From fraction division to multiplying by the reciprocal
Discover Mathematics P6: Term 2, Unit 8, Lessons 3–4
GoalDiscover that dividing by a fraction gives the same answer as multiplying by its reciprocal.
New words
reciprocal · the fraction turned upside down; a number times its reciprocal is 1divisor · the number we divide byrule · a pattern that works every time
Explanation
Look at the models from last week. 3 ÷ 1/4 = 12, and 3 × 4 = 12 too. 6 ÷ 1/2 = 12, and 6 × 2 = 12. 3/4 ÷ 1/8 = 6, and 3/4 × 8 = 6. In every case, dividing by a fraction gave the same answer as multiplying by the fraction turned upside down. The upside-down fraction is called the reciprocal: the reciprocal of 1/4 is 4, of 2/3 is 3/2, and of 5 is 1/5. A number multiplied by its reciprocal always makes 1. Why does the rule work? Dividing by 1/4 asks how many quarters fit, and each whole holds 4 quarters, so we multiply by 4. Dividing by a whole number also fits: 3/4 ÷ 3 = 3/4 × 1/3 = 1/4. Only the divisor, the number after ÷, is turned over.
★Remember
Dividing by a fraction = multiplying by its reciprocal. Turn over only the number after the ÷ sign.
Worked examples
Division
Multiply by the reciprocal
Answer
3 ÷ 1/4
3 × 4
12
6 ÷ 1/2
6 × 2
12
3/4 ÷ 1/8
3/4 × 8
6
Match the division with the multiplication. Both columns give the same answers, which we found with models.
Number
2/3
5
1/8
7/4
Reciprocal
3/2
1/5
8
4/7
Reciprocals. Swap the top and the bottom. A whole number is a fraction over 1, so 5 = 5/1 and its reciprocal is 1/5. The product of a number and its reciprocal is 1; for example 2/3 × 3/2 = 6/6 = 1.
Test the rule. 2/3 ÷ 1/6 = 2/3 × 6 = 12/3 = 4. Check with a model: 2/3 = 4/6, and 4 sixths ÷ 1 sixth = 4. The rule agrees. Another test: 1/2 ÷ 1/4 = 1/2 × 4 = 2, and indeed 2 quarters fit in a half.
Class activity
Flip game. Write ten numbers on cards (fractions, whole numbers, and a few mixed ones). A player draws a card and says its reciprocal; the other player checks by multiplying. Then they pick two cards and divide the first by the second using the rule.
Practice
1
Write the reciprocals of 3/5, 4, 1/7 and 9/2.
5/3, 1/4, 7 and 2/9.
2
Use the rule to find 5 ÷ 1/3.
5 × 3 = 15.
3
Find 2/3 ÷ 1/6.
2/3 × 6 = 12/3 = 4.
4
Test the rule on 1/2 ÷ 1/4 with a model and with multiplication.
Model: 2 quarters fit in a half. Rule: 1/2 × 4 = 2. Both give 2.
5
Hana wrote 3/4 ÷ 2/5 = 4/3 × 2/5. What is wrong, and what is the right answer?
She turned over the first fraction. Only the divisor is turned over: 3/4 × 5/2 = 15/8 = 1 7/8.
