Comparing Two Fractions

The two fractions are:

2                3
and    —–
5                7

The goal is to find which one is smaller.

If we replace

2                   2                      2
—-   with    —-   that is   ——–
5                   3                   5 – 2

and replace

3                  3                    3
—-   with  —-  that is   ——–
7                  4                  7 – 3

And then replace each of the two new fractions

2                  2                      2
—-   with   —-   that is    ——-
3                  1                    3 – 2

and

3                  3                    3
—-  with   —-  that is   ——
4                  1                 4 – 3

Since 2 is less than 3, then the first fraction is smaller.

Is this method always valid? Or was it just a coincidence?

math grad - Interest: Number theory
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6 Responses to Comparing Two Fractions

1. This would be much easier with LaTeX. I tried it in one of my blog posts and it works great.

2. Husain Alshehhi says:

It is always true as long as both fractions do not change signs or both of them change signs.