Year 10 Higher · Work with fractions
Add and subtract fractions
Learn how to add, subtract, multiply and divide both numeric and algebraic fractions, simplify them by factorising, and turn repeating decimals into exact fractions.
Definition
A fraction is one number divided by another, and nothing about that changes when the numbers become algebraic expressions: (x + 2)/3 is a value divided by 3, just as 53 is. Every fraction has endless equivalent forms, made by multiplying or dividing the top and bottom by the same non-zero quantity, and every method in this unit works by choosing the form that suits the job. Even a repeating decimal such as 0.727272... is a fraction in disguise, and a short piece of algebra recovers its exact form.
Why Do I Have to Learn This?
Fractions describe parts of a whole, like half a pizza or a quarter of an hour. They are everywhere in cooking, telling the time and sharing fairly.
Key Rules:
- 1. To add or subtract fractions, rewrite each over the lowest common denominator and combine the numerators, whether the fractions hold numbers or letters.
Remember:
- •Fractions with letters obey exactly the same rules as fractions with numbers, and only whole factors ever cancel.
Don’t Forget...
- •Cancelling a term instead of a factor, for example crossing out the two x terms in (x + 3)/x to leave 3. Cancelling means dividing the top and bottom by the same factor, and x is not a factor of x + 3, so removing it changes the value: with x = 1 the original fraction equals 4, not 3.
- •Adding numerators and denominators straight across, turning 3/(x + 1) + 2/(x - 1) into 5/(2x). Addition only combines parts of equal size, so each fraction must first be rewritten over a common denominator; adding tops and bottoms gives a value between the two fractions, not their sum.
- •Multiplying by 10 instead of 100 when a two-digit block repeats. The subtraction only cancels the infinite tail when both copies carry the repeating block in the same decimal positions, which needs a shift of one whole block, so 100 for a two-digit repeat.
Denominator
The bottom number of a fraction: how many equal parts the whole is split into.
In 34 the denominator is 4.
Worked Example:
Write 3/(x + 1) + 2/(x - 1) as a single fraction in its simplest form.
The lowest common denominator is (x + 1)(x - 1), the product of the two different denominators.
Rewrite both fractions over it: 3(x - 1)/((x + 1)(x - 1)) and 2(x + 1)/((x + 1)(x - 1)).
Add the numerators: 3(x - 1) + 2(x + 1) = 3x - 3 + 2x + 2 = 5x - 1.
Check for common factors: 5x - 1 shares no factor with (x + 1) or (x - 1), so nothing cancels.
Answer: (5x - 1)/((x + 1)(x - 1))
Why It Works:
Multiplying the numerator and denominator of a fraction by the same non-zero expression changes its appearance but not its value, which is why common denominators, cancelling whole factors and clearing fractions from an equation are all legitimate moves. The repeating decimal method works because 100x and x carry exactly the same infinite tail, so their difference is a whole number and x must be a ratio of whole numbers.
Another Example:
Write the repeating decimal 0.727272..., where the digits 72 repeat forever, as a fraction in its simplest form.
Let x = 0.727272..., so the unknown carries the whole repeating tail.
Two digits repeat, so multiply by 100: 100x = 72.727272...
Subtract the first line from the second: the identical tails cancel, leaving 99x = 72.
Solve and simplify: x = 7299, and dividing the top and bottom by 9 gives 811.
Answer: 811
Animated Example:
Free Add and subtract fractions Worksheet
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Before this: Fractions (Year 9)
Let’s Practise the Concept
Step 1 of 3
Put the fraction on the line
Tap where 6/10 belongs between 0 and 1.
Tap the line where 6/10 belongs.