Algebraic Fractions: Adding, Subtracting and Solving Equations: Fluency and Exam Drill - Worksheets, Questions and Revision

19 original exam-style questions - 4 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Algebra

A2D Algebraic Fractions: Adding, Subtracting and Solving Equations: Fluency and Exam Drill

AQA 8365 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Simplify x/4 + 3x/5

(Total for Question 1 is 1 mark)

Simplify 2x/3 - x/6

(Total for Question 2 is 1 mark)

Solve x/4 + x/6 = 10

(Total for Question 3 is 2 marks)

Solve 5/(x + 2) = 1

(Total for Question 4 is 1 mark)

Solve 8/(2x - 1) = 2

(Total for Question 5 is 1 mark)

Write 3/(x + 2) + 4/(x - 1) as a single fraction in its simplest form.

(Total for Question 6 is 1 mark)

Write 6/(x - 3) - 2/(x + 4) as a single fraction in its simplest form.

(Total for Question 7 is 1 mark)

Simplify (x² + 2x - 8)/(x² - 16) fully, stating any value(s) of x for which it is undefined.

(Total for Question 8 is 2 marks)

Write 4/(x² - 16) + 3/(x + 4) as a single fraction in its simplest form.

(Total for Question 9 is 2 marks)

Write 5/(x + 3) - 2/(x² + 3x) as a single fraction in its simplest form.

(Total for Question 10 is 2 marks)

Solve (3x + 1)/(x + 4) = (3x - 2)/(x + 1)

(Total for Question 11 is 2 marks)

Solve 4/(x + 1) = 3

(Total for Question 12 is 1 mark)

Simplify each fraction fully.

(a)Simplify (3x - 2)/6 - (x - 1)/4(1)

(b)Simplify (2x + 3)/5 + (x - 2)/10(1)

(Total for Question 13 is 2 marks)

Solve 5/(x - 1) - 3/(x + 1) = 1, giving your answers in exact surd form.

(Total for Question 14 is 3 marks)

Solve 6/(x - 2) - 4/(x + 2) = 1

(Total for Question 15 is 3 marks)

Solve 7/(x - 1) = x + 5, giving all solutions.

(Total for Question 16 is 3 marks)

Show that 4/(x - 3) - 1/(x + 2) = (3x + 11)/(x² - x - 6), stating any value(s) of x for which the expression is not defined.

(Total for Question 17 is 3 marks)

Solve (x + 4)/(x - 1) - (x - 2)/(x + 1) = 2, giving all solutions in exact surd form.

(Total for Question 18 is 4 marks)

Maya cycles 60 km from her home to a lake at an average speed of x km/h. Due to a strong headwind on the return journey, her average speed falls to (x - 3) km/h, and the return journey takes 1 hour longer than the outward journey.

(a)Show that x² - 3x - 180 = 0.(2)

(b)Solve the equation to find the value of x, given that x is greater than 3.(3)

(Total for Question 19 is 5 marks)

Mark scheme · A2D Algebraic Fractions: Adding, Subtracting and Solving Equations: Fluency and Exam Drill

Question 1

B1 17x/20 cao
Answer: 17x/20

Question 2

B1 x/2 cao
Answer: x/2

Question 3

M1 multiply through by common denominator 12: 3x + 2x = 120
A1 x = 24 cao
Answer: x = 24

Question 4

B1 x = 3 cao
Answer: x = 3

Question 5

B1 x = 5/2 oe cao, e.g. x = 2.5
Answer: x = 5/2

Question 6

B1 (7x+5)/(x²+x-2) oe cao
Answer: (7x + 5)/(x² + x - 2)

Question 7

B1 (4x+30)/(x²+x-12) oe cao
Answer: (4x + 30)/(x² + x - 12)

Question 8

M1 factorise numerator: (x - 2)(x + 4); factorise denominator: (x - 4)(x + 4)
A1 (x - 2)/(x - 4) oe cao, with x not equal to 4 or -4
Answer: (x - 2)/(x - 4), x not equal to 4 or -4

Question 9

M1 factorise x²-16 = (x-4)(x+4); rewrite second fraction with denominator (x-4)(x+4)
A1 (3x - 8)/(x² - 16) oe cao
Answer: (3x - 8)/(x² - 16)

Question 10

M1 factorise x²+3x = x(x+3); rewrite first fraction with denominator x(x+3)
A1 (5x - 2)/(x² + 3x) oe cao
Answer: (5x - 2)/(x² + 3x)

Question 11

M1 cross multiply and expand both brackets: (3x+1)(x+1) = (3x-2)(x+4)
A1 x = 3/2 oe cao, e.g. x = 1.5
Answer: x = 3/2

Question 12

B1 x = 1/3 oe cao
Answer: x = 1/3

Question 13

(a) B1 (3x - 1)/12 cao
(a) Answer: (3x - 1)/12
(b) B1 (5x + 4)/10 cao
(b) Answer: (5x + 4)/10

Question 14

M1 multiply through by (x-1)(x+1) to clear fractions: 5(x+1) - 3(x-1) = (x-1)(x+1)
M1 expand and rearrange to x² - 2x - 9 = 0
A1 x = 1 + √10 and x = 1 - √10 oe
Answer: x = 1 + √10 or x = 1 - √10

Question 15

M1 multiply through by (x-2)(x+2) to clear fractions: 6(x+2) - 4(x-2) = x² - 4
M1 expand and simplify to x² - 2x - 24 = 0, then factorise (x - 6)(x + 4) = 0
A1 x = 6 and x = -4 (both values) cao
Answer: x = 6 or x = -4

Question 16

M1 multiply both sides by (x - 1): 7 = (x + 5)(x - 1)
M1 expand and rearrange to x² + 4x - 12 = 0
A1 factorise (x + 6)(x - 2) = 0, giving x = -6 and x = 2
Answer: x = -6 or x = 2

Question 17

M1 common denominator (x-3)(x+2) = x² - x - 6 used
M1 numerator 4(x+2) - (x-3) expanded correctly
A1 (3x + 11)/(x² - x - 6) cso, answer printed, with x not equal to 3 or -2
Answer: (3x + 11)/(x² - x - 6) (shown); x not equal to 3 or -2

Question 18

M1 combine LHS over common denominator (x - 1)(x + 1), expanding (x+4)(x+1) and (x-2)(x-1) correctly
A1 numerator simplifies to 8x + 2
M1 form and solve the equation (8x+2)/(x²-1) = 2, i.e. 2x² - 8x - 4 = 0, i.e. x² - 4x - 2 = 0
A1 x = 2 + √6 and x = 2 - √6
Answer: x = 2 + √6 or x = 2 - √6

Question 19

(a) M1 sets up equation 60/(x-3) - 60/x = 1 and multiplies through by x(x-3)
(a) A1 x² - 3x - 180 = 0 cso, answer printed
(a) Answer: x² - 3x - 180 = 0 (shown)
(b) M1 factorise (x - 15)(x + 12) = 0
(b) A1 x = 15 and x = -12 found
(b) A1 reject x = -12 (speed must be positive/x > 3); x = 15 km/h cao, with units
(b) Answer: x = 15 km/h

Question 19

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Algebraic Fractions: Adding, Subtracting and Solving Equations: Fluency and Exam Drill - Worksheets, Questions and Revision

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A2D Algebraic Fractions: Adding, Subtracting and Solving Equations: Fluency and Exam Drill

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