Multiplication by Clever Form of 1 at a Glance

This super handy trick lets you simplify an expression without changing its value. Which is good, because fractions should not be subject to inflation.

Sample Problem

Here's a horrible fraction. A really horrible fraction. This thing is getting coal for Christmas.

If we multiply it by 1, we won't change its value. If we multiply it by anything equivalent to 1 (such as ), we also won't change its value. Since each of our two mini-fractions has a 3 in its denominator, what happens if we multiply the whole thing by ?

Our new numerator is , and our new denominator is . Now the fraction's value hasn't changed, and it looks way more manageable. It might even get an Xbox under the tree this year.

Sample Problem

We can also use this trick to rewrite decimal division problems:

^3.4/_7.8 × ¹⁰/₁₀ = ³⁴/₇₈

We sort of hate it when decimals and fractions are all mixed up like that, so we just multiply the whole fraction by ¹⁰/₁₀ (a.k.a. a clever form of 1), which turns the numerator and denominator both into nice, whole numbers. Ah, much better.