A fraction is just a way of talking about part of something. Cut a pizza into equal slices and take some — that's a fraction. Once you see fractions as parts of a whole, the rules for comparing, adding and multiplying them stop being magic and start making sense. This lesson builds that picture and shows you how to work with fractions confidently.
What the two numbers mean
A fraction has two numbers with a line between them, like 3/4. The bottom number, the denominator, tells you how many equal parts the whole is split into. The top number, the numerator, tells you how many of those parts you have.
So 3/4 means: split the whole into 4 equal parts, and take 3 of them. The bigger the denominator, the smaller each part — a pizza cut into 8 gives smaller slices than one cut into 4. Keeping this picture in your head is the key to everything else.
3/4: three of the four equal parts are shaded.
Equivalent fractions
Different-looking fractions can be equal. 1/2 is the same amount as 2/4 and 4/8 — half a pizza, whether you cut it into 2, 4 or 8. You get an equivalent fraction by multiplying (or dividing) the top and bottom by the *same* number.
This is why you can simplify a fraction: divide top and bottom by a number that goes into both. 6/8 divides by 2 to give 3/4 — the same amount, written more simply. Always give your answer in its simplest form when you can.
1/2 and 2/4 shade the same amount of the bar — they are equivalent.
Worked example
Simplify 8/12.
Find a number that divides both 8 and 12. Both divide by 4.
Divide the top: 8 ÷ 4 = 2.
Divide the bottom: 12 ÷ 4 = 3.
Answer: 8/12 = 2/3
Adding and subtracting fractions
You can only add fractions when their denominators are the same — you can't add quarters and thirds directly, just as you can't add slices of different sizes. So first rewrite them with a common denominator (a number both denominators divide into), then add the numerators and keep the denominator.
Subtracting works the same way: match the denominators, then subtract the numerators.
Worked example
Work out 1/4 + 2/3.
The denominators 4 and 3 both divide into 12, so use 12.
Rewrite: 1/4 = 3/12 (times top and bottom by 3).
Rewrite: 2/3 = 8/12 (times top and bottom by 4).
Add the numerators: 3/12 + 8/12 = 11/12.
Answer: 1/4 + 2/3 = 11/12
Multiplying fractions
Multiplying is actually the easy one — no common denominator needed. Just multiply the tops together and the bottoms together, then simplify. Multiplying by a fraction means 'take a part of', so 1/2 of 1/2 is 1/4 (half of a half).
To divide by a fraction, flip the second fraction upside down and multiply — but master multiplying first, and dividing will feel natural later.
Worked example
Work out 2/3 × 3/5.
Multiply the tops: 2 × 3 = 6.
Multiply the bottoms: 3 × 5 = 15.
That gives 6/15, which simplifies by dividing both by 3.
Answer: 2/3 × 3/5 = 6/15 = 2/5
Check yourself
Try these. Pick an answer to see whether it's right and why.
What does the denominator (bottom number) of a fraction tell you?
The denominator is how many equal parts the whole is divided into; the numerator (top) is how many of those parts you have.
Which fraction is equal to 1/2?
Multiply top and bottom of 1/2 by 4 to get 4/8 — the same amount written differently.
Simplify 9/12.
Both 9 and 12 divide by 3: 9 ÷ 3 = 3 and 12 ÷ 3 = 4, giving 3/4.
What is 1/3 + 1/6?
Use denominator 6: 1/3 = 2/6, so 2/6 + 1/6 = 3/6, which simplifies to 1/2.
What is 1/2 × 4/5?
Multiply tops (1 × 4 = 4) and bottoms (2 × 5 = 10) to get 4/10, which simplifies to 2/5.
In a nutshell
A fraction is part of a whole: the denominator is how many equal parts, the numerator is how many you have.
Equivalent fractions are equal — multiply or divide top and bottom by the same number; simplify by dividing.
To add or subtract, first make the denominators the same, then add or subtract the numerators.
To multiply, multiply the tops together and the bottoms together, then simplify.
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