How many fractions can we find between 1/3 and 1/4?

Dec 12, 2016

There are infinite number of fractions between $\frac{1}{3}$ and $\frac{1}{4}$ like $\frac{7}{24}$ and $\frac{2}{7}$.

Explanation:

There are infinite fractions between any two fractions. Let the fractions be $\frac{a}{b}$ and $\frac{c}{d}$.

Then picking up any two positive numbers $m$ and $n$, the number (m×a/b+n×c/d)/(m+n) always lies between $\frac{a}{b}$ and $\frac{c}{d}$.

Further (m×a+n×c)/(m×b+n×d) also lie between $\frac{a}{b}$ and $\frac{c}{d}$.

For example choosing $m = 1$ and $n = 1$, the number between $\frac{1}{3}$ and $\frac{1}{4}$ using first method is $\frac{\frac{1}{3} + \frac{1}{4}}{1 + 1} = \frac{7}{24}$, which lies between $\frac{1}{3}$ and $\frac{1}{4}$. Similarly using second method $\frac{1 + 1}{3 + 4} = \frac{2}{7}$ lies between $\frac{1}{3}$ and $\frac{1}{4}$.