Question
The compound interest on a certain sum of money for 2 years at 5% per annum is Rs 410. The simple interest on the same sum at the same rate and for the same time is = ?
Answer: Option A
$$\eqalign{
& {\text{Rate of interest 5}}\% \cr
& = \frac{1}{{20}} \cr
& {\text{Let principal}} \cr
& {\text{ = }}{\left( {20} \right)^2}{\text{ = 400 units}} \cr
& \Rightarrow {\text{ Total compound interest }} \cr
& {\text{41 Units }} \to {\text{Rs. 410 }} \cr
& {\text{1 Units }} \to {\text{Rs. 10 }} \cr
& {\text{400 Units }} \to {\text{Rs. 400 }} \cr
& {\text{Total simple interest}} \cr
& {\text{ = Rs. 400}} \cr} $$
Alternate
Total compound interest for 2 years at 5% p.a.
$$\eqalign{
& {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr
& {\text{Total simple interest}} \cr
& = 10\% \cr
& \Rightarrow 10.25\% \to 410 \cr
& \Rightarrow 10\% \to 400 \cr
& {\text{Simple interest}} \cr
& {\text{ = Rs}}{\text{. 400}} \cr} $$
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$$\eqalign{
& {\text{Rate of interest 5}}\% \cr
& = \frac{1}{{20}} \cr
& {\text{Let principal}} \cr
& {\text{ = }}{\left( {20} \right)^2}{\text{ = 400 units}} \cr
& \Rightarrow {\text{ Total compound interest }} \cr
& {\text{41 Units }} \to {\text{Rs. 410 }} \cr
& {\text{1 Units }} \to {\text{Rs. 10 }} \cr
& {\text{400 Units }} \to {\text{Rs. 400 }} \cr
& {\text{Total simple interest}} \cr
& {\text{ = Rs. 400}} \cr} $$
Alternate
Total compound interest for 2 years at 5% p.a.
$$\eqalign{
& {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr
& {\text{Total simple interest}} \cr
& = 10\% \cr
& \Rightarrow 10.25\% \to 410 \cr
& \Rightarrow 10\% \to 400 \cr
& {\text{Simple interest}} \cr
& {\text{ = Rs}}{\text{. 400}} \cr} $$
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