Question

Which of the following represents a set of $4$ rational numbers between $\frac{1}{2}$ and $\frac{1}{3}?$

• A. $\frac{1}{30}$, $\frac{11}{30}$, $\frac{10}{30}$, $\frac{12}{30}$
• B. $\frac{11}{30}$, $\frac{12}{30}$, $\frac{13}{30}$, $\frac{14}{30}$
• C. $\frac{11}{30}$, $\frac{22}{30}$, $\frac{33}{30}$, $\frac{40}{30}$
• D. $\frac{1}{30}$, $\frac{2}{30}$, $\frac{3}{30}$, $\frac{4}{30}$

Answer 1

The correct option is B $\frac{11}{30}$, $\frac{12}{30}$, $\frac{13}{30}$, $\frac{14}{30}$

The given two rational numbers are $\frac{1}{2}$ and $\frac{1}{3}$
Let us convert these numbers into fractions with equal denominators,
$\frac{1}{2}=\frac{1\times 3}{2\times 3}=\frac{3}{6}$
And,$\frac{1}{3}=\frac{1\times 2}{3\times 2}=\frac{2}{6}$
Here 2 and 3 on the numerators are consecutive natural numbers.
Now, let us multiply the numerators and denominators of both the fractions by 5.
So we get,
$\frac{3\times 5}{6\times 5}=\frac{15}{30}$ and $\frac{2\times 5}{6\times 5}=\frac{10}{30}$
Now, we can have 4 consecutive integers between 10 and 15.
So, the required set of 4 numbers between $\frac{1}{2}$ and $\frac{1}{3}$ are $\frac{11}{30}$, $\frac{12}{30}$, $\frac{13}{30}$, $\frac{14}{30}$.