Question

Mark the correct alternative in each of the following:

The smallest of the fractions $\frac{2}{3},\frac{4}{7},\frac{8}{11}$ and $\frac{5}{9}$ is

(a) $\frac{2}{3}$ (b) $\frac{4}{7}$ (c) $\frac{8}{11}$ (d) $\frac{5}{9}$

Answer 1

The given fractions are $\frac{2}{3},\frac{4}{7},\frac{8}{11}$ and $\frac{5}{9}$.

LCM of 3, 7, 9 and 11 = 693

Firstly, convert the fractions into equivalent fractions with denominator 693.

$$\begin{gathered} \frac{2}{3}=\frac{2\times 231}{3\times 231}=\frac{462}{693} \\ \frac{4}{7}=\frac{4\times 99}{7\times 99}=\frac{396}{693} \\ \frac{8}{11}=\frac{8\times 63}{11\times 63}=\frac{504}{693} \\ \frac{5}{9}=\frac{5\times 77}{9\times 77}=\frac{385}{693} \end{gathered}$$

Now,

385 < 396 < 462 < 504

$$\begin{gathered} \therefore \frac{385}{693}<\frac{396}{693}<\frac{462}{693}<\frac{504}{693} \\ \Rightarrow \frac{5}{9}<\frac{4}{7}<\frac{2}{3}<\frac{8}{11} \end{gathered}$$

Thus, the smallest of the given fractions is $\frac{5}{9}$.

Hence, the correct answer is option (d).