Ex.8.3 Q8 Comparing Quantities Solutions - NCERT Maths Class 8

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Question

Find the amount and the compound interest on \(\rm{Rs}\, 10,000 \) for \((1\begin{align}\frac{1}{2})\end{align}\) years at \(10\%\) per annum, compounded half yearly. Would this interest be more than the interest he would get if it was compounded annually?

Text Solution

What is known?

Principal, Time Period and Rate of Interest

What is unknown?

Amount and Compound Interest (C.I.)

Reasoning:

\(\begin{align}{A = P}\left( {{1 + }\frac{{r}}{{{100}}}} \right)^{\rm{n}}\end{align} \)

 \(P=\rm{Rs}\, 10,000\)

\(N= (1\begin{align}\frac{1}{2})\end{align}\) years

\(R= 10\%\) p.a. compounded annually and half-yearly

Steps:

For calculation of C.I. compounded half yearly, we will take Interest rate as \(5\% \)

\[\begin{align}A &= P\left( {{1 + }\frac{{r}}{{{100}}}} \right)^{n}  \\ &= 10000\left( {{1 + }\frac{{5}}{{{100}}}} \right)^{3}  \\ &= 10000\left( {{1 + }\frac{{1}}{{{20}}}} \right)^{3}  \\ & = 10000\left( {\frac{{21}}{{20}}} \right)^3  \\ &= 10000\times \frac{{21}}{{20}} \times \frac{{21}}{{20}} \times \frac{{21}}{{20}} \\ &= 10000 \times \frac{{9261}}{{8000}} \\ &= {5} \times \frac{{9261}}{4} \\ &= 11576.25 \\ \end{align}\]

Interest earned at \(10\%\) p.a. compounded half-yearly \(= 11576.25 \,– \,10000=\rm{Rs}\, 1576.25\)

The amount earned at \(10\%\) p.a. compounded half-yearly \(= 11576.25\)

The C.I. earned at \(10\%\) p.a. compounded half-yearly \(= 1576.25\)

The above interest earned being compounded half-yearly would be more than the interest compounded annually since interest compounded half yearly is always more than compounded annually at the same rate of interest.