Lesson 7 of 10 · Grades 3 to 5
Fractions of a whole number
A fraction can also ask a question about a group: what is two thirds of 18? To find out, first find one equal share of the group, then take as many shares as the numerator says. Two steps and you are done.
A fraction asking a question
Fractions can do more than name a slice of a shape. They can also ask for a part of a group of things: what is two thirds of 18 pencils? What is three quarters of 20 minutes? The word "of" is the clue. It tells you that the fraction is a share of the whole number that follows it.
The picture below shows the idea. Twelve dots are split into three equal groups. Each group is one third of 12, and each group holds 4 dots. If you want two thirds, take two of the groups, which gives 8 dots.
Two steps: divide, then multiply
The denominator tells you how many equal shares to make, so it always comes first. The numerator tells you how many of those shares to take, so it comes second. That gives a reliable two-step method:
- Divide the whole number by the denominator to find one share.
- Multiply that share by the numerator to take the number of shares you need.
For 2/3 of 12: divide 12 by 3 to get 4, then multiply 4 by 2 to get 8. For 3/4 of 20: divide 20 by 4 to get 5, then multiply 5 by 3 to get 15. The division always finds the size of one share, which is the same as 1 over the denominator of the number.
Why dividing first is easier
You could multiply first and divide after: for 3/4 of 20, work out 3 x 20 = 60, then divide 60 by 4 to get 15. The answer is the same, but the numbers get bigger along the way. Dividing first keeps the numbers small and makes the mental arithmetic friendlier. Both orders are correct; choose the one that feels easier for the numbers in front of you.
Worked example
A school has 24 students in a class. Five sixths of them walk to school. How many students walk? Step 1: find one sixth. Divide 24 by 6 to get 4 students. Step 2: take five shares. Multiply 4 by 5 to get 20. So 20 students walk. Check the leftovers: 24 - 20 = 4, which is one sixth of the class, as expected.
Watch out for this
A common slip is dividing by the numerator instead of the denominator. For 2/3 of 12, dividing by 2 gives 6 and then multiplying by 3 gives 18, which is more than the original group of 12. That cannot be right: a fraction of something must be less than the whole (unless the fraction is worth more than 1). Check your answer against the original number. One third of 12 must be smaller than 12, and two thirds must be smaller than 12 as well.
Remember
- Divide by the denominator to find one equal share.
- Multiply by the numerator to take the number of shares.
- A fraction of a whole number is never larger than the whole number.
- Whole numbers can be split into equal groups to check your work.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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What is 1/4 of 20?
Show answer
5. 20 divided by 4 equals 5.
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What is 2/5 of 15?
Show answer
6. 15 divided by 5 equals 3, and 2 x 3 = 6.
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What is 3/4 of 12?
Show answer
9. 12 divided by 4 equals 3, and 3 x 3 = 9.
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What is 5/8 of 24?
Show answer
15. 24 divided by 8 equals 3, and 5 x 3 = 15.
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There are 18 cupcakes at a party. Two thirds of them are chocolate. How many chocolate cupcakes are there?
Show answer
12. 18 divided by 3 equals 6, and 2 x 6 = 12.
Puzzle · Marble bag
Puzzle time
In a bag of marbles, 1/2 are red and 1/4 are blue. The other 6 marbles are green. How many marbles are in the bag?
Show a hint
Red and blue together are 1/2 + 1/4 of the bag. What fraction of the bag is green?
Show the answer
Answer: 24. The bag holds 24 marbles. Red is 2/4 of the bag and blue is 1/4, so green is the last 1/4. If 1/4 of the bag is 6 marbles, the whole bag is 4 × 6 = 24. Check: 12 red + 6 blue + 6 green = 24.