Hidden fractions

Let’s write $\frac{1}{2}$ as a fraction with denominator $100$. This can be done by multiplying both the numerator and the denominator by 50:
$$\frac{1}{2}=\frac{1\times 50}{2\times 50}=\frac{50}{100}$$We want a fraction that is smaller than this, but not too small. So let’s choose $\frac{49}{100}$. We now have to check that it is greater than $\frac{1}{3}$. The problem is that $\frac{1}{3}$ cannot be written as a fraction with denominator 100.
But that’s fine: we can still rewrite both fractions to a common denominator, say, 300 :
$$\frac{49}{100}=\frac{49 \times 3}{100 \times 3}=\frac{147}{300} \qquad\quad \frac{1}{3}=\frac{1 \times 100}{3 \times 100}=\frac{100}{300}$$So $\frac{49}{100}$ is between $\frac{1}{3}$ and $\frac{1}{2}$.