Formulae & algebraic fractions · Ages 13–14
Adding and subtracting algebraic fractions
Common denominator first; bracket a top that is taken away.
How to do it
Method
Common denominator first; bracket a top that is taken away
2x + 1 + 3x − 2 = 5x − 1(x + 1)(x − 2)
Another way
Check with a number
x = 3: 24 + 31 = 72, 144 × 1 = 72
In full
Bring them to a common denominator: the L.C.M. of the denominators. Multiply each fraction's top and bottom by the same expression, then add or subtract the tops. For 2x + 1 + 3x − 2 it is (x + 1)(x − 2), and the top is 2(x − 2) + 3(x + 1) = 5x − 1. When a fraction is taken away, its top goes in a bracket and every term changes sign: (x + 5) − (x − 2) = 7, not 3. When two denominators differ only by a sign (x − 2 and 2 − x), write 2 − x as −(x − 2) first, and the minus turns into a plus. Then check with a number.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
1x + 97x = x
Fill in each blank
Show the answer and working
Answer: 16, 7
The common denominator is 7x: 1x = 77x, and the top is 7 + 9 = 16.
Example 2
5x + 7 + 8x + 7 = 13x + 7.
True or false?
Show the answer and working
Answer: True
The denominators match, so add the tops only: 5 + 8 = 13; the bottom x + 7 stays.
Example 3
6x + 5 + 3x − 7 = x + (x + 5)(x − 7)
Fill in each blank
Show the answer and working
Answer: 9, −27
Over the common denominator, the top is 6(x − 7) + 3(x + 5) = 9x − 27.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.4x + 8 + 3x + 8 = 7x + 8.
True or false?
Answer:
2.6x − 8 + 78 − x = x − 8
Answer:
3.2x + 7 + 2x − 4 = x + (x + 7)(x − 4)
Fill in each blank
Answer:
4.7x + 1 + 5(x + 1)2 = x + (x + 1)2
Fill in each blank
Answer:
5.What is 9x − 6 − 1x + 3?
- A. 82x − 3
- B. 8(x − 6)(x + 3)
- C. 8x + 33(x − 6)(x + 3)
- D. 8x + 21(x − 6)(x + 3)
Choose the right answer
Answer:
All the answers
- 1. True
- 2. −1
- 3. 4, 6
- 4. 7, 12
- 5. C. 8x + 33(x − 6)(x + 3)
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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