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GCSE Level 2 Certificate · Algebra

A1 Algebraic Fractions: Simplifying, Multiplying and Dividing

AQA 8365 · Calculator allowed · about 100 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Which of the following is the fully simplified form of (15x^3y^2)/(20x^5y^4)?

A) 3/(4x^2y^2)
B) 3x^2y^2/4
C) 3/(4xy²)
D) 15/(20x^2y^2)

(Total for Question 1 is 1 mark)

Simplify each fraction fully.

(a)Simplify fully (24x^6y^3)/(16x^2y^5)(2)

(b)Simplify fully (18a^5b^2)/(27a^2b^7)(2)

(Total for Question 2 is 4 marks)

Simplify each fraction fully by factorising first.

(a)Simplify fully (5x+15)/(x²+3x)(3)

(b)Simplify fully (8x-24)/(x²-3x)(3)

(Total for Question 3 is 6 marks)

Simplify fully (x²+6x+8)/(x²+2x-8)

(Total for Question 4 is 3 marks)

Simplify each fraction fully, using difference of two squares where needed.

(a)Simplify fully (x²-49)/(x²+4x-21)(3)

(b)Simplify fully (9x²-4)/(3x²+4x-4)(3)

(Total for Question 5 is 6 marks)

Simplify each product fully.

(a)Simplify fully (4x/3y) x (9y³/8x²)(3)

(b)Simplify fully (10a²/7b³) x (14b⁵/25a⁴)(3)

(Total for Question 6 is 6 marks)

Simplify each quotient fully.

(a)Simplify fully (6x³/5y²) / (9x/10y)(3)

(b)Simplify fully (8a⁴/3b) / (12a/9b⁴)(3)

(Total for Question 7 is 6 marks)

Simplify fully (x+4)/(x²-9) x (x-3)/(3x+12)

(Total for Question 8 is 4 marks)

Simplify fully (x²-4)/(x+5) / (x-2)/(4x+20)

(Total for Question 9 is 4 marks)

Priya is asked to simplify (x²-16)/(x-4) and writes the answer as x-16.

(a)Explain the mistake Priya has made.(1)

(b)Work out the correct fully simplified answer.(2)

(Total for Question 10 is 3 marks)

Show that (2x²+9x+4)/(x²-16) simplifies to (2x+1)/(x-4)

(Total for Question 11 is 3 marks)

Simplify fully (10-2x)/(x²-25)

(Total for Question 12 is 3 marks)

Simplify fully (2x²-5x-3)/(x²-9)

(Total for Question 13 is 3 marks)

Simplify fully (8+2x-x²)/(x²-x-12)

(Total for Question 14 is 4 marks)

Simplify fully (x²+7x+12)/(x²-16) x (x²-4x)/(x²+x-6)

(Total for Question 15 is 4 marks)

Simplify fully (2x²-3x-2)/(x²-4) / (2x+1)/(x-2)

(Total for Question 16 is 4 marks)

Simplify fully (x⁴-81)/(x²-9)

(Total for Question 17 is 4 marks)

A rectangular tile has area (x²-4)/(x+3) cm² and length (x-2)/(x²-9) cm. Find an expression for the width of the tile in its simplest form.

(Total for Question 18 is 5 marks)

(x²-2x-15)/(x²-25) x (x²+5x)/(x²+8x+15)

(a)Simplify the expression above fully.(5)

(b)Hence find the value of the expression when x=10, giving your answer as a fraction in its simplest form.(2)

(Total for Question 19 is 7 marks)

Simplify fully (x²-6x+9)/(x²-9) / (3-x)/(x+3)

(Total for Question 20 is 5 marks)

Mark scheme · A1 Algebraic Fractions: Simplifying, Multiplying and Dividing

Question 1

B1 correct option identified (A) oe
Answer: A) 3/(4x^2y^2)

Question 2

(a) M1 numeric coefficient simplified to 3/2 oe
(a) A1 3x⁴/(2y²) cao
(a) Answer: 3x⁴/(2y²)
(b) M1 numeric coefficient simplified to 2/3 oe
(b) A1 2a³/(3b⁵) cao
(b) Answer: 2a³/(3b⁵)

Question 3

(a) M1 numerator factorised as 5(x+3)
(a) M1 denominator factorised as x(x+3)
(a) A1 5/x cao
(a) Answer: 5/x
(b) M1 numerator factorised as 8(x-3)
(b) M1 denominator factorised as x(x-3)
(b) A1 8/x cao
(b) Answer: 8/x

Question 4

M1 numerator factorised as (x+2)(x+4)
M1 denominator factorised as (x+4)(x-2)
A1 (x+2)/(x-2) cao
Answer: (x+2)/(x-2)

Question 5

(a) M1 numerator factorised as (x-7)(x+7)
(a) M1 denominator factorised as (x+7)(x-3)
(a) A1 (x-7)/(x-3) cao
(a) Answer: (x-7)/(x-3)
(b) M1 numerator factorised as (3x-2)(3x+2)
(b) M1 denominator factorised as (3x-2)(x+2)
(b) A1 (3x+2)/(x+2) cao
(b) Answer: (3x+2)/(x+2)

