What you'll learn
A fraction describes a number of equal parts of a whole: in 3/4 the whole has been cut into 4 equal parts and 3 of them are being counted. This topic covers naming fractions, finding equivalent fractions, comparing and ordering them, adding and subtracting, finding a fraction of a quantity, and simple multiplication and division. Fractions run right through the SEA Mathematics paper — they appear in Number questions in their own right, and again inside money, measurement and problem-solving items. The word "equal" is the one to hold on to: parts that are not equal are not fractions of anything.
Key terms and definitions
Numerator — the top number of a fraction. It counts how many parts are being taken.
Denominator — the bottom number. It says how many equal parts the whole has been cut into.
Proper fraction — a fraction worth less than one whole, where the numerator is smaller than the denominator, such as 3/5.
Improper fraction — a fraction worth one whole or more, where the numerator is equal to or larger than the denominator, such as 7/4.
Mixed number — a whole number and a proper fraction written together, such as 2 3/4.
Equivalent fractions — different fractions that name the same amount, such as 1/2, 2/4 and 5/10.
Simplest form (lowest terms) — the fraction written with the smallest possible numbers, reached by dividing top and bottom by the same number.
Common denominator — the same bottom number given to two or more fractions so they can be added, subtracted or compared.
Core concepts
What the two numbers actually mean
The denominator sets the size of each piece and the numerator counts the pieces. This explains something children often find strange: the bigger the denominator, the smaller each piece. A roti cut into 8 gives smaller pieces than the same roti cut into 4, so 1/8 is less than 1/4.
That single idea resolves most confusion in this topic. When comparing 1/6 and 1/2, the child who thinks "6 is bigger than 2, so 1/6 is bigger" has counted the pieces instead of measuring them.
Equivalent fractions
Multiplying or dividing both the numerator and the denominator by the same number gives an equivalent fraction. The amount does not change, only the way it is written.
3/4 = 6/8 = 9/12 = 12/16 (multiplying top and bottom by 2, 3 and 4)
To simplify, go the other way. For 18/24, both numbers divide by 6, giving 3/4.
The rule that matters: whatever you do to the top, you must do to the bottom. Multiplying only the denominator makes the pieces smaller without counting more of them, which changes the value.
Comparing and ordering fractions
If the denominators already match, compare the numerators: 5/8 is more than 3/8 because 5 pieces beats 3 pieces of the same size.
If the denominators differ, rewrite them over a common denominator first. To compare 2/3 and 3/5, use fifteenths: 2/3 = 10/15 and 3/5 = 9/15, so 2/3 is larger.
A shortcut worth knowing: a fraction is exactly one half when the numerator is half the denominator. So 2/5 must be less than a half, because half of 5 is two and a half and the numerator is only 2.
Adding and subtracting
With the same denominator, add or subtract the numerators only and leave the denominator alone:
2/6 + 3/6 = 5/6 and 7/9 − 4/9 = 3/9
The denominator does not change because the size of the pieces has not changed — you are simply counting more or fewer of them. Adding the denominators as well (2/6 + 3/6 = 5/12) is the single most common error in the topic, and it produces an answer smaller than one of the fractions you started with.
With different denominators, find a common denominator first:
1/2 + 1/4 → 2/4 + 1/4 = 3/4
5/6 − 1/3 → 5/6 − 2/6 = 3/6 = 1/2
Improper fractions and mixed numbers
To change a mixed number to an improper fraction, work out how many pieces the whole numbers hold and add the extra pieces. For 2 3/4: each whole holds 4 quarters, so 2 wholes are 8 quarters, and 8 + 3 = 11 quarters, giving 11/4.
To change back, divide. For 17/5: 17 ÷ 5 = 3 remainder 2, so the answer is 3 2/5.
Fractions of a quantity
To find a fraction of an amount, divide by the denominator and multiply by the numerator.
3/8 of 24 marbles: 24 ÷ 8 = 3, then 3 × 3 = 9 marbles.
Dividing first keeps the numbers small and is far less error-prone than multiplying first.
Multiplying and dividing
To multiply two fractions, multiply the numerators together and the denominators together: 1/2 × 3/5 = 3/10.
To multiply a fraction by a whole number, find one part and scale up: 2/3 × 12 means 12 ÷ 3 = 4, then 4 × 2 = 8.
