Divide by the denominator, then multiply by the numerator
The denominator: find one share
Find of 12. The denominator, 3, says to split the whole amount into 3 equal shares. Draw 12 as a bar and cut it into 3 equal parts. One part is 12 ÷ 3 = 4, so of 12 is 4.
The bar of 12 is cut into 3 equal parts of 4. One part, of 12, is colored.
The numerator: take that many shares
The numerator, 2, says how many of the shares to take. Each share is 4, so 2 shares are 2 × 4 = 8. of 12 = 8.
So to find a fraction of an amount, divide the amount by the denominator, then multiply by the numerator.
2 of the 3 parts are colored. Each holds 4, so together they are 8.
Larger amounts
The amount does not have to be something you can count one by one. Find of 60 minutes. Divide by the denominator: 60 ÷ 4 = 15, so a quarter of an hour is 15 minutes. Multiply by the numerator: 3 × 15 = 45. So of 60 minutes is 45 minutes.
To check, add on the part you did not take. The quarter left over is 15 minutes, and 45 + 15 = 60, the whole amount again.
When the numerator is 1, dividing is the whole job: of 35 is 35 ÷ 5 = 7.
The bar of 60 is cut into 4 equal parts of 15. Three of them make 45, and the fourth, 15, makes the total up to 60.
3/4 × 60 =
The button cuts the 60 into equal parts, each labeled with what it holds; the handle takes parts from the left. Quarters hold 15 each, and three of them make 45.
When the division is not exact
Find of 10. Dividing first gives 10 ÷ 3, which does not come out exactly. Instead, multiply first: 2 × 10 = 20, then divide: .
Either order gives the same answer. Dividing first, one third of 10 is , and two thirds is . Divide first when the division is exact, because the numbers stay small.
Cut 10 into 3 equal parts and each part is . Two parts make .
The usual mistakes
Taking the wrong part. In of 12, the 2 shares taken are 8, and the share left over is 4. Answering 4 gives the part that was not taken.
Taking one share too many. The numerator says exactly how many shares to take: is 2 shares of 4, which is 8, not 3 shares, which would be all 12.
Fractions · Egg Tarts and Cream Puffs, Each Sold in Its Own Fraction
QuestionA baker made 240 pastries. 38 of them were egg tarts and the rest were cream puffs. He sold 56 of the egg tarts at $2 each and 45 of the cream puffs at $3 each. (a) How many pastries did he sell? (b) How much money did he collect?
1.Draw the 240 pastries as a bar of 8 equal units. One unit is 240 ÷ 8 = 30 pastries.
The 240 pastries as 8 equal units: 240 ÷ 8 = 30 in each. 2.The egg tarts are 3 units, 3 × 30 = 90, and the cream puffs are the other 5 units, 5 × 30 = 150.
The egg tarts are 3 units, 90, and the cream puffs the other 5 units, 150. 3.He sold 56 of the 90 egg tarts: 90 ÷ 6 = 15, and 5 × 15 = 75 egg tarts.
Sixths of the 90 egg tarts are 15 each, and 5 sixths were sold: 5 × 15 = 75. 4.He sold 45 of the 150 cream puffs: 150 ÷ 5 = 30, and 4 × 30 = 120 cream puffs.
Fifths of the 150 cream puffs are 30 each, and 4 fifths were sold: 4 × 30 = 120. 5.(a) He sold 75 + 120 = 195 pastries.
(a) He sold 75 + 120 = 195 pastries. 6.(b) He collected 75 × $2 + 120 × $3 = $150 + $360 = $510. Check: he did not sell 90 − 75 = 15 egg tarts and 150 − 120 = 30 cream puffs, and 195 + 15 + 30 = 240.
(b) Only the pastries sold bring in money: 75 × $2 + 120 × $3 = $510.
Answer: (a) 195 pastries; (b) $510
Watch out for
- Taking 56 of all 240 pastries as the egg tarts sold. The 56 is a fraction of the 90 egg tarts only, so he sold 56 × 90 = 75 of them.
- Pricing the pastries left instead of the pastries sold, 15 × $2 + 30 × $3. Only the pastries sold bring in money: 75 egg tarts and 120 cream puffs.
Your turn
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