IGCSE · 4MA1 · Topic

Algebraic Fractions: past paper questions & worked solutions

Every Algebraic Fractions question from the 4MA1 papers in our bank, 2020–2026, sorted from easiest to hardest. Try each one first, then open the worked solution.

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28questions

108marks in total

46%of papers (25 of 54) include it

0/21/7easy / medium / hard

Questions, easiest first

Level is estimated from marks: 1–2 easy, 3–4 medium, 5+ hard.

1. Jun 2026 P1H · Q18

Medium3 marks

Jun 2026 P1H Question 18 (3 marks): Algebraic Fractions – 18 Simplify fully 4 9 2 5 12 2 2 x x x − − − Show clear algebraic working. (Total for Question 18 is 3 marks)

Show worked solution

Solution | Step 1
\[4x^2-9=(2x-3)(2x+3),\qquad2x^2-5x-12=(2x+3)(x-4).\]
Use difference of squares for the numerator and factorise the denominator.
Solution | Step 2
\[\frac{4x^2-9}{2x^2-5x-12}=\frac{(2x-3)(2x+3)}{(2x+3)(x-4)}=\frac{2x-3}{x-4}.\]
Cancel the common factor, not individual terms.
Solution | Step 3
The original fraction requires $x\ne-\frac32$ and $x\ne4$.
Cancellation does not restore excluded values.

Final answer

$\frac{2x-3}{x-4}$, $x\ne-\frac32,4$.

2. Nov 2025 P1H · Q23

Medium3 marks

Nov 2025 P1H Question 23 (3 marks): Algebraic Fractions – 23 a x x x y y 2 5 1 5 2 3 Write a in the form m ny p y ( )1 where m, n and p are integers. Show your working clearly. a = (Total for Question 23 is 3 marks)

Show worked solution

Split the ratio
\[ a=\frac{2x+5}{1-x},\quad x=\frac{5-2y}{3y} \]
Split the ratio
Substitute for $x$.
Numerator
Numerator:
\[ 2x+5=2\left(\frac{5-2y}{3y}\right)+5 \]
\[ =\frac{10-4y+15y}{3y} =\frac{10+11y}{3y} \]
Denominator
Denominator:
\[ 1-x=1-\frac{5-2y}{3y} \]
\[ =\frac{3y-5+2y}{3y} =\frac{5y-5}{3y} =\frac{5(y-1)}{3y} \]
Split the ratio
Therefore
\[ a=\frac{\frac{10+11y}{3y}}{\frac{5(y-1)}{3y}} =\frac{10+11y}{5(y-1)} \]

Final answer

$\dfrac{10+11y}{5(y-1)}$

3. Nov 2025 4WM1H · Q25

Medium3 marks

Nov 2025 4WM1H Question 25 (3 marks): Algebraic Fractions – 25 a x x x y y 2 5 1 5 2 3 Write a in the form m ny p y ( )1 where m, n and p are integers. Show your working clearly. a = (Total for Question 25 is 3 marks)

Show worked solution

Split the ratio
\[ a=\frac{2x+5}{1-x},\quad x=\frac{5-2y}{3y} \]
Split the ratio
Substitute for $x$.
Numerator
Numerator:
\[ 2x+5=2\left(\frac{5-2y}{3y}\right)+5 \]
\[ =\frac{10-4y+15y}{3y} =\frac{10+11y}{3y} \]
Denominator
Denominator:
\[ 1-x=1-\frac{5-2y}{3y} \]
\[ =\frac{3y-5+2y}{3y} =\frac{5y-5}{3y} =\frac{5(y-1)}{3y} \]
Split the ratio
Therefore
\[ a=\frac{\frac{10+11y}{3y}}{\frac{5(y-1)}{3y}} =\frac{10+11y}{5(y-1)} \]

Final answer

$\dfrac{10+11y}{5(y-1)}$

4. Jun 2025 P1H · Q12

Medium3 marks

$Jun 2025 P1H Question 12 (3 marks): Algebraic Fractions – 12 Express 5 4 3 6 x x as a single fraction in its simplest form. (Total for Question 12 is 3 marks)$

Show worked solution

Simplify fraction
\[ \frac54+\frac{x-3}{6x} \]
The common denominator is
The common denominator is $12x$:
\[ \frac54=\frac{15x}{12x} \]
\[ \frac{x-3}{6x}=\frac{2x-6}{12x} \]
\[ \frac{15x}{12x}+\frac{2x-6}{12x} =\frac{17x-6}{12x} \]

Final answer

$\frac{17x-6}{12x}$.

