Question

Find the decimal expansion of $\frac{98}{280}$ if it has, without performing the long division.

Answer 1

Step 1: State the given data and find factors

The given fraction is $\frac{98}{280}$,

In order to express the given fraction into a decimal without performing the long division, it is required to convert the denominator into a multiple of $10$.

First, by prime factorization,

So, $98=2\times 7\times 7$

And, $280=2\times 2\times 2\times 5\times 7$

Step 2: Convert into a decimal by shifting the decimal method

Then, the fraction be written in its factor form as,

$\frac{98}{280}=\frac{2\times 7\times 7}{2\times 2\times 2\times 5\times 7}$

$$\begin{gathered} \Rightarrow \frac{98}{280}=\frac{1\times 1\times 7}{1\times 2\times 2\times 5\times 1} \\ \Rightarrow \frac{98}{280}=\frac{7}{2\times 2\times 5} \\ \Rightarrow \frac{98}{280}=\frac{7}{2\times 2\times 5}\times \frac{5}{5} \end{gathered}$$

$\Rightarrow \frac{98}{280}=\frac{7\times 5}{2\times 2\times 5\times 5}$ $\left[\because \frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}\right]$

$$\begin{gathered} \Rightarrow \frac{98}{280}=\frac{35}{100} \\ \Rightarrow \frac{98}{280}=0.35 \end{gathered}$$

Hence, the decimal expansion of $\frac{98}{280}$ is $0.35$.