Division asks how many of the small pieces fit inside. 3/4 ÷ 1/8 asks how many eighths fit into three quarters. Since 3/4 is the same as 6/8, the answer is 6. This is why dividing by a fraction smaller than one gives an answer larger than the number you started with — a result that surprises most children the first time they meet it.
Worked examples
Example 1: Adding fractions with different denominators
Anisa ate 1/3 of a roti and her brother ate 1/4 of the same roti. What fraction did they eat altogether?
The denominators differ, so rewrite both over twelfths: 1/3 = 4/12 and 1/4 = 3/12.
Now add the numerators only: 4/12 + 3/12 = 7/12.
If the question had gone on to ask what was left, the answer would be 12/12 − 7/12 = 5/12.
Example 2: Fraction of a quantity in context
A maxi taxi has 24 seats. Two thirds of the seats are taken. How many seats are empty?
Find two thirds of 24 first: 24 ÷ 3 = 8, then 8 × 2 = 16 seats taken.
The question asks for the empty seats, so 24 − 16 = 8 seats.
Stopping at 16 answers a question that was not asked. Reading the final sentence again before writing the answer is what prevents this.
Example 3: Working backwards from a remainder
Kern read 1/5 of his book on Monday and 1/4 on Tuesday. He has 44 pages left. How many pages are in the book?
Over twentieths he read 4/20 and 5/20, which is 9/20 altogether. So the 44 pages left must be 11/20 of the book.
If 11 twentieths is 44 pages, one twentieth is 44 ÷ 11 = 4 pages. The whole book is 20 twentieths, so 4 × 20 = 80 pages.
Common mistakes and how to avoid them
Adding the denominators. Writing 1/4 + 1/4 = 2/8. Fix: check the answer against common sense — two quarters must be a half, and 2/8 is only a quarter.
Thinking a bigger denominator means a bigger fraction. Choosing 1/6 over 1/2. Fix: picture the roti. More cuts means smaller pieces.
Multiplying only the bottom when making equivalent fractions. Turning 3/4 into 3/12. Fix: say "whatever I do to the top, I do to the bottom" every time.
Writing the digits side by side for mixed numbers. Turning 2 3/4 into 23/4. Fix: work out how many quarters are in 2 wholes first.
Answering the first step instead of the question. Finding the seats taken when the question asked for the seats empty. Fix: underline the final question before starting.
Assuming division always makes things smaller. Expecting 6 ÷ 1/2 to be 3. Fix: read it as "how many halves fit into 6?" — the answer is 12.
How parents can help at home
Fractions are easier to feel than to explain, and a kitchen is a better classroom than a worksheet.
Cut something and talk about it. A roti, a cake, an orange, a bar of chocolate. Ask "if I cut this into 8 and you take 3, what fraction do you have?" Then ask the harder question: "is that more or less than a half?"
Use measuring cups. Half a cup, a quarter cup, and how many quarter cups fill one whole cup. This makes division by a fraction obvious rather than mysterious.
Ask the comparison question. "Which is bigger, 1/3 or 1/4?" is worth asking often, because the wrong instinct is so strong. Every time your child answers correctly and explains why, the idea gets a little more secure.
If your child adds the denominators, resist correcting it straight away. Instead ask, "so is 1/4 plus 1/4 more or less than a half?" They will usually spot the problem themselves, and self-corrected errors stay corrected.
Exam technique for fractions
Simplify your final answer unless the question tells you otherwise. An unsimplified answer is usually still accepted at SEA level, but simplifying makes checking easier and matches the answer options in multiple-choice items.
Where a question gives a fraction of an amount, divide before you multiply. The numbers stay small and the arithmetic stays clean.
Check every fraction answer for sense. If you add two fractions and get something smaller than one of them, you have added the denominators. If you take a fraction of an amount and get a bigger number than you started with, you have multiplied where you should have divided.
In multiple-choice items the distractors are built from the standard errors, so an answer that matches one of them exactly deserves a second look.
Quick revision summary
- The denominator sets the size of the pieces; the numerator counts them.
- A bigger denominator means smaller pieces, so 1/8 is less than 1/4.
- Multiply or divide top and bottom by the same number to make equivalent fractions.
- Same denominator: add or subtract the numerators only, leaving the denominator alone.
- Different denominators: rewrite over a common denominator first.
- A fraction equals one half when the numerator is half the denominator.
- Fraction of an amount: divide by the denominator, then multiply by the numerator.
- Multiply fractions across the top and across the bottom.
- Dividing by a fraction less than one gives a larger answer.