Topics · Primary Maths · Fractions
Fractions: equal parts of a whole
A fraction counts equal parts. The bottom number says how many parts the whole was cut into; the top number says how many you have. Everything else follows from those two sentences.
- 431 exam papers analysed
- appears in 87% of P6 Maths papers
- ~4 min read
01 Equal parts, and the word doing the work
Cut something into parts that are the same size, and a fraction tells you how many you took. The word equal is the whole definition — parts of different sizes are not fractions of anything.
The bottom number is the denominator: how many equal parts the whole was cut into. The top is the numerator: how many of them you have.
02 Equivalent fractions
Those two bars above are shaded to exactly the same length. 3/4 and 6/8 are the same amount — the second was simply cut into twice as many pieces, each half the size.
That is what makes fractions manageable. Multiply top and bottom by the same number and the fraction is unchanged, because you have cut every piece the same way. Divide them both and you are gluing pieces back together, which is simplifying.
A fraction is in its simplest form when no number divides both top and bottom. 6/8 divides by 2 to give 3/4, and 3/4 will not go further.
03 Adding and subtracting
You can only add parts that are the same size, which is why fractions with the same denominator are easy: 1/5 + 2/5 = 3/5. One part plus two parts is three parts.
Different denominators cannot be added as they stand — one-half plus one-third is neither two-fifths nor anything you can read off. Cut both into the same-sized pieces first, then add.
- Find a common denominator. A number both denominators divide into. For halves and thirds, sixths work.
- Rewrite each fraction over that denominator. 1/2 becomes 3/6; 1/3 becomes 2/6.
- Add or subtract the numerators only. 3/6 + 2/6 = 5/6. The denominator says what size the pieces are and does not change.
- Simplify at the end if anything divides both.
04 A fraction of a quantity
Three quarters of 240 means: cut 240 into 4 equal parts and take 3 of them. Divide by the bottom, multiply by the top.
240 ÷ 4 = 60 for one part, and 3 × 60 = 180. This is the same one-unit move as a bar model word problem, and it is worth noticing that they are the same idea — the denominator tells you how many units, the numerator how many to take.
05 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- fractionsayFRAK-shuhnmore than onefractions
A way of counting equal parts of a whole, written with one number above another.
- numeratorsayNYOO-muh-ray-tuhmore than onenumerators
The top number — how many equal parts you have.
- denominatorsaydi-NOM-i-nay-tuhmore than onedenominators
The bottom number — how many equal parts the whole was cut into.
- equivalentsayi-KWIV-uh-luhnt
Equal in value though written differently. 3/4 and 6/8 are equivalent — the same amount, cut up differently.
- simplest formmore than onesimplest forms
A fraction where no number divides both the top and the bottom. 6/8 simplifies to 3/4.
- improper fractionsayim-PROP-uhmore than oneimproper fractions
A fraction whose top is bigger than its bottom, like 7/4. It is more than one whole, and can be written as the mixed number 1 3/4.
- mixed numbermore than onemixed numbers
A whole number with a fraction beside it, like 1 3/4.
06 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P2
Halves and quarters of a shape, and sharing equally.
Practise P2 Fractions — 10 questions →66% of P2 papers
- P3
Naming fractions, comparing them, and equivalent fractions on a picture.
Practise P3 Fractions — 10 questions →53% of P3 papers
- P4
Adding and subtracting with the same denominator, mixed numbers and improper fractions.
Practise P4 Fractions — 10 questions →75% of P4 papers
- P5
Different denominators, fraction of a quantity, and multiplying a fraction by a whole number.
Practise P5 Fractions — 10 questions →63% of P5 papers
- P6
Dividing with fractions, and fractions inside multi-step word problems where a quantity changes.
Practise P6 Fractions — 10 questions →87% of P6 papers
07 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“1/2 + 1/3 = 2/5.”
Adding the tops and the bottoms is the commonest fraction error there is. Halves and thirds are different-sized pieces; cut both into sixths first and it is 3/6 + 2/6 = 5/6. Notice 2/5 is smaller than the 1/2 you started with, which cannot be right.
“1/8 is bigger than 1/4 because 8 is bigger than 4.”
The bigger the bottom number, the more pieces the whole was cut into, so the smaller each piece is. Eighths are smaller than quarters.
“The parts do not have to be exactly equal.”
They do. A shape cut into four unequal pieces has no quarters in it at all.
“To find 3/4 of 240 I multiplied by 4 and divided by 3.”
The other way round: divide by the bottom, multiply by the top. 240 ÷ 4 = 60, then 3 × 60 = 180.
“7/4 is wrong because the top is bigger.”
It is an improper fraction and perfectly correct. It means more than one whole — the same as 1 3/4.
Now try it
50 questions across 5 levels, marked as you go with the working shown.