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

## 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.