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Types of Numbers

Types of Numbers

At a Glance - Multiplication by Clever Form of 1

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.

Example 1

What is 1.52/3.1 as a fraction of two integers?

Exercise 1

Find the cleverly disguised form of 1 that will let you simplify the rotten fraction below, then simplify. And quickly, please—it hurts our eyes just to look at it. 

Exercise 2

What's the cleverly disguised form of 1 that will let you simplify ? What's the simplified version?

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