To clear the denominators of x/6 + x/9 = 7, what is the handiest number to multiply both sides by? A. 15 B. 54 C. 18 D. 6 Choose the right answer

C. 18. The L.C.M. of 6 and 9 is 18; times 18 clears both denominators.

4/9 x = −9, x = □ Answer in simplest form

−81/4. Multiply both sides by the reciprocal, 9/4: x = −9 × 9/4 = −81/4. Or times 9 first, 4x = −81, then divide by 4.

(x − 9)/2 − (x − 9)/5 = 9, x = □

39. Times 10: 5(x − 9) − 2(x − 9) = 90. The second numerator has a minus in front, so it is a bracket and every term flips. This gives x = 39.

Linear equations in one unknown · Ages 12–13

Fractions and decimals in equations

Multiply every term; a numerator is a bracket.

How to do it

Method

Multiply every term; a numerator is a bracket

x2 + x3 = 5, 3x + 2x = 30, x = 6

Another way

Add the fractions

x2 + x3 = 5x6, 5x6 = 5, x = 6

In full

One way: clear the denominators. Multiply both sides by the L.C.M. of the denominators, every term, even one with no denominator: x2 + x3 = 5; times 6, 3x + 2x = 30, x = 6. A numerator with two terms needs a bracket: x − 13 times 3 is x − 1, and with a minus in front, −(x − 1). Another way: add the fractions as they are: x2 + x3 = 5x6, so 5x6 = 5, x = 6. With one fraction on each side you can cross-multiply. Decimals work alike: times 10 or 100 makes them whole, so 0.2x + 1 = 3 becomes 2x + 10 = 30.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
To clear the denominators of x6 + x9 = 7, what is the handiest number to multiply both sides by?
- A. 15
- B. 54
- C. 18
- D. 6
Choose the right answer
Show the answer and working
Answer: C. 18
The L.C.M. of 6 and 9 is 18; times 18 clears both denominators.
Example 2
49 x = −9, x =
Answer in simplest form
Show the answer and working
Answer: −814
Multiply both sides by the reciprocal, 94: x = −9 × 94 = −814. Or times 9 first, 4x = −81, then divide by 4.
Example 3
x − 92 − x − 95 = 9, x =
Show the answer and working
Answer: 39
Times 10: 5(x − 9) − 2(x − 9) = 90. The second numerator has a minus in front, so it is a bracket and every term flips. This gives x = 39.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.x9 + 9 = 15 times 9 on both sides is x + 9 = 135.
True or false?
Answer:
2.To clear the denominators of x10 + x15 = 9, what is the handiest number to multiply both sides by?
- A. 25
- B. 30
- C. 150
- D. 10
Choose the right answer
Answer:
3.x2 + 2 = 6: times 2 on both sides, x + = , so x = .
Fill in each blank
Answer:
4.x4 + x5 = −36, x =
Answer:
5.0.6x + 3 = 7.2, x =
Answer:

All the answers

1. False
2. B. 30
3. 4, 12, 8
4. −80
5. 7

Free worksheet: Fractions and decimals in equationsWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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