Question

Find the difference between simple interest and compound interest on \[Rs.20000\] at \[8\] p.c.p.a.

Answer 1

Hint: To find the difference we subtract the simple interest from the compound interest of the same principal and as for the time variable let us take the time period as \[T=2\] as the first year the interest is often the same in both cases. Hence, the formula of difference used in this case is:
The difference between simple interest and compound interest is:
Difference \[=P\left[ \left[ {{\left( 1+\dfrac{R}{100} \right)}^{T}}-1 \right]-\dfrac{RT}{100} \right]\].
where \[P\] is the principal, \[R\] is the rate percentage and \[T\] is the time period in years (here \[T=2\text{ years}\] as time period is not given).

Complete step-by-step answer:
Placing the values as given in the question in the formula for the principal \[P=Rs.20000\], \[T=2\text{ years}\] and \[R=8%\] we have:
Difference \[=P\left[ \left[ {{\left( 1+\dfrac{R}{100} \right)}^{T}}-1 \right]-\dfrac{RT}{100} \right]\]
\[=20000\left[ \left[ {{\left( 1+\dfrac{8}{100} \right)}^{2}}-1 \right]-\dfrac{8\times 2}{100} \right]\]
\[=20000\left[ \left[ {{\left( \dfrac{108}{100} \right)}^{2}}-1 \right]-\dfrac{16}{100} \right]\]
\[=20000\left[ \left[ {{\left( 1.08 \right)}^{2}}-1 \right]-0.16 \right]\]
\[=20000\left[ \left[ 1.1664-1 \right]-0.16 \right]\]
\[=20000\times 0.0064\]
\[=Rs.128\]

Hence, the difference between simple interest and compound interest on \[Rs.20000\] at \[8%\] per annum is given as \[Rs.128\].

Note: Another method to find the difference is by shortcut method when the difference is asked for two years we use this formula:
\[\dfrac{P\times {{R}^{2}}\times \left( 300+R \right)}{100}\]
Placing the values in the formula of \[R=8%\] and \[P=Rs.20000\] we get:
\[=\dfrac{P\times {{8}^{2}}\times \left( 300+8 \right)}{100}\]
\[=\dfrac{20000\times {{8}^{2}}\times \left( 300+8 \right)}{100}\]
\[=\dfrac{64\times 2\times 308}{100}\]
\[=Rs.128\]