \frac{R}{{100}}} \right)^T}\)
Where,
A = Amount
P = Principal
R = Rate of interest (half yearly) = 6/2 = 3%
T = Time period (half yearly) = 2 half years
C.I. = A – P
\(\begin{array}{l} C.I.\; = P \times {\left( {1\; + \frac{3}{{100}}} \right)^2}-P\\ \Rightarrow C.I.\; = P \times \left( {\frac{{103}}{{100}}} \right) \times \;\left( {\frac{{103}}{{100}}} \right)-P \end{array}\)
⇒ C.I. = P × (609/10000)
Given That, C.I. – S.I. = 618
\(\Rightarrow \frac{{609P}}{{10000}}-\frac{{3P}}{{100}} = 618\)
⇒ P = (618 × 10000)/309
⇒ P = 20,000
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