☰ Year plan · Mathematics

Rational numbers

Week 5 · Rational numbers: their forms and the four operations · 3 lessons
3

Multiplying and dividing rational numbers by the multiplicative inverse

Mathematics Prep 1 Term 1, Unit 1, Lesson 5
GoalMultiply rational numbers, find the multiplicative inverse and divide by multiplying by it.
New words
multiplicative inverse · the number that gives 1 when multiplied: 5/3 for 3/5cancel · divide a top and a bottom by the same numbermixed number · a whole number and a fraction together, as 1.5 = 3/2divisor · the number you divide by
Explanation

To multiply fractions, multiply the tops and multiply the bottoms, and cancel first to keep numbers small. The sign follows the integer rules: 2/3 × (−9/10) = −(2 × 9)/(3 × 10) = −3/5. The multiplicative inverse of a number is the number that gives 1 when you multiply them. The inverse of −3/5 is −5/3, because (−3/5) × (−5/3) = 1. The sign stays the same. Zero has no inverse. Division is multiplying by the inverse of the divisor: 3/4 ÷ 3/8 = 3/4 × 8/3 = 2. Always take the inverse of the number you divide BY, the second one. Change a mixed number into an improper fraction first.

NumberMultiplicative inverse
3/55/3
−3/5−5/3
21/2
0none
Worked examples
The inverse of −3/5 is −5/3. Check: (−3/5) × (−5/3) = 15/15 = 1. Zero has no inverse, because 0 × anything is 0, never 1.
Find 3/4 ÷ 3/8. Multiply by the inverse of 3/8: 3/4 × 8/3. Cancel 3 with 3 and 8 with 4: 1 × 2 = 2. So 3/4 ÷ 3/8 = 2. Check: 2 × 3/8 = 3/4.
A water tank is 3/5 full and holds 120 L. How much does it hold when full? Divide 120 by 3/5: 120 × 5/3 = 200 L. Check: 3/5 of 200 = 120.
Class activity

Inverse pairs. With a grown-up, write six fractions on cards and write the inverse of each on the back. Check each pair by multiplying. Try a card with 0: what happens?

Practice
1
Find 2/5 × (−3/4).
2
Write the multiplicative inverse of −7/2.
3
Find 5/6 ÷ 5/12.
4
Find 4 ÷ 2/3.
5
Mona works out 3/4 ÷ 1/2 as 4/3 × 1/2 = 2/3. What is wrong?