5. May 2025 4WM1H · Q19

Medium3 marks

May 2025 4WM1H Question 19 (3 marks): Algebraic Fractions – 19 Simplify fully 3 3 5 5 10 2 2 2 2 x y x y xy x y xy Show clear algebraic working. (Total for Question 19 is 3 marks)

Show worked solution

Simplify fraction
\[ \frac{3x^2-3y^2}{5x+5y}\div\frac{xy^2-x^2y}{10xy} \]
Factorise
Factorise.
\[ =\frac{3(x-y)(x+y)}{5(x+y)}\div\frac{xy(y-x)}{10xy} \]
\[ =\frac{3(x-y)}5\div\frac{y-x}{10} \]
\[ =\frac{3(x-y)}5\times\frac{10}{y-x} \]
Simplify fraction
Since $y-x=-(x-y)$, the expression is
\[ -6 \]

Final answer

$-6$

6. May 2025 4WM1H · Q20

Medium3 marks

May 2025 4WM1H Question 20 (3 marks): Algebraic Fractions – 20 Given that k m n and m n = 3 4 express 7 5 k m − in the form a bn cn d 2 where a, b and c are integers and d is prime. (Total for Question 20 is 3 marks)

Show worked solution

Simplify fraction
\[ k=m+n,\qquad m=\frac{3}{4n} \]
Simplify fraction
So
\[ k=\frac{3}{4n}+n=\frac{3+4n^2}{4n} \]
Simplify fraction
Then
\[ 7k=\frac{21+28n^2}{4n} \]
Simplify fraction
and
\[ 5-m=5-\frac{3}{4n}=\frac{20n-3}{4n} \]
Simplify fraction
Therefore
\[ \frac{7k}{5-m}=\frac{\frac{21+28n^2}{4n}}{\frac{20n-3}{4n}} \]
\[ =\frac{21+28n^2}{20n-3} \]

Final answer

$\frac{21+28n^2}{20n-3}$

7. Jun 2024 P2H · Q20

Medium3 marks

Jun 2024 P2H Question 20 (3 marks): Algebraic Fractions – 20 Given that k = x - y and x = 1 4y express 5 2 k x + in the form a by c dy 2 where a, b, c and d are integers. (Total for Question 20 is 3 marks)

Show worked solution

Simplify fraction
\[ k=x-y,\quad x=\frac{1}{4y} \]
Simplify fraction
So
\[ k=\frac{1}{4y}-y=\frac{1-4y^2}{4y} \]
Simplify fraction
Also
\[ x+2=\frac{1}{4y}+2=\frac{1+8y}{4y} \]
Simplify fraction
Therefore
\[ \frac{5k}{x+2} =\frac{5\left(\frac{1-4y^2}{4y}\right)}{\frac{1+8y}{4y}} \]
Cancel
Cancel $4y$:
\[ \frac{5(1-4y^2)}{1+8y} \]
\[ =\frac{5-20y^2}{1+8y} \]

Final answer

$\dfrac{5-20y^2}{1+8y}$

8. Nov 2023 P1H · Q11

Medium3 marks

$Nov 2023 P1H Question 11 (3 marks): Algebraic Fractions – 11 Express 3 4 5 6 x x as a single fraction in its simplest terms. (Total for Question 11 is 3 marks)$

Show worked solution

Simplify fraction
\[ \frac34+\frac{5-x}{6x} \]
The common denominator is
The common denominator is $12x$:
\[ \frac34=\frac{9x}{12x} \]
\[ \frac{5-x}{6x}=\frac{10-2x}{12x} \]
\[ \frac{9x+10-2x}{12x} =\frac{7x+10}{12x} \]

Final answer

$\frac{7x+10}{12x}$.