Question 6

(a) M1 multiplies numerators and denominators: 36xy³/(24x^2y) oe
(a) M1 numeric coefficient simplified to 3/2 oe
(a) A1 3y²/(2x) cao
(a) Answer: 3y²/(2x)
(b) M1 multiplies numerators and denominators: 140a^2b^5/(175a^4b^3) oe
(b) M1 numeric coefficient simplified to 4/5 oe
(b) A1 4b²/(5a²) cao
(b) Answer: 4b²/(5a²)

Question 7

(a) M1 inverts second fraction and multiplies: (6x³/5y²)(10y/9x) oe
(a) M1 numeric coefficient simplified to 4/3 oe
(a) A1 4x²/(3y) cao
(a) Answer: 4x²/(3y)
(b) M1 inverts second fraction and multiplies: (8a⁴/3b)(9b⁴/12a) oe
(b) M1 numeric coefficient simplified to 2 oe
(b) A1 2a^3b^3 cao
(b) Answer: 2a^3b^3

Question 8

M1 x²-9 factorised as (x-3)(x+3)
M1 3x+12 factorised as 3(x+4)
M1 cancels (x+4) and (x-3) correctly
A1 1/(3x+9) oe cao
Answer: 1/(3x+9)

Question 9

M1 inverts second fraction and factorises x²-4 as (x-2)(x+2)
M1 4x+20 factorised as 4(x+5)
M1 cancels (x-2) and (x+5) correctly
A1 4x+8 oe cao
Answer: 4x+8

Question 10

(a) B1 identifies that x²-16 must first be factorised as (x-4)(x+4) as a difference of two squares before any cancelling, rather than cancelling the 16 and the 4 as separate terms oe
(a) Answer: Priya cancelled individual terms (16 and 4) instead of factorising the numerator first.
(b) M1 factorises x²-16 as (x-4)(x+4)
(b) A1 x+4 cao
(b) Answer: x+4

Question 11

M1 numerator factorised as (2x+1)(x+4)
M1 denominator factorised as (x-4)(x+4)
A1 cancels (x+4) correctly to reach (2x+1)/(x-4) cso
Answer: (2x+1)/(x-4)

Question 12

M1 numerator factorised as -2(x-5)
M1 denominator factorised as (x-5)(x+5)
A1 -2/(x+5) oe cao
Answer: -2/(x+5)

Question 13

M1 numerator factorised as (2x+1)(x-3)
M1 denominator factorised as (x-3)(x+3)
A1 (2x+1)/(x+3) cao
Answer: (2x+1)/(x+3)

Question 14

M1 rewrites numerator as -(x²-2x-8)
M1 factorises x²-2x-8 as (x-4)(x+2) and x²-x-12 as (x-4)(x+3)
M1 cancels (x-4) correctly, retaining the negative sign
A1 -(x+2)/(x+3) oe cao
Answer: -(x+2)/(x+3)

Question 15

M1 factorises x²+7x+12 as (x+3)(x+4) and x²-16 as (x-4)(x+4)
M1 factorises x²-4x as x(x-4) and x²+x-6 as (x+3)(x-2)
M1 cancels (x+3), (x+4) and (x-4) correctly
A1 x/(x-2) oe cao
Answer: x/(x-2)

Question 16

M1 inverts second fraction and factorises 2x²-3x-2 as (2x+1)(x-2)
M1 factorises x²-4 as (x-2)(x+2)
M1 cancels (2x+1) and one factor of (x-2) correctly
A1 (x-2)/(x+2) oe cao
Answer: (x-2)/(x+2)

Question 17

M1 recognises x⁴-81 as a difference of two squares (x²)² - 9²
M1 factorises to (x²-9)(x²+9)
M1 cancels (x²-9) with the denominator
A1 x²+9 cao
Answer: x²+9

Question 18

M1 sets up width = area / length
M1 factorises x²-4 as (x-2)(x+2) and x²-9 as (x-3)(x+3)
M1 inverts the length fraction and multiplies correctly
M1 cancels (x-2) and (x+3) correctly
A1 (x+2)(x-3) oe, or x²-x-6, cao
Answer: (x+2)(x-3), which expands to x²-x-6

Question 19

(a) M1 factorises x²-2x-15 as (x-5)(x+3) and x²-25 as (x-5)(x+5)
(a) M1 factorises x²+5x as x(x+5) and x²+8x+15 as (x+3)(x+5)
(a) M1 cancels (x-5) and (x+3) correctly
(a) M1 cancels one remaining factor of (x+5) correctly
(a) A1 x/(x+5) oe cao
(a) Answer: x/(x+5)
(b) M1 substitutes x=10 into x/(x+5) ft from part (a)
(b) A1 2/3 oe cao
(b) Answer: 2/3

Question 20

M1 factorises x²-6x+9 as (x-3)²
M1 factorises x²-9 as (x-3)(x+3)
M1 recognises 3-x = -(x-3)
M1 inverts the second fraction and multiplies, cancelling (x-3)² and (x+3) correctly
A1 -1 cao
Answer: -1

Algebraic Fractions: Simplifying, Multiplying and Dividing - Worksheets, Questions and Revision