Fractions · Technique guide
Fractions and Simplifying in Your Head
By Robert Hansson, Kung Fu Math
Fractions scare many people, yet they follow just a few rules. The trick to doing them in your head is to keep the numbers small: cancel early, choose the smallest common denominator and know the common fraction–decimal pairs by heart.
In Kung Fu Math you even earn a bonus star when you give a fraction in its simplest form and simplifying was actually needed. The techniques below show you how to get there in one clean step.
01
Find the smallest common denominator
To add or subtract fractions, they need the same denominator. Multiplying the denominators always works, but often gives big numbers. Instead, look at the larger denominator and count up its multiples until the smaller one fits.
For 5/6 + 3/8: multiples of 8 are 8, 16, 24. 24 is divisible by 6, so 24 is the common denominator.
Worked example
5/6 + 3/8
- → Common denominator: 24
- → 5/6 = 20/24
- → 3/8 = 9/24
- → 20/24 + 9/24 = 29/24
Answer: 29/24
Practice: Try 1/4 + 1/6, 2/9 + 1/6 and 7/10 − 1/4.
02
Cross multiply for two fractions
For a quick sum of two fractions, use a/b + c/d = (ad + bc)/bd. It is one line of thinking: cross multiply for the top, multiply the bottoms for the bottom. Simplify at the end.
This is especially fast when the numerators are 1: 1/a + 1/b = (a + b)/ab. So 1/3 + 1/4 = 7/12 at once.
Worked example
2/3 + 3/5
- → Top: 2 · 5 + 3 · 3 = 19
- → Bottom: 3 · 5 = 15
- → 19/15
Answer: 19/15
Practice: Try 1/5 + 1/7, 3/4 − 2/5 and 2/7 + 1/3.
03
Cancel before you multiply
When multiplying fractions, any numerator may cancel against any denominator. Do this before multiplying and the numbers stay tiny. You also land directly on the simplest form.
For 4/9 × 15/8: 4 cancels with 8 to 1/2, and 15 cancels with 9 to 5/3. Now 1/3 × 5/2 = 5/6.
Worked example
4/2 × 3/4
- → 4 on top cancels with 4 below: 1/2 × 3/1
- → Multiply: 3/2
- → Without cancelling you get 12/8, which also simplifies to 3/2
Answer: 1.5
Practice: Try 6/7 × 14/9, 5/12 × 8/15 and 9/10 × 25/27.
04
Flip and multiply to divide
Dividing by a fraction is the same as multiplying by its reciprocal. So a/b ÷ c/d = a/b × d/c. Then cancel as usual.
A useful sanity check: dividing by a fraction smaller than 1 makes the answer bigger.
Worked example
3/4 ÷ 3/8
- → Flip the second fraction: 3/4 × 8/3
- → Cancel the 3s and 4 with 8: 1/1 × 2/1
- → Answer: 2
Answer: 2
Practice: Try 2/5 ÷ 4/15, 5/6 ÷ 10/9 and 7 ÷ 1/3.
05
Know the fraction–decimal–percent pairs
Memorise a small table: 1/2 = 0.5 = 50 %, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, 1/3 ≈ 0.333 and 1/6 ≈ 0.167. Every other common fraction is a multiple of one of these.
So 3/8 is 3 × 0.125 = 0.375 = 37.5 %, and 5/6 is 1 − 1/6 ≈ 0.833.
Worked example
7/8 as a percentage
- → 1/8 = 12.5 %
- → 7/8 = 7 × 12.5 %
- → = 87.5 %
Answer: 87.5
Practice: Convert 3/5, 5/8 and 2/3 to percentages.
Common mistakes
- Adding denominators: 1/2 + 1/3 is not 2/5.
- Cancelling across a plus sign. Only factors cancel, never terms of a sum.
- Flipping the wrong fraction when dividing. Flip the one you divide by.
- Stopping before the simplest form. Check whether top and bottom share a factor.
Summary
- Use the smallest common denominator to keep numbers small.
- a/b + c/d = (ad + bc)/bd in one line.
- Cancel before multiplying and you land on the simplest form.
- To divide, flip the second fraction and multiply.
Into the dojo
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