Question : Find the Rate percent per annum, if Rs. 2000 amounts to Rs. 2315.25 in a year and a half, with interest being compounded half-yearly.
Option 1: 11.5%
Option 2: 10%
Option 3: 5%
Option 4: 20%
Correct Answer: 10%
Solution :
Amount(A) = Rs. 2315.25
Principal(P) = Rs. 2000
Time = 1.5 years ($n$ = 3)
By applying the formula: Amount = P[$(1+\frac{r}{100})^n$] where P is principal, $r$ is the rate of interest per term for $n$ terms.
Since CI is compounded half-yearly.
Let the rate be $R\%$ per annum. So, $r=\frac{R}{2}$
Putting the value in the formula of CI,
$2315.25=2000(1+\frac{\frac{R}{2}}{100})^{3}$
⇒ $\frac{2315.25}{2000}=(1+\frac{\frac{R}{2}}{100})^{3}$
⇒ $\frac{231525}{200000}=(1+\frac{\frac{R}{2}}{100})^{3}$
⇒ $\frac{9261}{8000}=(1+\frac{\frac{R}{2}}{100})^{3}$
⇒ $(\frac{21}{20})^{3}=(1+\frac{R}{100})^{3}$
Comparing both sides, we get,
⇒ $1+\frac{\frac{R}{2}}{100}=\frac{21}{20}$
⇒ $1+\frac{\frac{R}{2}}{100}=1+\frac{1}{20}$
⇒ $\frac{R}{200}=\frac{1}{20}$
⇒ $R=\frac{200}{20}$ = 10%
Hence, the correct answer is 10%
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