9. Jun 2022 P2H · Q19

Medium3 marks

Jun 2022 P2H Question 19 (3 marks): Algebraic Fractions – 19 a = 14 3 7 x - x = 7 4 3 y - Express a in the form py q ry s + + where p, q, r and s are integers. Give your answer in its simplest form. a = (Total for Question 19 is 3 marks)

Show worked solution

Simplify fraction
\[ a=\frac{14}{3x-7}, \qquad x=\frac{7}{4y-3} \]
Simplify fraction
Substitute for $x$: \[ 3x-7=\frac{21}{4y-3}-7 \]
\[ =\frac{21-7(4y-3)}{4y-3} =\frac{42-28y}{4y-3} \]
Simplify fraction
So \[ a=\frac{14}{\frac{42-28y}{4y-3}} =\frac{4y-3}{3-2y} \]

Final answer

$a=\frac{4y-3}{3-2y}$

10. Jun 2022 P2HR · Q23

Medium3 marks

$Jun 2022 P2HR Question 23 (3 marks): Algebraic Fractions – 23 Express 20 36 2 6 1 4 2 x x x - - -    × - as a single fraction in its simplest form. (Total for Question 23 is 3 marks)$

Show worked solution

Use trigonometry
\[ \left(\frac{20}{x^2-36}-\frac2{x-6}\right)\times\frac1{4-x} \]
Factor
Factor:
\[ x^2-36=(x-6)(x+6) \]
Use trigonometry
So
\[ \frac{20}{(x-6)(x+6)}-\frac2{x-6} \]
Use the common denominator
Use the common denominator $(x-6)(x+6)$:
\[ \frac{20-2(x+6)}{(x-6)(x+6)} \]
\[ =\frac{20-2x-12}{(x-6)(x+6)} \]
\[ =\frac{8-2x}{(x-6)(x+6)} \]
\[ =\frac{-2(x-4)}{(x-6)(x+6)} \]
Use trigonometry
Now multiply by
\[ \frac1{4-x}=-\frac1{x-4} \]
\[ \frac{-2(x-4)}{(x-6)(x+6)}\times\left(-\frac1{x-4}\right) \]
Cancel
Cancel $x-4$:
\[ \frac{2}{(x-6)(x+6)} \]
\[ =\frac2{x^2-36} \]

Final answer

$\displaystyle \boxed{\frac2{x^2-36}}$

11. Specimen 4WM1H · Q21

Medium3 marks

$Specimen 4WM1H Question 21 (3 marks): Algebraic Fractions – 21 Express (20/(x^2 - 36) - 2/(x - 6)) x 1/(4 - x) as a single fraction in its simplest form. (Total for Question 21 is 3 marks)$

Show worked solution

Factorise the denominator
\[ x^2-36=(x-6)(x+6) \]
Simplify the bracket
\[ \frac{20}{(x-6)(x+6)}-\frac{2}{x-6} =\frac{20-2(x+6)}{(x-6)(x+6)} \]
\[ =\frac{8-2x}{(x-6)(x+6)} =\frac{-2(x-4)}{(x-6)(x+6)} \]
Cancel the common factor
Since $4-x=-(x-4)$,
\[ \frac{-2(x-4)}{(x-6)(x+6)}\times\frac{1}{4-x} =\frac{2}{(x-6)(x+6)} \]

Final answer

$\frac{2}{x^2-36}$.

