#### Determine the rate of interest for a sum that becomes 216/125 times of itself in 1 ½ years, compounded semi-annually.

**Solution:**

Consider principal (P) = x

Amount (A) = 216/125 x

Period (n) = 1 ½ years = 3 half years

Take rate percent per year = 2r% and r% half yearly

We know that

A/P = (1 + r/100)^{n}

Substituting the values

216x/125x = (1 + r/100)^{3}

By further calculation

(1 + r/100)^{3} = 216/125 = (6/5)^{3}

So we get

1 + r/100 = 6/5

r/100 = 6/5 – 1 = 1/5

By cross multiplication

r = 100 × 1/5 = 20%

So the rate percent per year = 2 × 20 = 40%

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