How many fit?
Division can ask how many times one amount fits into another. 12 ÷ 3 asks how many 3s fit into 12, and the answer is 4. Dividing by a fraction asks the same question.
asks how many halves fit into 3 wholes. Draw the 3 wholes as 3 bars of the same size, and fill them with half-bars.
Three whole bars, each the same size, with nothing laid in yet.
Lay the halves in
Lay the half-bars in one at a time. The first 2 fill the first bar, and the third fills half of the second bar. After 3 halves, only a bar and a half is full, so the answer must be more than 3.
3 half-bars fill the first bar and half of the second. A bar and a half is still empty.
Each whole holds 2 halves
Keep going until all 3 bars are full. Each bar holds 2 halves, so 3 bars hold 3 × 2 = 6 halves. So .
Dividing by gives the same answer as multiplying by 2. A half is smaller than a whole, so it fits into each whole more than once, and the answer is bigger than the number you started with.
3 ÷ 1/2 = 3 × 2 = 6: a smaller piece fits more times, which is why dividing by a smaller fraction gives a bigger answer
Lay 1/2 pieces in until they fill 3: 3 ÷ 1/2 = ?
Drag the handle to lay halves into the 3 bars. The bars are full at 6 halves: .
Smaller pieces fit more times
The smaller the piece, the more of them fit. A quarter is smaller than a half, and each whole holds 4 quarters. So : 8 quarters fit into 2 wholes.
In the same way, each whole holds 3 thirds, so .
To check a division, multiply back. 8 quarters make wholes, which is what you started with.
2 ÷ 1/2 = 2 × 2 = 4: a smaller piece fits more times, which is why dividing by a smaller fraction gives a bigger answer
Lay 1/4 pieces in until they fill 2: 2 ÷ 1/4 = ?
Choose the quarter piece and fill the 2 bars. Each bar holds 4 quarters, so 8 fit: .
The usual mistakes
Adding the two numbers. is not 3 + 2 = 5. The question asks how many halves fit, and each of the 3 wholes holds 2 of them, so the numbers multiply: 3 × 2 = 6.
Dividing by 2 instead of by . shares 3 bars between 2 people. Dividing by a half counts how many halves fit, and small pieces fit many times, so the answer grows.
Worked example: Fractional Pieces Cut from a Whole Length
Question A carpenter cuts a plank 5 m long into pieces that are each 34 m long. (a) What is the greatest number of complete pieces he can cut? (b) What length of plank is left over, and what fraction of one piece is it?
1.Cut each meter of the plank into quarters. The plank is 5 × 4 = 20 quarters long and one piece is 3 quarters long.
Measure both in quarters of a meter: 5 × 4 = 20 quarters, and 3 quarters in a piece. 2.Put the quarters into groups of 3: 20 ÷ 3 = 6 remainder 2. This is the division 5 ÷ 34 = 5 × 43 = 203 = 623.
20 ÷ 3 = 6 remainder 2, which is 5 ÷ 34 = 623. 3.(a) The greatest number of complete pieces is 6.
(a) 6 complete pieces can be cut. 4.The 6 pieces use 6 × 34 = 184 = 412 m of the plank, so 2 quarters are left over.
The 6 pieces use 6 × 34 = 412 m. 5.(b) The length left over is 5 − 412 = 12 m. It is 2 of the 3 quarters in a piece, so it is 23 of a piece.
(b) 5 − 412 = 12 m is left, which is 2 of the 3 quarters in a piece: 23 of a piece.
Answer: (a) 6 pieces; (b) 12 m, which is 23 of a piece
Common mistakes
- Reading the 23 in 623 as 23 m. The quotient counts pieces, so 23 means two thirds of a piece, and two thirds of 34 m is 12 m.
- Multiplying instead of dividing: 5 × 34 = 334. The pieces are shorter than 1 m, so there must be more than 5 of them.
More multiplying and dividing fractions problems, worked step by step →