Year 9 · Unit 7 Equations and linear systems · Lesson 2 of 7

Multiplying the fractions away

A fraction in an equation is a nuisance you can remove: multiply every term on both sides by a common multiple of the denominators and the balance is unchanged.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Solve 4(x − 1) = 20.
    Answerx = 6, because 4x − 4 = 20 and 4x = 24.
  2. What is the lowest common multiple of 4 and 6?
    Answer12, the first number that both 4 and 6 divide into.
  3. Simplify 3x/4 × 4.
    Answer3x: the 4 on the bottom cancels the 4 you multiply by.
Words used in this lesson
denominator
the number on the bottom of a fraction. For example: in x/4 the denominator is 4.
numerator
the number or expression on top. For example: in (x + 1)/3 the numerator is x + 1.
common multiple
a number that two numbers both divide into. For example: 12 is a common multiple of 4 and 6.
term
one part of an expression, separated by + or −. For example: 3x/4 − 5 has two terms.
clear
remove the fractions from an equation. For example: multiply every term by 6.
balance
both sides stay equal. For example: multiply both sides by the same number.

Half of the batch and a third of the batch

A women's soap cooperative in Kumasi sells half of a batch at the market and a third of the batch to a hotel. The equation x/2 + x/3 = 5 is a smaller version of the same idea. Fractions are awkward to carry through every step. Is there a number you can multiply by so that both fractions vanish?

Predict first: which multiplier clears both fractions in x/2 + x/3 = 5: 2, 3, 5 or 6? Slide the multiplier and watch the terms. Then try the other two equations.

x/2 + x/3 = 5
1
x/2 + x/3 = 5the equation now
2terms with a fraction

Times 1: x/2 + x/3 = 5. 2 terms still have a fraction.

Show me the steps
  1. x/2 + x/3 = 5 times 1 is unchanged: both terms on the left have a fraction.
  2. Times 2: x + 2x/3 = 10. The x/2 has gone, but 2x/3 is still a fraction.
  3. Times 6: 3x + 2x = 30. No fractions, so 5x = 30 and x = 6.
  4. 3x/4 − x/6 = 7 times 12: 9x − 2x = 84, so 7x = 84 and x = 12.
  5. x/5 + 2x/3 = 13 times 15: 3x + 10x = 195, so 13x = 195 and x = 15.

Multiply every term, on both sides, by a common multiple of the denominators. Each fraction is divided by its own denominator exactly and disappears. Because both sides were multiplied by the same number, the balance and the solution are unchanged. The lowest common multiple keeps the numbers smallest.

Every term means every term: the plain numbers as well as the fractions, and the whole of a numerator that has a bracket in it.

A common mistake: multiplying only the fraction. In (3x + 1)/2 + 3 = 2x, multiplying by 2 gives 3x + 1 + 6 = 4x, not 3x + 1 + 3 = 4x. Test the wrong version: it gives x = 4, and (12 + 1)/2 + 3 = 9.5 is not 2 × 4 = 8.

Worked examples

The soap batch: half the batch plus a third of the batch is 25 bars. Find the batch.

  1. x/2 + x/3 = 25. The lowest common multiple of 2 and 3 is 6.
  2. Multiply every term by 6: 3x + 2x = 150.
  3. 5x = 150, so x = 30.
  4. Check: 30 ÷ 2 + 30 ÷ 3 = 15 + 10 = 25. 30 bars.

Solve (3x + 1)/2 + 3 = 2x.

  1. Multiply every term by 2: (3x + 1) + 6 = 4x. The 2 cancels the denominator and multiplies the 3 and the 2x.
  2. 3x + 7 = 4x, so 7 = x.
  3. Check: (21 + 1) ÷ 2 + 3 = 14 and 2 × 7 = 14. x = 7.

Your go. The first step is given. Solve (x + 4)/3 = x − 2.

  1. Multiply both sides by 3: x + 4 = 3(x − 2).
  2. Now expand the bracket and solve, then check.
Check the answer

x + 4 = 3x − 6, so 10 = 2x and x = 5. Check: 9 ÷ 3 = 3 and 5 − 2 = 3.

Check before you go on

1. What is the smallest whole number that clears both fractions in x/6 + x/9 = 10?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Solve x/3 + 4 = 9.

3. Solve (x + 5)/2 = 7.

Level 2 · Building

4. Solve x/2 + x/5 = 14.

5. A coach says: one third of the squad plus 5 players is the same number as half the squad. How many players are in the squad?

Level 3 · Stretch

6. Solve (2x − 3)/5 + 2 = x/2.

7. Solve (x + 2)/3 − (x − 1)/4 = 2.

Something to think about

Solve x/4 + x/6 = 10 twice: once by multiplying every term by 12, and once by 24. Do the two routes give the same x? Which line of working was shorter?

One way to think about it

Times 12: 3x + 2x = 120, so x = 24. Times 24: 6x + 4x = 240, so x = 24. The same answer, because both sides were scaled by the same amount. The lowest common multiple, 12, keeps the numbers smaller.

Exit ticket

8. Every term of (x − 1)/4 + 3 = 3x is multiplied by 4. Which line is correct?