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Question
In how many years will ₹ 3375 become ₹ 4096 at `13 1/3` p.a if the interest is compounded half-yearly?
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Solution
Principal = ₹ 3375
Amount = ₹ 4096
r = `13 1/3% "p.a"`
= `40/3% "p.a"`
Compounded half-yearly r = `(40/3)/2 = 20/3%` half-yearly
Let no. of years be n
For compounding half-yearly, formula is
A = `"P"(1 + "r"/100)^(2"n")`
∴ 4096 = `3375 + (1 + (20/3)/100)^(2"n")`
∴ `4096/3375 = (1 + 20/(3 xx 100))^(2"n")`
= `(1 + 1/15)^(2"n")`
`((15 + 1)/15)^(2"n") = (16/15)^(2"n")`
⇒ `4096/3375 = (16/15)^(2"n")`
Taking cubic root on both sides,
`(16/15)^((2"n")/3) = root(3)(4096)/(root(3)(3375)) = 16/15`
∴ `(2"n")/3` = 1
∴ n = `3/2` = 1.5 years
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