Dividing a whole number by a fraction: how many pieces fit?
Discover Mathematics P6: Term 2, Unit 8, Lessons 1–2
GoalDivide a whole number by a fraction by asking how many of that fraction fit, and see that the answer is bigger.
New words
how many fit · the meaning of dividing by a piece sizenumber line · a line with equal marks that shows numbersdivision sentence · an equation with ÷, such as 3 ÷ 1/4 = 12
Explanation
When we divide a whole number by a fraction, we ask: how many pieces of this size fit in the whole number? For 3 ÷ 1/4, how many quarters fit in 3? Each whole holds 4 quarters, so 3 wholes hold 3 × 4 = 12 quarters. For 4 ÷ 2/3, each whole holds 3 thirds, so 4 wholes hold 12 thirds. The pieces are 2 thirds long, so 12 ÷ 2 = 6 pieces fit. When the pieces are smaller than 1, more than the whole number of them fit, so the answer is bigger than the number we started with. Be careful: 6 ÷ 1/2 = 12, but 1/2 ÷ 6 = 1/12 is something quite different.
★Remember
Whole ÷ fraction: how many pieces of that size fit? Each whole holds as many pieces as the bottom number says. The answer is bigger than the whole number.
Worked examples
3 ÷ 1/4: the number line from 0 to 3 is cut into quarters. Count the quarter steps: 4 in each whole, 12 in all. So 3 ÷ 1/4 = 12.
Division
Meaning
Answer
6 ÷ 1/2
how many halves in 6
12
1/2 ÷ 6
share a half among 6
1/12
Compare two divisions that look alike. 6 ÷ 1/2 asks how many halves fit in 6, and the answer is 12. 1/2 ÷ 6 asks to share a half between 6, and the answer is 1/12.
A rope is 4 m long and pieces of 2/3 m are cut from it. In each metre there are 3 thirds, so 4 m is 12 thirds. Pieces of 2 thirds each: 12 ÷ 2 = 6. So 4 ÷ 2/3 = 6 pieces.
Class activity
Pour and count. Take a cup that holds about 1/4 of a litre and a bottle of water. Count how many cups you can fill from 2 litres (use a litre jug if you have one). Write the division sentence and compare your count with the answer.
Practice
1
Find 5 ÷ 1/3.
Each whole has 3 thirds, so 5 × 3 = 15.
2
Find 8 ÷ 1/4.
Each whole has 4 quarters, so 8 × 4 = 32.
3
Find 3 ÷ 3/4.
3 wholes are 12 quarters. Groups of 3 quarters: 12 ÷ 3 = 4.
4
A 2 L bottle of milk is poured into cups of 1/4 L. How many cups are filled?
2 ÷ 1/4 = 2 × 4 = 8 cups.
5
Without calculating, say whether the answer is bigger or smaller than the first number: (a) 5 ÷ 1/2, (b) 1/2 ÷ 5.
