Fractions

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").

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.

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.

2. Equivalent & Simplified Fractions

What are equivalent fractions?

Equivalent fractions are different-looking fractions that represent the exact same amount. Going back to the pizza analogy: cutting a pizza into 2 slices and eating 1 (1/2) gives you exactly the same amount of pizza as cutting it into 12 slices and eating 6 (6/12). Nothing about the actual amount of pizza changed — you just sliced it differently.

Simplifying (cancelling) fractions

A simplified fraction is one written with the smallest possible whole numbers — you can't shrink it any further. If both the numerator and denominator share a common factor (a number that divides into both exactly), you can divide both by it to simplify.