12. Specimen 4WM2H · Q21

Medium3 marks

Specimen 4WM2H Question 21 (3 marks): Algebraic Fractions – 21 a = 14/(3x - 7) and x = 7/(4y - 3). Express a in the form (py + q)/(ry + s), where p, q, r and s are integers. Give your answer in its simplest form. (Total for Question 21 is 3 marks)

Show worked solution

Simplify fraction
\[ a=\frac{14}{3x-7},\qquad x=\frac{7}{4y-3} \]
Simplify fraction
Substitute for $x$:
\[ a=\frac{14}{3\left(\frac{7}{4y-3}\right)-7} \]
Simplify the denominator
Simplify the denominator:
\[ 3\left(\frac{7}{4y-3}\right)-7 =\frac{21}{4y-3}-7 \]
\[ =\frac{21-7(4y-3)}{4y-3} =\frac{42-28y}{4y-3} \]
Simplify fraction
So
\[ a=\frac{14}{\frac{42-28y}{4y-3}} =\frac{14(4y-3)}{42-28y} \]
\[ a=\frac{4y-3}{3-2y} \]

Final answer

$\displaystyle a=\frac{4y-3}{3-2y}$.

13. May 2026 4WM1H · Q20

Medium4 marks

$May 2026 4WM1H Question 20 (4 marks): Algebraic Fractions – 20 Express 5 4 7 3 3 5 9 16 6 7 5 2 2 x x x x x as a single fraction in its simplest form. Show clear algebraic working. (Total for Question 20 is 4 marks)$

Show worked solution

Subtract the fractions
\[\frac{5(3x-5)-3(4x-7)}{(4x-7)(3x-5)}=\frac{3x-4}{(4x-7)(3x-5)}\]
Factorise the second fraction
\[9x^2-16=(3x-4)(3x+4),\qquad 6x^2-7x-5=(3x-5)(2x+1)\]
Divide = multiply by the reciprocal
\[\frac{3x-4}{(4x-7)(3x-5)}\times\frac{(3x-5)(2x+1)}{(3x-4)(3x+4)}\]
Cancel
\[=\frac{2x+1}{(4x-7)(3x+4)}\]

Final answer

$\dfrac{2x+1}{(4x-7)(3x+4)}$

14. Nov 2025 4WM1H · Q28

Medium4 marks

Nov 2025 4WM1H Question 28 (4 marks): Algebraic Fractions – 28 Solve 6 5 4 1 15 2 8 4 2 3 2 2 2 x x x x x x x x Show clear algebraic working. x = (Total for Question 28 is 4 marks)

Show worked solution

Factor each expression
Factor each expression:
\[ \frac{x(3x+1)(2x+1)}{(2x-1)(2x+1)}\div\frac{(5x-2)(3x+1)}{4(2x-1)}=2x \]
\[ \frac{x(3x+1)}{2x-1}\times\frac{4(2x-1)}{(5x-2)(3x+1)}=2x \]
\[ \frac{4x}{5x-2}=2x \]
\[ 4x=2x(5x-2) \]
\[ 10x^2-8x=0 \]
\[ 2x(5x-4)=0 \]
The original fractions do not
The original fractions do not allow $x=0$, so
\[ x=\frac45 \]

Final answer

$x=\frac45$.

15. Jun 2025 P2H · Q23

Medium4 marks

Jun 2025 P2H Question 23 (4 marks): Algebraic Fractions – 23 4 4 120 5 180 5 10 60 2 2 2 2 x x x x x x x p where p is an integer. Find the value of p Show clear algebraic working. p = (Total for Question 23 is 4 marks)

Show worked solution

Split the ratio
\[ \frac{4x^2-4x-120}{5x^2-180}\div \frac{x^2+5x}{10x^2+60x} \]
Factorise
Factorise:
\[ 4x^2-4x-120=4(x-6)(x+5) \]
\[ 5x^2-180=5(x-6)(x+6) \]
\[ x^2+5x=x(x+5) \]
\[ 10x^2+60x=10x(x+6) \]
Split the ratio
So
\[ \frac{4(x-6)(x+5)}{5(x-6)(x+6)} \div \frac{x(x+5)}{10x(x+6)} \]
\[ =\frac{4(x+5)}{5(x+6)}\times \frac{10(x+6)}{x+5}=8 \]

Final answer

$p=8$

16. May 2025 4WM1HR · Q24

Medium4 marks

$May 2025 4WM1HR Question 24 (4 marks): Algebraic Fractions – 24 Write 2 1 2 5 12 16 4 6 11 3 2 2 2 2 x x x x x x x x as a single fraction in its simplest form. Show clear algebraic working. (Total for Question 24 is 4 marks)$

Show worked solution

Factor the fractional part first
Factor the fractional part first:
\[ \frac{2x^2-5x-12}{x^2-16}\times\frac{x^2+4x}{6x^2+11x+3} \]
\[ =\frac{(2x+3)(x-4)}{(x-4)(x+4)}\times\frac{x(x+4)}{(3x+1)(2x+3)} \]
\[ =\frac{x}{3x+1} \]
Evaluate fraction
So
\[ (2x-1)+\frac{x}{3x+1} =\frac{(2x-1)(3x+1)+x}{3x+1} \]
\[ =\frac{6x^2-1}{3x+1} \]

Final answer

$\dfrac{6x^2-1}{3x+1}$.

17. Jun 2024 P2HR · Q26

Medium4 marks

$Jun 2024 P2HR Question 26 (4 marks): Algebraic Fractions – 26 Write 4 3 5 3 10 4 1 2 x x x x as a single fraction in its simplest form. (Total for Question 26 is 4 marks)$

Show worked solution

Use trigonometry
\[ 4-\left[(3x-5)\div\frac{3x^2+x-10}{4x-1}\right] \]
Factor the quadratic
Factor the quadratic:
\[ 3x^2+x-10=(3x-5)(x+2) \]
Use trigonometry
So the expression inside the square brackets is
\[ (3x-5)\div\frac{(3x-5)(x+2)}{4x-1} \]
Dividing by a fraction means
Dividing by a fraction means multiplying by its reciprocal:
\[ (3x-5)\times \frac{4x-1}{(3x-5)(x+2)} \]
Cancel
Cancel $3x-5$:
\[ \frac{4x-1}{x+2} \]
Use trigonometry
Therefore the full expression is
\[ 4-\frac{4x-1}{x+2} \]
Use the common denominator
Use the common denominator $x+2$:
\[ \frac{4(x+2)-(4x-1)}{x+2} \]
\[ =\frac{4x+8-4x+1}{x+2} \]
\[ =\frac9{x+2} \]

Final answer

$\displaystyle \boxed{\frac9{x+2}}$

18. May 2023 P1HR · Q23

Medium4 marks

May 2023 P1HR Question 23 (4 marks): Algebraic Fractions – 23 Simplify x x x x x 2 2 4 4 7 2 2 Give your answer in the form ax bx c 2 + where a, b and c are integers. (Total for Question 23 is 4 marks)

Show worked solution

Simplify surd
\[ (x^2-4)\div \left(\frac{4x^2-7x-2}{x}\right)-2x \]
Change division into multiplication by
Change division into multiplication by the reciprocal:
\[ (x^2-4)\times \frac{x}{4x^2-7x-2}-2x \]
Factorise
Factorise:
\[ x^2-4=(x-2)(x+2) \]
\[ 4x^2-7x-2=(4x+1)(x-2) \]
Simplify surd
So
\[ \frac{x(x-2)(x+2)}{(4x+1)(x-2)}-2x =\frac{x(x+2)}{4x+1}-2x \]
\[ \frac{x(x+2)-2x(4x+1)}{4x+1} =\frac{x^2+2x-8x^2-2x}{4x+1} =\frac{-7x^2}{4x+1} \]

Final answer

$\dfrac{-7x^2}{4x+1}$

19. May 2021 P1H · Q21

Medium4 marks

$May 2021 P1H Question 21 (4 marks): Algebraic Fractions – 21 Given that x y y a = + - 5 9 5 5 5 2 and that = find an expression for x in terms of a. Give your expression as a single fraction in its simplest form. (Total for Question 21 is 4 marks)$

Show worked solution

Given
\[ x=\frac5{9y+5} \]
Use trigonometry
and
\[ y=\frac5{5a-2} \]
Use cosine rule
Substitute $y=\frac5{5a-2}$ into the expression for $x$:
\[ x=\frac5{9\left(\frac5{5a-2}\right)+5} \]
\[ x=\frac5{\frac{45}{5a-2}+5} \]
Write the denominator as a
Write the denominator as a single fraction:
\[ \frac{45}{5a-2}+5 =\frac{45+5(5a-2)}{5a-2} \]
\[ =\frac{45+25a-10}{5a-2} \]
\[ =\frac{25a+35}{5a-2} \]
Use trigonometry
So
\[ x=5\div\frac{25a+35}{5a-2} \]
\[ x=5\times\frac{5a-2}{25a+35} \]
\[ x=\frac{5(5a-2)}{5(5a+7)} \]
\[ x=\frac{5a-2}{5a+7} \]

Final answer

$\displaystyle \boxed{\frac{5a-2}{5a+7}}$

20. Jan 2021 P2H · Q21

Medium4 marks

$Jan 2021 P2H Question 21 (4 marks): Algebraic Fractions – 21 Write 25 64 5 13 6 8 15 5 8 7 2 2 2 x x x x x x x - - - × - + + - - ( ) as a single fraction in its simplest form. Show clear algebraic working. (Total for Question 21 is 4 marks)$

Show worked solution

Use trigonometry
\[ \frac{25x^2-64}{5x^2-13x-6} \times \frac{x^2-8x+15}{5x+8} -(x-7) \]
Factor
Factor:
\[ 25x^2-64=(5x-8)(5x+8) \]
\[ 5x^2-13x-6=(5x+2)(x-3) \]
\[ x^2-8x+15=(x-3)(x-5) \]
Use trigonometry
So
\[ \frac{(5x-8)(5x+8)}{(5x+2)(x-3)} \times \frac{(x-3)(x-5)}{5x+8} -(x-7) \]
Cancel common factors
Cancel common factors:
\[ \frac{(5x-8)(x-5)}{5x+2}-(x-7) \]
Use the common denominator
Use the common denominator $5x+2$:
\[ \frac{(5x-8)(x-5)-(x-7)(5x+2)}{5x+2} \]
Expand
Expand:
\[ (5x-8)(x-5)=5x^2-33x+40 \]
\[ (x-7)(5x+2)=5x^2-33x-14 \]
Use trigonometry
So the numerator is
\[ 5x^2-33x+40-(5x^2-33x-14)=54 \]

Final answer

$\displaystyle \boxed{\frac{54}{5x+2}}$

21. Jan 2020 P2H · Q24

Medium4 marks

$Jan 2020 P2H Question 24 (4 marks): Algebraic Fractions – 24 Express 4 2 5 3 2 3 9 4 6 17 5 3 2 x x x x x x - - -    ÷ - - + as a single fraction in its simplest form. (Total for Question 24 is 4 marks)$

Show worked solution

Use trigonometry
\[ \left(\frac4{2x-5}-\frac3{2x-3}\right)\div \frac{9x-4x^3}{6x^2-17x+5} \]
Use trigonometry
First simplify the bracket:
\[ \frac4{2x-5}-\frac3{2x-3} =\frac{4(2x-3)-3(2x-5)}{(2x-5)(2x-3)} \]
\[ =\frac{8x-12-6x+15}{(2x-5)(2x-3)} \]
\[ =\frac{2x+3}{(2x-5)(2x-3)} \]
Use trigonometry
Now factor the second fraction:
\[ 9x-4x^3=-x(2x-3)(2x+3) \]
\[ 6x^2-17x+5=(2x-5)(3x-1) \]
Dividing by a fraction means
Dividing by a fraction means multiplying by its reciprocal:
\[ \frac{2x+3}{(2x-5)(2x-3)} \times \frac{(2x-5)(3x-1)}{-x(2x-3)(2x+3)} \]
Cancel common factors
Cancel common factors:
\[ =-\frac{3x-1}{x(2x-3)^2} \]
\[ =\frac{1-3x}{x(2x-3)^2} \]

Final answer

$\displaystyle \boxed{\frac{1-3x}{x(2x-3)^2}}$

22. May 2024 P1H · Q12

Hard5 marks

$May 2024 P1H Question 12 (5 marks): Algebraic Fractions – 12 (a) Factorise fully 6y 2 - 5y - 4 (2) (b) Express 2 1 4 7 5 3 x x x x as a single fraction in its simplest form. (3) (Total for Question 12 is 5 marks)$

Show worked solution

(a)
\[ 6y^2-5y-4=(3y-4)(2y+1) \]
(b)
\[ \frac{2x+1}{4x}+\frac{7-5x}{3x} \]
Use denominator
Use denominator $12x$: \[ \frac{3(2x+1)+4(7-5x)}{12x} \]
\[ =\frac{6x+3+28-20x}{12x}=\frac{31-14x}{12x} \]

Final answer

(a) $(3y-4)(2y+1)$, (b) $\frac{31-14x}{12x}$

23. Nov 2023 P2H · Q24

Hard5 marks

Nov 2023 P2H Question 24 (5 marks): Algebraic Fractions – 24 Solve 45 80 3 4 1 3 4 1 4 2 5 8 3 2 x x x x x x x x ( ) Show clear algebraic working. x = (Total for Question 24 is 5 marks)

Show worked solution

Solve quadratic equation
\[ \frac{45x^3-80x}{3x^2+x-4} \left(\frac{1}{3x-4}+\frac{1}{x}\right) =\frac{4(x+2)}{5x-8} \]
Factorise
Factorise:
\[ 45x^3-80x=5x(3x-4)(3x+4) \]
\[ 3x^2+x-4=(3x+4)(x-1) \]
Solve quadratic equation
Also,
\[ \frac{1}{3x-4}+\frac{1}{x} =\frac{x+3x-4}{x(3x-4)} =\frac{4(x-1)}{x(3x-4)} \]
Solve quadratic equation
So the left side becomes
\[ \frac{5x(3x-4)(3x+4)}{(3x+4)(x-1)} \times \frac{4(x-1)}{x(3x-4)} =20 \]
Solve quadratic equation
Therefore,
\[ 20=\frac{4(x+2)}{5x-8} \]
\[ 20(5x-8)=4(x+2) \]
\[ 100x-160=4x+8 \]
\[ 96x=168 \]
\[ x=\frac74 \]

Final answer

$x=\dfrac74$

24. Jan 2022 P1HR · Q12

Hard5 marks

$Jan 2022 P1HR Question 12 (5 marks): Algebraic Fractions – 12 (a) Expand and simplify n(n - 4)(3n + 5) (2) (b) Express 3 2 2 1 4 x x x + + + as a single fraction in its simplest form. (3) (Total for Question 12 is 5 marks)$

Show worked solution

(a)
\[ n(n-4)(3n+5)=n(3n^2-7n-20)=3n^3-7n^2-20n \]
(b)
\[ \frac3x+\frac{x+2}{2x}+\frac14 \]
Use denominator
Use denominator $4x$: \[ \frac{12}{4x}+\frac{2(x+2)}{4x}+\frac{x}{4x} =\frac{12+2x+4+x}{4x} \]
\[ =\frac{3x+16}{4x} \]

Final answer

(a) $3n^3-7n^2-20n$, (b) $\frac{3x+16}{4x}$

25. Jan 2021 P2HR · Q18

Hard5 marks

Jan 2021 P2HR Question 18 (5 marks): Algebraic Fractions – 18 Solve the equation 5 2 3 2 2 2 x x x + + + = Show clear algebraic working. (Total for Question 18 is 5 marks)

Show worked solution

Solve equation
\[ \frac{5}{x+2}+\frac{3}{x^2+2x}=2 \]
Solve equation
Since \[ x^2+2x=x(x+2) \]
multiply by
multiply by $x(x+2)$: \[ 5x+3=2x(x+2) \]
\[ 5x+3=2x^2+4x \]
\[ 2x^2-x-3=0 \]
Factorise
Factorise: \[ (2x-3)(x+1)=0 \]
\[ x=\frac32 \quad \text{or} \quad x=-1 \]
Both values are allowed because
Both values are allowed because $x\ne0$ and $x\ne-2$.

Final answer

$x=\frac32$ or $x=-1$.

26. Specimen 4WM1H · Q12

Hard5 marks

$Specimen 4WM1H Question 12 (5 marks): Algebraic Fractions – 12 (a) Factorise 9x^2 - 4y^2. (b) Express 7/8 - (x + 3)/(4x) as a single fraction in its simplest form. (Total for Question 12 is 5 marks)$

Show worked solution

Use difference of two squares
\[ 9x^2-4y^2=(3x)^2-(2y)^2 \]
\[ =(3x-2y)(3x+2y) \]
Use a common denominator
\[ \frac78-\frac{x+3}{4x} =\frac{7x}{8x}-\frac{2(x+3)}{8x} \]
Simplify the numerator
\[ \frac{7x-2x-6}{8x} =\frac{5x-6}{8x} \]

Final answer

$(3x-2y)(3x+2y)$, and $\frac{5x-6}{8x}$.

27. May-Nov 2020 P1H · Q21

Hard5 marks

$May-Nov 2020 P1H Question 21 (5 marks): Algebraic Fractions – 21 Express 1 3 2 9 4 3 13 10 7 1 2 2 x x x x x - × - - - - - as a single fraction in its simplest form. (Total for Question 21 is 5 marks)$

Show worked solution

Use trigonometry
\[ \frac1{3x-2}\times\frac{9x^2-4}{3x^2-13x-10} -\frac7{x-1} \]
Factor
Factor:
\[ 9x^2-4=(3x-2)(3x+2) \]
\[ 3x^2-13x-10=(3x+2)(x-5) \]
Use trigonometry
So the first product becomes
\[ \frac1{3x-2}\times \frac{(3x-2)(3x+2)}{(3x+2)(x-5)} \]
Cancel common factors
Cancel common factors:
\[ \frac1{x-5} \]
Use trigonometry
Now subtract:
\[ \frac1{x-5}-\frac7{x-1} \]
Use the common denominator
Use the common denominator $(x-5)(x-1)$:
\[ \frac{x-1-7(x-5)}{(x-5)(x-1)} \]
\[ =\frac{x-1-7x+35}{(x-5)(x-1)} \]
\[ =\frac{34-6x}{(x-5)(x-1)} \]

Final answer

$\displaystyle \boxed{\frac{34-6x}{(x-5)(x-1)}}$

28. Jun 2026 P1HR · Q16

Hard6 marks

$Jun 2026 P1HR Question 16 (6 marks): Algebraic Fractions – 16 (a) Express 5 8 2 4 � � x x as a single fraction in its simplest form. (3) (b) Expand and simplify 3x(2x – 1)(x + 4) (3) (Total for Question 16 is 6 marks)$

Show worked solution

Part (a) | Step 1
\[\frac58-\frac{x+2}{4x}=\frac{5x}{8x}-\frac{2(x+2)}{8x},\qquad x\ne0.\]
Use 8x as a common denominator.
Part (a) | Step 2
\[\frac{5x-2(x+2)}{8x}=\frac{5x-2x-4}{8x}=\frac{3x-4}{8x}.\]
Subtract the entire numerator of the second fraction and collect like terms.
Part (b) | Step 1
\[(2x-1)(x+4)=2x^2+8x-x-4=2x^2+7x-4.\]
Multiply each term in the first bracket by each term in the second.
Part (b) | Step 2
\[3x(2x^2+7x-4)=6x^3+21x^2-12x.\]
Multiply every term by 3x.

Final answer

(a) $\frac{3x-4}{8x}$

(b) $6x^3+21x^2-12x$

Algebraic Fractions: past paper questions & worked solutions

Questions, easiest first

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