Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 1 of this unit) Write 18/45 in simplest form.
Answer
2/5. The greatest common factor of 18 and 45 is 9. - (Unit 2, lesson 3) Find 30% of 80.
Answer
24. 10% is 8, and 3 × 8 = 24. - (Lesson 4 of this unit) Work out 3/10 + 1/5.
Answer
1/2. 1/5 = 2/10, and 3/10 + 2/10 = 5/10 = 1/2.
Words used in this lesson
- product
- the result of multiplying. For example: the product of 1/2 and 1/3 is 1/6.
- of
- multiplied by, when a fraction comes before it. For example: half of 12 is 1/2 × 12.
- cancel
- divide a top and a bottom by the same number. For example: in 4/9 × 3/8, cancel 4 and 8.
- area model
- a picture where a fraction of a fraction is an overlap. For example: columns for one, rows for the other.
Columns and rows
A third of the plot behind the school is for vegetables. Half of that vegetable section is for chillies. What fraction of the whole plot is chillies? Predict first: is it more or less than 1/3? Then test it: set the first fraction to 1/3 and the second to 1/2. The figure cuts the plot into columns for the first fraction and rows for the second, and you look at the overlap.
2/5 as columns and 1/2 as rows: the overlap is 2 cells out of 5 × 2 = 10, so 2/5 × 1/2 = 2/10 = 1/5.
Show me the steps
- 1/3 × 1/2: 1 cell out of 6, so 1/6.
- 2/3 × 3/4: 2 × 3 = 6 cells out of 3 × 4 = 12, so 6/12 = 1/2.
- 3/5 × 2/3: 6 cells out of 15, so 6/15 = 2/5.
- 5/6 × 3/5: 15 cells out of 30, so 15/30 = 1/2.
- 1/4 × 1/4: 1 cell out of 16, so 1/16: smaller than both.
Multiplying by a fraction means taking that fraction of a quantity. Cut the whole into b columns and d rows: there are b × d equal cells. a/b shades a columns, c/d shades c rows, and the overlap has a × c cells. So a/b × c/d = (a × c)/(b × d): multiply the numerators (tops) and multiply the denominators (bottoms).
Simplify before multiplying: divide any top and any bottom by a factor they share. For mixed numbers, write them as improper fractions first. A whole number is a fraction over 1, so 5 × 3/8 = 15/8.
The product of two positive fractions that are each less than 1 is less than both of them, because you take a part of a part.
Back to the plot: half of a third is 1 cell out of 6, so chillies cover 1/6 of the plot, which is less than the 1/3 of vegetables.
A common mistake: “Multiplying always makes the number bigger.” Multiplying a positive number by something greater than 1 makes it bigger, by 1 leaves it the same, and by something between 0 and 1 makes it smaller. Taking 1/2 of something makes it smaller.
Simplify first
Multiplying tops and bottoms can make big numbers that you then have to simplify. Instead, look for a top and a bottom (from either fraction) that share a factor. The two colours show which top goes with which bottom.
8/15 × 5/12: 8 and 12 both divide by 4, and 5 and 15 both divide by 5, so the product is (2 × 1)/(3 × 3) = 2/9.
Show me the steps
- 8/15 × 5/12: 8 and 12 share 4, 5 and 15 share 5, so (2 × 1)/(3 × 3) = 2/9. Multiplying first gives 40/180.
- 9/10 × 4/15: 9 and 15 share 3, 4 and 10 share 2, so (3 × 2)/(5 × 5) = 6/25. Multiplying first gives 36/150.
- 6/7 × 14/15: 6 and 15 share 3, 14 and 7 share 7, so (2 × 2)/(1 × 5) = 4/5. Multiplying first gives 84/105.
- 10/21 × 7/25: 10 and 25 share 5, 7 and 21 share 7, so (2 × 1)/(3 × 5) = 2/15. Multiplying first gives 70/525.
- 12/25 × 5/8: 12 and 8 share 4, 5 and 25 share 5, so (3 × 1)/(5 × 2) = 3/10. Multiplying first gives 60/200.
- 3/4 × 8/9: 3 and 9 share 3, 8 and 4 share 4, so (1 × 2)/(1 × 3) = 2/3. Multiplying first gives 24/36.
Worked examples
Work out 3/8 × 4/9.
- 4 and 8 share 4: 4 ÷ 4 = 1 and 8 ÷ 4 = 2. Now it is 3/2 × 1/9.
- 3 and 9 share 3: 3 ÷ 3 = 1 and 9 ÷ 3 = 3. Now it is 1/2 × 1/3.
- Multiply: 1 × 1 = 1 and 2 × 3 = 6. 1/6.
Work out 1 1/3 × 2 1/4.
- Improper fractions: 1 1/3 = 4/3 and 2 1/4 = 9/4.
- 4 and 4 share 4, and 9 and 3 share 3: 4/3 × 9/4 becomes 1/1 × 3/1.
- 3. Check by estimating: 1 1/3 × 2 is 2 2/3, and 2 1/4 is a bit more than 2, so the answer is a bit more than 2 2/3, which is about 3.
Your go. The first step is given. Work out 4/9 × 15/16.
- 4 and 16 both divide by 4, so the 4 becomes 1 and the 16 becomes 4.
- Now look for another pair, then multiply and check.
Check the answer
15 and 9 both divide by 3: 15 becomes 5 and 9 becomes 3. Now it is 1/3 × 5/4, and (1 × 5)/(3 × 4) = 5/12.
Check before you go on
1. What is 2/5 × 3/4?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. 1/2 × 1/5. What is the bottom number of the answer?
3. A tank holds 28 litres. It is 3/4 full. How many litres of water are in it?
Level 2 · Building
4. Work out 5/6 × 9/10 and write it in simplest form. What is the top number of the answer?
5. A batch of rice needs 2 1/2 cups. Aroha makes 3/5 of a batch. How many cups does she need? Give your answer as a decimal.
6. A bag holds 48 sweets. One third of one quarter of the sweets are red. How many red sweets are there?
Level 3 · Stretch
7. 3/5 of a number is 18. What is 2/3 of that number?
8. n is a whole number and n × 3/7 is less than 10. What is the greatest value n can have?
Something to think about
Does 2/3 of 3/4 equal 3/4 of 2/3? Use the area model: what happens to the rows and the columns when you swap the two fractions?
One way to think about it
Both are 6/12 = 1/2. Swapping the two fractions turns the picture on its side: the columns become rows and the overlap is the same cells, since a × c = c × a. The order of multiplying does not change the product.
Exit ticket
9. Which statement about 5/6 × 3/7 is true?