Discover Mathematics P5: Term 2, Unit 9, Lessons 5–8
GoalMultiply a fraction by a mixed number in two ways and simplify the answer.
New words
mixed number · a whole number and a fraction together, such as 3 1/2improper fraction · a fraction whose numerator is greater than or equal to its denominator, such as 7/2distributive property · multiply each part of a sum and then add: a × (b + c) = a × b + a × c
Explanation
To find a fraction of a mixed number you can use two ways. Way 1: change the mixed number into an improper fraction, then multiply the numerators and the denominators. 2/3 × 1 1/2 = 2/3 × 3/2 = 6/6 = 1. Way 2: split the mixed number into its whole part and its fraction part, multiply each part, then add. 2/3 × 1 = 2/3 and 2/3 × 1/2 = 2/6 = 1/3, so 2/3 + 1/3 = 1. Both ways give the same answer. Way 1 is quicker when the numbers are small or when numerators and denominators cancel. Way 2 shows what is happening: you take the same fraction of every piece. Always simplify the answer at the end, and write it as a mixed number if it is greater than 1.
2/3 × 1 1/2 = 1
Way
Work
1: improper fraction
1 1/2 = 3/2, 2/3 × 3/2 = 6/6 = 1
2: split the mixed number
2/3 × 1 = 2/3, 2/3 × 1/2 = 1/3, 2/3 + 1/3 = 1
Worked examples
Way 1
Way 2
Step 1
3 1/2 = 7/2
1/2 × 3 = 3/2
Step 2
1/2 × 7/2 = 7/4
1/2 × 1/2 = 1/4
Answer
1 3/4
3/2 + 1/4 = 1 3/4
Work out 1/2 × 3 1/2. Way 1: 3 1/2 = 7/2, so 1/2 × 7/2 = 7/4 = 1 3/4. Way 2: 1/2 × 3 = 3/2 and 1/2 × 1/2 = 1/4, so 3/2 + 1/4 = 6/4 + 1/4 = 7/4 = 1 3/4. The two ways agree.
Work out 3/4 × 2 2/3. Change 2 2/3 into 8/3. Then 3/4 × 8/3 = 24/12 = 2. Check with way 2: 3/4 × 2 = 6/4 = 3/2 and 3/4 × 2/3 = 6/12 = 1/2, and 3/2 + 1/2 = 2.
A recipe needs 2 1/2 cups of flour. Hana makes only 3/5 of the recipe. She needs 3/5 × 5/2 = 15/10 = 3/2 = 1 1/2 cups of flour. The 5s can be cancelled first: 3/5 × 5/2 = 3/2.
Class activity
Folded paper bars. Cut a long paper strip and fold it into 7 equal parts: each part is 1/2, so the whole strip is 3 1/2 (7 halves). Fold every part in half again, so the strip shows 14 quarters. To find 1/2 of the strip, cut it in half. How many quarters are in one half (7)? Write 1/2 × 3 1/2 = 7/4 = 1 3/4 and check by laying the pieces next to a strip of length 1.
Practice
1
Work out 1/2 × 4 1/2.
4 1/2 = 9/2, so 1/2 × 9/2 = 9/4 = 2 1/4.
2
Work out 2/3 × 3 3/4 and simplify.
3 3/4 = 15/4. 2/3 × 15/4 = 30/12 = 5/2 = 2 1/2.
3
Work out 3/8 × 2 2/3.
2 2/3 = 8/3. 3/8 × 8/3 = 24/24 = 1.
4
Is 3/4 × 5 1/2 greater or less than 5 1/2? Then work it out.
Less, because 3/4 is less than 1. 5 1/2 = 11/2 and 3/4 × 11/2 = 33/8 = 4 1/8.
5
Nour has a ribbon 2 1/4 m long and uses 2/3 of it for a gift. How long is the piece she uses?
