Question

Mark the correct alternative in each of the following:

Which of the following fractions lies between $\frac{2}{3}$ and $\frac{5}{7}$?

(a) $\frac{3}{4}$ (b) $\frac{4}{5}$ (c) $\frac{5}{6}$ (d) None of these

Answer 1

Consider the fractions $\frac{2}{3}$, $\frac{5}{7}$, $\frac{3}{4}$, $\frac{4}{5}$ and $\frac{5}{6}$.

LCM of 3, 4, 5, 6 and 7 = 420

Firstly, convert the fractions into equivalent fractions with denominator 420.

$$\begin{gathered} \frac{2}{3}=\frac{2\times 140}{3\times 140}=\frac{280}{420} \\ \frac{5}{7}=\frac{5\times 60}{7\times 60}=\frac{300}{420} \\ \frac{3}{4}=\frac{3\times 105}{4\times 105}=\frac{315}{420} \\ \frac{4}{5}=\frac{4\times 84}{5\times 84}=\frac{336}{420} \\ \frac{5}{6}=\frac{5\times 70}{6\times 70}=\frac{350}{420} \end{gathered}$$

Now,

280 < 300 < 315 < 336 < 350

$$\begin{gathered} \therefore \frac{280}{420}<\frac{300}{420}<\frac{315}{420}<\frac{336}{420}<\frac{350}{420} \\ \Rightarrow \frac{2}{3}<\frac{5}{7}<\frac{3}{4}<\frac{4}{5}<\frac{5}{6} \end{gathered}$$

Thus, none of the fractions $\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{6}$ lies between the fractions $\frac{2}{3}$ and $\frac{5}{7}$.

Hence, the correct answer is option (d).