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What is the difference between the compound interests (In ₹ ) on ₹ $5000$ for $1 \frac{1}{2}$ years at $4 \%$ per annum compounded yearly end half-yearly?
$2.04$
$3.06$
$4.80$
$8.30$
Solution
Difference in C.I. $=5000\left[\left(1+\frac{2}{100}\right)^{3}-1\right]-5000\left[\left(1+\frac{4}{100}\right)\left(1+\frac{2}{100}\right)-1\right]$
$=5000\left[\left\{\left(\frac{51}{50}\right)^{3}-1\right\}-\left(\frac{26}{25} \times \frac{51}{50}-1\right)\right]$
$=5000\left[\left(\frac{51^{3}-50^{3}}{50^{3}}\right)-\frac{\left(13 \times 51-25^{2}\right)}{25^{2}}\right]$
$=5000\left[\frac{51^{2}+50^{2}+51(50)}{50^{3}}-\frac{(663-625)}{25^{2}}\right]$
$=5000\left[\frac{2601+2500+2550}{50^{3}}-\frac{38}{25^{2}}\right]=5000\left[\frac{7651}{50^{3}}-\frac{38}{25^{2}}\right]$
$=\frac{5000}{50^{3}}[7651-38 \times 4 \times 50]=\frac{2}{50}[7651-7600]$
$=\frac{2}{50} \times 51=₹ 2.04$