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GCSE · Algebra

A2 Algebraic Fractions: Adding, Subtracting and Solving Equations

AQA 8365 · Calculator allowed · about 105 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write each expression as a single fraction in its simplest form.

(a)Simplify x/5 + 2x/3.(2)

(b)Simplify (3x - 1)/4 - (x + 2)/6.(3)

(Total for Question 1 is 5 marks)

Solve x/3 + x/5 = 16.

(Total for Question 2 is 3 marks)

Solve 4/(x - 3) = 6/(x + 1), stating any value(s) of x for which the equation is not defined.

(Total for Question 3 is 4 marks)

Write 2/(x + 4) + 5/(x - 2) as a single fraction in its simplest form.

(Total for Question 4 is 3 marks)

Write 5/(x - 1) - 3/(x + 2) as a single fraction in its simplest form.

(Total for Question 5 is 3 marks)

Solve 7/(2x + 1) = 2.

(Total for Question 6 is 3 marks)

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

(Total for Question 7 is 5 marks)

Simplify (x² + 5x + 6)/(x² - 4) fully, stating any value(s) of x for which it is undefined.

(Total for Question 8 is 3 marks)

Write 3/(x² - 9) + 2/(x + 3) as a single fraction in its simplest form.

(Total for Question 9 is 4 marks)

Write 4/(x + 5) - 1/(x² + 5x) as a single fraction in its simplest form.

(Total for Question 10 is 4 marks)

Solve (2x + 3)/(x + 5) = (2x - 1)/(x + 1).

(Total for Question 11 is 4 marks)

Show that 3/(x - 2) - 1/(x + 1) = (2x + 5)/(x² - x - 2), stating any value(s) of x for which the expression is not defined.

(Total for Question 12 is 3 marks)

Solve 6/(x + 1) = x - 4, giving all solutions.

(Total for Question 13 is 4 marks)

Solve 5/(x + 2) + 3/(x - 1) = 4, giving your answers correct to 3 significant figures.

(Total for Question 14 is 5 marks)

Write 2/(x² - x - 6) + 3/(x - 3) as a single fraction in its simplest form.

(Total for Question 15 is 4 marks)

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

(Total for Question 16 is 5 marks)

Solve (x + 3)/(x - 2) - (x - 1)/(x + 2) = 3, giving all solutions as exact fractions.

(Total for Question 17 is 6 marks)

Show that (2x² + 7x + 3)/(2x² - 7x - 4) simplifies to (x + 3)/(x - 4), stating any value(s) of x for which the fraction is undefined.

(Total for Question 18 is 5 marks)

Priya cycles 24 km from Leeds to Harrogate at an average speed of x km/h. Due to a strong headwind on the return journey, her average speed falls to (x - 2) km/h, and the return journey takes 1 hour longer than the outward journey.

(a)Show that x² - 2x - 48 = 0.(3)

(b)Solve the equation to find the value of x, given that x is greater than 2.(3)

(Total for Question 19 is 6 marks)

Write 2/(x + 1) - 3/(x - 1) + 4/(x² - 1) as a single fraction in its simplest form.

(Total for Question 20 is 5 marks)

Mark scheme · A2 Algebraic Fractions: Adding, Subtracting and Solving Equations

Question 1

(a) M1 common denominator 15 used, e.g. 3x/15 + 10x/15
(a) A1 13x/15 oe cao
(a) Answer: 13x/15
(b) M1 common denominator 12 used
(b) M1 numerator 3(3x-1) - 2(x+2) formed and expanded
(b) A1 7(x-1)/12 oe cao, e.g. (7x-7)/12
(b) Answer: 7(x - 1)/12

Question 2

M1 multiply through by common denominator 15: 5x + 3x = 240
M1 collect to 8x = 240
A1 x = 30 cao
Answer: x = 30

Question 3

M1 cross multiply: 4(x + 1) = 6(x - 3)
M1 expand and collect: 4x + 4 = 6x - 18
A1 x = 11 cao
B1 states x is not equal to 3 and x is not equal to -1
Answer: x = 11 (x not equal to 3 or -1)

Question 4

M1 common denominator (x+4)(x-2) used
M1 numerator 2(x-2) + 5(x+4) expanded correctly
A1 (7x + 16)/(x² + 2x - 8) oe cao, e.g. (7x+16)/((x+4)(x-2))
Answer: (7x + 16)/(x² + 2x - 8)

Question 5

M1 common denominator (x-1)(x+2) used
M1 numerator 5(x+2) - 3(x-1) expanded correctly
A1 (2x + 13)/(x² + x - 2) oe cao
Answer: (2x + 13)/(x² + x - 2)

Question 6

M1 multiply both sides by (2x + 1): 7 = 2(2x + 1)
M1 expand and rearrange: 4x = 5
A1 x = 5/4 oe cao, e.g. x = 1.25
Answer: x = 5/4

