What fraction is exactly between 1/3 and 1/7?

 What fraction is exactly between 1/3 and 1/7?👇

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To find the fraction exactly between $\frac{1}{3}$ and $\frac{1}{7}$, we need to first determine the arithmetic mean of these two fractions. The arithmetic mean of two numbers $𝑎$ and $𝑏$ is given by $\frac{𝑎+𝑏}{2}$.

Let's denote the two fractions as follows: $𝑎=\frac{1}{3}$ $𝑏=\frac{1}{7}$

First, we need to add these two fractions. To do that, we need a common denominator. The least common multiple of the denominators 3 and 7 is 21.

Convert each fraction to have the common denominator of 21: $\frac{1}{3}=\frac{7}{21}$ $\frac{1}{7}=\frac{3}{21}$

Now, add these fractions: $\frac{7}{21}+\frac{3}{21}=\frac{7+3}{21}=\frac{10}{21}$

Next, we find the arithmetic mean of $\frac{7}{21}$ and $\frac{3}{21}$ by dividing the sum by 2: $\text{Mean}=\frac{\frac{10}{21}}{2}=\frac{10}{21}\times \frac{1}{2}=\frac{10\cdot 1}{21\cdot 2}=\frac{10}{42}$

Simplify $\frac{10}{42}$ by dividing the numerator and the denominator by their greatest common divisor, which is 2: $\frac{10}{42}=\frac{10÷2}{42÷2}=\frac{5}{21}$

Therefore, the fraction exactly between $\frac{1}{3}$ and $\frac{1}{7}$ is: $\boxed{{\displaystyle \frac{5}{21}}}$