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Question
A sum of money was lent for 2 years at 20% compounded annually. If the interest is payable half-yearly instead of yearly, then the interest is Rs 482 more. Find the sum.
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Solution
\[A = P \left( 1 + \frac{R}{100} \right)^n \]
Also,
P = A - CI
Let the sum of money be Rs x.
If the interest is compounded annually, then:
\[ A_1 = x \left( 1 + \frac{20}{100} \right)^2 \]
\[ = 1 . 44x\]
\[ \therefore CI = 1 . 44x - x\]
\[ = 0 . 44x . . . (1)\]
If the interest is compounded half - yearly, then:
\[ A_2 = x \left( 1 + \frac{10}{100} \right)^4 \]
\[ = 1 . 4641x\]
∴ CI = 1 . 4641x - x
\[ = 0 . 4641x . . . (2)\]
It is given that if interest is compounded half - yearly, then it will be Rs 482 more.
∴ 0 . 4641x = 0 . 44x + 482 [From (1) and (2)]
\[0 . 4641x - 0 . 44x = 482\]
\[0 . 0241x = 482\]
\[x = \frac{482}{0 . 0241}\]
\[ = 20, 000\]
Thus, the required sum is Rs 20, 000.
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