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Question
A sum of money is lent out at compound interest for two years at 20% p.a., being reckoned yearly. If the same sum of the money was lent Gut at compound interest of the same rate of percent per annum C.I., being reckoned half yearly would have fetched Rs. 482 more by way of interest. Calculate the sum of money lent out.
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Solution
Let the principal be ₹ 100
For first case r = 20% p.a.
n = 2 years
A = P`(1 + r/100)^n`
= ₹ 100`(1 + 20/100)^2`
= ₹ 100`(1 + 1/5)^2`
= ₹ 100`(6/5)^2`
= ₹ `(100 xx 6 xx 6)/(5 xx 5)`
= ₹ `(3,600)/(25)`
= ₹ 144
∴ C.I. = A - P
= ₹ 144 - ₹ 100
= ₹ 44
For second case
r = 20% p.a. = `(20)/(2)`% half yearly = 10% semi annual
Time = 2 years = 4 half years
A = P`(1 + r/100)^n`
= ₹100`(1 + 10/100)^4`
= ₹100`(1 + 1/10)^4`
= ₹100`(11/10)^4`
= ₹`(100 xx 11 xx 11 xx 11 xx 11)/(10 xx 10 xx 10 xx 10)`
= ₹`(121 xx 121)/(10 xx 10)`
= ₹`(14,641)/(100)`
= ₹ 146·41
C.I. = A - P
= 146·41 - 100
= ₹ 46·41
Difference between two interest
= ₹ 46·41 - ₹ 44·00
= ₹ 2·41
If difference is ₹2·41 then principal be ₹100
If difference is ₹482 then principal will be
= ₹ `(100)/(2·41) xx 482`
= ₹ `(100 xx 482)/(241) xx 100`
= ₹ 100 x 100 x 2
= ₹ 20,000
∴ Sum is ₹ 20,000
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