A Fraction of a Group

The whole can be a group

A fraction does not only cut up one shape. It can also describe part of a group of things. Here are 8 counters, drawn as 8 squares, and the question is: what is three quarters of 8?

This time the whole is all 8 counters together. The bottom number of $\frac{3}{4}$3/4 is 4, and it says to split the whole into 4 equal groups. 8 counters in 4 equal groups make 2 counters in each group.

The 8 counters split into 4 equal groups, with 2 counters in each group.

Take as many groups as the top number says

The top number of $\frac{3}{4}$3/4 is 3, and it says how many of those groups to take. Each group holds 2 counters, so 3 groups hold 2 + 2 + 2 = 6 counters. Three quarters of 8 is 6.

3 of the 4 groups are colored in. They hold 6 of the 8 counters.

Deal the counters into groups

When the equal groups are hard to see, make them by dealing. For three quarters of 12, the bottom number says 4 groups. Put the counters into the 4 groups one at a time, in turn, until all 12 are used. Every group ends up with 3 counters, because 12 ÷ 4 = 3.

So one quarter of 12 is 3. The top number says to take 3 of the groups: 3 + 3 + 3 = 9. Three quarters of 12 is 9.

Deal the 12 counters one at a time into the 4 groups. After each full round the groups are level, and when the pile is gone each group has 3.

3 of the 4 groups of 3 are colored in: three quarters of 12 is 9.

Each number has its own job

The usual mistake is to answer with the top number: three quarters of 8 is not 3. The 3 counts groups, not counters, and each group holds 2 counters, so the answer is 6. Another mistake is to take the bottom number away, as in 8 − 4 = 4. The 4 is not a number of counters. It says how many equal groups to make.

To check, add the group you did not take. Three quarters of 8 is 6, one quarter is 2, and 6 + 2 = 8, the whole group again.