Fractions of a collection
Theory
Finding a fraction of a collection is a Year 3 maths skill: sharing a group of objects, such as counters or apples, into equal groups. This page explains how to find unit fractions like one half, one third and one quarter of a set, and fractions such as two thirds and three quarters, by dividing to make equal groups and then taking the groups you need. It also shows how to name the fraction of a set that is shaded and how to work out "the rest". Curriculum: VC2M3N03 / AC9M3N02.
A collection is a group of things you can count, like counters, marbles or apples. A fraction of a collection tells you to share that group into equal parts and take some of them.
To find a fraction of a set, share it into equal groups. The bottom number (denominator) says how many equal groups to make, and the top number (numerator) says how many groups to keep. For \(\dfrac{1}{4}\) of \(20\) you share \(20\) into 4 equal groups of 5, so \(\dfrac{1}{4}\) of \(20 = 5\).
A unit fraction has a 1 on top, like \(\dfrac{1}{2}\), \(\dfrac{1}{3}\) or \(\dfrac{1}{5}\). It takes just one equal group. A fraction like \(\dfrac{3}{4}\) takes more than one group: share into 4 equal groups, then keep 3 of them.
You can also work backwards. If some objects in a set are shaded, sort the whole set into equal groups the same size as the shaded group. The number of groups is the bottom number, and the shaded groups give the top number.
Two facts help you find any fraction of a collection. Divide by the bottom number to make equal groups, then multiply by the top number.
How to find a fraction of a collection
- Read the bottom number. It tells how many equal groups to make. Share the collection by dividing by that number.
- Find one group. Work out how many objects are in each equal group.
- Take the top number of groups. Keep as many groups as the top number says. For a unit fraction that is just one group.
How to name the fraction that is shaded
- Count the shaded objects. This is the size of one group if the shaded objects make one group.
- Make equal groups of that size. Count how many groups the whole collection makes. That number is the bottom number.
- Write the fraction. One shaded group out of that many groups is a unit fraction, like \(\dfrac{1}{5}\).
How to find "the rest"
- Find each part. Work out each fraction of the collection.
- Add and take away. Add the parts you found, then take that total away from the whole collection.
| \(\dfrac{1}{4}\text{ of }20\) | \(=\) | \(20 \div 4\) |
| \(20 \div 4\) | \(=\) | \(5\) |
Share \(20\) into 4 equal groups of 5. So \(\dfrac{1}{4}\) of \(20\) is \(5\).
| \(16 \div 4\) | \(=\) | \(4\text{ in each group}\) |
| \(3 \times 4\) | \(=\) | \(12\) |
Divide by 4, then take 3 groups. So \(\dfrac{3}{4}\) of \(16\) is \(12\).
| \(15\text{ counters}\) | \(\to\) | \(5\text{ equal groups}\) |
| \(2\text{ groups shaded}\) | \(=\) | \(\dfrac{2}{5}\) |
Count the equal groups, not single counters. The shaded fraction is \(\dfrac{2}{5}\).
| \(\$25 \div 5\) | \(=\) | \(\$5\text{ in each group}\) |
| \(3 \times \$5\) | \(=\) | \(\$15\) |
Share \(\$25\) into 5 equal groups, then take 3 groups. She spends \(\$15\).
| \(\text{groups of }3\) | \(:\) | \(15 \div 3 = 5\text{ groups}\) |
| \(\text{red}\) | \(=\) | \(1\text{ of }5\text{ groups}\) |
| \(\text{fraction}\) | \(=\) | \(\dfrac{1}{5}\) |
The red pencils make 1 of 5 equal groups, so \(\dfrac{1}{5}\) of the pencils are red.
(i) How many are ripe?
(ii) How many are hard?
(iii) The rest are soft. How many are soft?
| \(\text{(i)}\;\; 36 \div 4\) | \(=\) | \(9\) |
| \(\text{(ii)}\;\; 36 \div 3\) | \(=\) | \(12\) |
| \(\text{(iii)}\;\; 36 - (9 + 12)\) | \(=\) | \(36 - 21 = 15\) |
(i) 9 are ripe. (ii) 12 are hard. (iii) 15 are soft.
Common pitfalls
Frequently asked questions
What does a fraction of a collection mean?
It is a fraction of a group of objects, like counters or apples. You share the group into equal groups and take some of them. For example, \(\dfrac{1}{3}\) of \(15\) counters is \(5\).
How do you find one third of a number?
Share the collection into 3 equal groups by dividing by 3. So \(\dfrac{1}{3}\) of \(21\) is \(21 \div 3 = 7\).
How do you find three quarters of a number?
First divide by the bottom number to make equal groups, then multiply by the top number. For \(\dfrac{3}{4}\) of \(16\), do \(16 \div 4 = 4\), then \(4 \times 3 = 12\).
What do the top and bottom numbers of a fraction tell you?
The bottom number (denominator) tells how many equal groups to share into. The top number (numerator) tells how many of those groups to keep.
How do you know which fraction of a group of counters is shaded?
Count how many equal groups there are, then how many are shaded. If \(15\) counters make 5 equal groups and 2 are shaded, the shaded fraction is \(\dfrac{2}{5}\).
Is half of a collection always a whole number?
Only when the collection shares evenly into 2 equal groups. Half of \(14\) is \(7\), but \(9\) does not share into two equal whole groups.