A fraction between two given positive true fractions

Problem

How can we find a fraction p/q between two given positive true fractions, say:

Analysis

There are many such fractions, as many as we wish. However, we want just any convenient one.

L.C.M.

One way is to find the L.C.M. of the denominators of the two given fractions:

Since 221 = 13 ´ 17 and 253 = 11 ´ 23

The L.C.M. of 221 and 253 = 13 ´ 17 ´ 11 ´ 23 = 55913

It is unfortunate that there is no common factor in the denominators of the two given fractions.

Therefore,

Any fractions between the above two fractions may serve our purpose, we choose for convenience, any one of the following:

A better way

If we add up the numerators and the denominators separately and then form the fraction:

Then:

!

Why?

Suppose :

Take :

Then:

Similarly,

Therefore: