Summary — What This Chapter Covers
- Basic fractions: what numerator and denominator actually mean, and three different methods for finding a fraction of an amount.
- Equivalent & simplified fractions: how to scale a fraction up or down without changing its value, and how to reduce it to its simplest form.
- Mixed numbers & improper fractions: converting between "whole number + fraction" form and "top-heavy" form fluently, in both directions.
- Adding & subtracting fractions: why you need a common denominator, and how to handle mixed numbers in the process.
- Multiplying fractions: cancelling common factors first, then multiplying straight across.
- Dividing fractions: the "flip and times" rule (multiplying by the reciprocal).
1. Basic Fractions
What is a fraction, really?
Think of a fraction like a pizza. If you cut a pizza into 5 equal slices and you eat 2 of them, you've eaten 2/5 of the pizza. The bottom number (the denominator) tells you how many slices the whole pizza was cut into. The top number (the numerator) tells you how many slices you actually have.
This gives you a quick way to judge the size of any fraction just by glancing at it:
- If the numerator is smaller than the denominator (like 2/5), the fraction is less than 1 — you have less than a whole pizza.
- If the numerator is bigger than the denominator (like 7/5), the fraction is more than 1 — you have more than one whole pizza (you'd need a second pizza to make up the extra slices).
Finding a fraction of an amount
This is one of the most common exam tasks, and there are three completely valid ways to do it. It's worth knowing all three because different numbers make different methods faster.
Method 1 — Divide then multiply
Divide by the denominator first (this finds the value of "one part"), then multiply by the numerator (this finds the value of "however many parts you need").
Worked Example
Find 2/5 of 60.
Step 1 — Divide by the denominator: 60 ÷ 5 = 12 (this is what "one fifth" is worth)
Step 2 — Multiply by the numerator: 12 × 2 = 24
Answer: 24
Method 2 — Convert to a decimal, then multiply
Turn the fraction into a decimal first, then just multiply normally. Handy when the fraction converts to a "nice" decimal.
Worked Example
Find 9/10 of 15.
Step 1 — Convert to decimal: 9/10 = 0.9
Step 2 — Multiply: 0.9 × 15 = 13.5
Answer: 13.5
Method 3 — Write both as fractions and multiply
Turn the whole number into a fraction (over 1) and multiply the two fractions together, cancelling where you can.
Worked Example
Find 5/6 of 8.
Step 1 — Write 8 as a fraction: 8 = 8/1
Step 2 — Multiply: 5/6 × 8/1 = 40/6
Step 3 — Simplify: 40/6 = 20/3
Answer: 20/3 (or 6⅔)
Practice Question 1
Find 3/4 of 48.
Practice Question 2
Find 7/8 of 32.
2. Equivalent & Simplified Fractions
What are equivalent fractions?
exact same amount
Worked Example
Find three fractions equivalent to 5/6.
5×2 / 6×2 = 10/12