Question 7

M1 multiply through by (x+1)(x-1) to clear fractions
A1 correct expanded equation 5x - 1 = x² - 1
M1 rearrange to x² - 5x = 0
M1 factorise x(x - 5) = 0
A1 x = 0 and x = 5
Answer: x = 0 or x = 5

Question 8

M1 factorise numerator: (x + 2)(x + 3)
M1 factorise denominator: (x - 2)(x + 2)
A1 (x + 3)/(x - 2) oe cao, with x not equal to 2 or -2
Answer: (x + 3)/(x - 2), x not equal to 2 or -2

Question 9

M1 factorise x² - 9 = (x - 3)(x + 3)
M1 rewrite second fraction with denominator (x-3)(x+3), e.g. 2(x-3)/[(x-3)(x+3)]
M1 combine numerators: 3 + 2(x - 3)
A1 (2x - 3)/(x² - 9) oe cao
Answer: (2x - 3)/(x² - 9)

Question 10

M1 factorise x² + 5x = x(x + 5)
M1 rewrite first fraction with denominator x(x+5), e.g. 4x/[x(x+5)]
M1 combine numerators: 4x - 1
A1 (4x - 1)/(x² + 5x) oe cao
Answer: (4x - 1)/(x² + 5x)

Question 11

M1 cross multiply: (2x + 3)(x + 1) = (2x - 1)(x + 5)
M1 expand both brackets correctly
M1 x² terms cancel, leaving 5x + 3 = 9x - 5
A1 x = 2 cao
Answer: x = 2

Question 12

M1 common denominator (x-2)(x+1) = x² - x - 2 used
M1 numerator 3(x+1) - (x-2) expanded correctly
A1 (2x + 5)/(x² - x - 2) cso, answer printed, with x not equal to 2 or -1
Answer: (2x + 5)/(x² - x - 2) (shown); x not equal to 2 or -1

Question 13

M1 multiply both sides by (x + 1): 6 = (x - 4)(x + 1)
M1 expand and rearrange to x² - 3x - 10 = 0
M1 factorise (x - 5)(x + 2) = 0
A1 x = 5 and x = -2
Answer: x = 5 or x = -2

Question 14

M1 multiply through by (x+2)(x-1) to clear fractions
M1 expand both sides correctly: 8x + 1 = 4x² + 4x - 8
M1 rearrange to 4x² - 4x - 9 = 0
M1 apply quadratic formula with a=4, b=-4, c=-9
A1 x = 2.08 and x = -1.08 (both awrt 3 s.f.)
Answer: x = 2.08 or x = -1.08 (3 s.f.)

Question 15

M1 factorise x² - x - 6 = (x - 3)(x + 2)
M1 rewrite second fraction with denominator (x-3)(x+2), e.g. 3(x+2)/[(x-3)(x+2)]
M1 combine numerators: 2 + 3(x + 2)
A1 (3x + 8)/(x² - x - 6) oe cao
Answer: (3x + 8)/(x² - x - 6)

Question 16

M1 multiply through by (x-1)(x+1) to clear fractions
M1 expand and simplify to a three-term quadratic equal to zero, e.g. x² - x - 6 = 0
M1 factorise (x - 3)(x + 2) = 0
A1 x = 3
A1 x = -2
Answer: x = 3 or x = -2

Question 17

M1 combine LHS over common denominator (x - 2)(x + 2)
M1 expand (x+3)(x+2) and (x-1)(x-2) correctly
A1 numerator simplifies to 8x + 4
M1 form quadratic equation 3x² - 8x - 16 = 0 (or equivalent) from (8x+4)/(x²-4) = 3
M1 factorise (3x + 4)(x - 4) = 0
A1 x = 4 and x = -4/3
Answer: x = 4 or x = -4/3

Question 18

M1 factorise numerator: (2x + 1)(x + 3)
M1 factorise denominator: (2x + 1)(x - 4)
M1 cancel common factor (2x + 1)
A1 (x + 3)/(x - 4) cso, answer printed
B1 states x not equal to -1/2 and x not equal to 4
Answer: (x + 3)/(x - 4) (shown); x not equal to -1/2 or 4

Question 19

(a) M1 form time expressions 24/x (outward) and 24/(x-2) (return)
(a) M1 set up equation 24/(x-2) - 24/x = 1 and multiply through by x(x-2)
(a) A1 x² - 2x - 48 = 0 cso, answer printed
(a) Answer: x² - 2x - 48 = 0 (shown)
(b) M1 factorise (x - 8)(x + 6) = 0
(b) A1 x = 8 and x = -6 found
(b) A1 reject x = -6 (speed must be positive/x > 2); x = 8 km/h cao, with units
(b) Answer: x = 8 km/h

Question 20

M1 identify common denominator (x - 1)(x + 1) for all three terms
M1 combine numerators correctly: 2(x-1) - 3(x+1) + 4
M1 simplify numerator to -x - 1 (or -(x+1))
M1 factorise numerator and cancel common factor (x + 1) with denominator
A1 -1/(x - 1) oe cao, e.g. 1/(1 - x)
Answer: -1/(x - 1)

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