Question

Mark the correct alternative in each of the following:

The difference between the greatest and the least fractions out of $\frac{6}{7},\frac{7}{8},\frac{8}{9}$ and $\frac{9}{10}$ is

(a) $\frac{3}{10}$ (b) $\frac{1}{56}$ (c) $\frac{1}{40}$ (d) $\frac{1}{72}$

Answer 1

Consider the fractions $\frac{6}{7},\frac{7}{8},\frac{8}{9}$ and $\frac{9}{10}$.

LCM of 7, 8, 9 and 10 = 2520

Firstly, convert the fractions into equivalent fractions with denominator 2520.

$$\begin{gathered} \frac{6}{7}=\frac{6\times 360}{7\times 360}=\frac{2160}{2520} \\ \frac{7}{8}=\frac{7\times 315}{8\times 315}=\frac{2205}{2520} \\ \frac{8}{9}=\frac{8\times 280}{9\times 280}=\frac{2240}{2520} \\ \frac{9}{10}=\frac{9\times 252}{10\times 252}=\frac{2268}{2520} \end{gathered}$$

Now,

2160 < 2205 < 2240 < 2268

$$\begin{gathered} \therefore \frac{2160}{2520}<\frac{2205}{2520}<\frac{2240}{2520}<\frac{2268}{2520} \\ \Rightarrow \frac{6}{7}<\frac{7}{8}<\frac{8}{9}<\frac{9}{10} \end{gathered}$$

So,

Greatest fraction = $\frac{9}{10}$

Least fraction = $\frac{6}{7}$

∴ Required difference

$$\begin{gathered} =\frac{9}{10}-\frac{6}{7} \\ =\frac{2268}{2520}-\frac{2160}{2520} \\ =\frac{2268-2160}{2520} \\ =\frac{108}{2520} \end{gathered}$$

$$\begin{gathered} =\frac{108÷36}{2520÷36}\left(\mathrm{HCF}\mathrm{of}108\mathrm{and}2520=36\right) \\ =\frac{3}{70} \end{gathered}$$


Disclaimer: None of the options given in the question matches with the answer.