Algebraic expressions · Ages 13–14

Adding and subtracting algebraic fractions worksheet

Part of: Formulae & algebraic fractions

In the print window, choose “Save as PDF” to download it. The answers print on the last page.

Algebraic expressions · Ages 13–14

Numsprite
numsprite.com

Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

13.It is 10 km from home to the hilltop. Up the hill at x km/h, down at 3x km/h. When x = 22, what is the average speed for the round trip, in km/h?
Working:
Answer:
14.Put the steps of working out 9x − 9 − 7x + 9 in order.
- A. 9(x + 9) − 7(x − 9)x² − 81 = 9x + 81 − 7x + 63x² − 81
- B. 9x − 9 − 7x + 9 = 9(x + 9) − 7(x − 9)x² − 81
- C. 9x + 81 − 7x + 63x² − 81 = 2x + 144x² − 81
Write the letters in the right order
Working:
Answer:
15.A cycle to a beach 30 km away at x km/h, and back 4 km/h slower. Total time = 30x + 30x − 4 = x − x(x − 4) hours; when x = 10, it is hours.
Fill in each blank
Working:
Answer:
16.A $120 prize was to be shared by x people, then 2 more joined. Each gets 120x − 120x + 2 = x(x + 2) dollars less; when x = 4, that is dollars less.
Fill in each blank
Working:
Answer:
17.Alone, A finishes a jigsaw in x minutes and B in 2x minutes. Together, each minute they do 1x + 12x = x; when x = 24, together they take minutes.
Fill in each blank
Working:
Answer:
18.A boat does x km/h in still water, and the current runs at 5 km/h. It goes 18 km downstream and comes back. How many hours does the round trip take?
- A. 180x² − 25
- B. 36xx² + 25
- C. 36xx² − 25
- D. 362x
Choose the right answer
Working:
Answer:

Show the answers and working

Adding and subtracting algebraic fractions worksheet: Answers and working

1. 22, 3
The common denominator is 3x: 5x = 153x, and the top is 15 + 7 = 22.
2. 41, 5
The common denominator is 5x: 8x = 405x, and the top is 40 + 1 = 41.
3. −4
6 − x = −(x − 6), so 56 − x = −5x − 6: the top is 1 − 5 = −4.
4. 10, −46
Over the common denominator, the top is 6(x − 9) + 4(x + 2) = 10x − 46.
5. 2
1 − x = −(x − 1), so 21 − x = −2x − 1: the top is 4 − 2 = 2.
6. 2, 19
The common denominator is (x + 5)², not (x + 5)³: the top is 2(x + 5) + 9 = 2x + 19.
7. 3, 30
The common denominator is (x + 9)², not (x + 9)³: the top is 3(x + 9) + 3 = 3x + 30.

8. False
The denominators match, so add the tops only: 8 + 1 = 9; the bottom x + 8 stays.
9. True
The whole top taken away goes in a bracket: 5 − (x − 8) = 5 − x + 8.
10. True
The whole top taken away goes in a bracket: 5 − (x − 5) = 5 − x + 5.
11. B. 5x + 23(x − 5)(x + 1)
Top = 8(x + 1) − 3(x − 5) = 8x + 8 − 3x + 15 = 5x + 23.
12. A. 3x + 39(x − 9)(x + 2)
Top = 6(x + 2) − 3(x − 9) = 6x + 12 − 3x + 27 = 3x + 39.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 33 km/h
Total time = 10x + 103x, total distance 20 km. At x = 22: average speed = 20 ÷ (1022 + 1066) = 33 km/h, not (22 + 66) ÷ 2.
14. B → A → C
Bring them over the common denominator (x − 9)(x + 9) = x² − 81; open the top, every term of −7(x − 9) changing sign; then collect like terms.
15. 60, 120, 8
Over the common denominator, the top is 30(x − 4) + 30x = 60x − 120. At x = 10: 3010 + 3010 − 4 = 8 hours.
16. 240, 10
The top is 120(x + 2) − 120x = 240. At x = 4: 1204 − 1204 + 2 = 10 dollars.
17. 3, 2, 16
1x = 22x, so each minute they do 32x. At x = 24 that is 348 a minute, so together 48 ÷ 3 = 16 minutes.
18. C. 36xx² − 25
Downstream x + 5, upstream x − 5: 18x + 5 + 18x − 5 = 18(x − 5) + 18(x + 5)x² − 25 = 36xx² − 25.

Numsprite · numsprite.com/en/worksheets/formulae-and-algebraic-fractions/adding-and-subtracting-algebraic-fractions