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15 Algebra Questions And Practice Problems (KS3 & KS4): Harder GCSE Exam Style Questions Included

Algebra questions involve using letters or symbols to represent unknown values or values that can change. Here you will find 15 algebra questions to test your knowledge and show you the different ways that algebra can be used to solve a problem to find an unknown value or to make generalisations.


Algebra in KS3 and KS4

There are many topics and techniques within algebra. In KS3 we learn to write and manipulate basic algebraic expressions and linear equations. In KS4 we develop these techniques to allow us to deal with more complicated algebra problems such as ones that involve quadratic equations or a system of equations.


How to solve algebraic questions

When you are presented with an algebraic problem it is important to establish what you are being asked to do. Here are some of the key terms along with what they mean:

  • Solve the equation – find out the value of the unknown
  • Substitute – put the values you have been given into the algebraic expression
  • Simplify – collect together like terms to make the expression or equation look simpler
  • Expand brackets – multiply out the brackets
  • Factorise – put into brackets
  • Make x the subject – rewrite the equation in the form x =…..


Remember, when working with algebra, we must still apply BODMAS / BIDMAS. i.e. Brackets, Indices (powers, exponents, square roots), Division, Multiplication, Addition, Subtraction.

When working with algebraic expressions and equations we must consider carefully which operations to deal with first.

Set of 25 printable GCSE maths questions

Try these challenging GCSE maths questions with your students in class. They all include answers and are organised by difficulty!


Algebra in KS2

The ideas of writing and simplifying expressions, solving equations and substitution are introduced in KS2. Here are some example KS2 algebra questions:


Algebra questions KS2

1. A chocolate bar costs c pence and a drink costs d pence. Write down an expression for the cost of 2 chocolate bars and 2 drinks.

c+d
GCSE Quiz False

2c+d
GCSE Quiz False

2c+2d
GCSE Quiz True

2c-2d
GCSE Quiz False

2 chocolate bars would cost 2 lots of c, or 2c, and 2 drinks would cost 2 lots of d, or 2d.

2. Simplify the expression 4m+5+2m-1

10m
GCSE Quiz False

6m+4
GCSE Quiz False

6m+6
GCSE Quiz True

6m-4
GCSE Quiz False

We need to collect together like terms here so 4m + 2m = 6m and 5 – 1 = 4 (watch out for the negative).


Algebra questions KS3

In KS3 we learn a variety of different algebra techniques to answer algebra questions and to practise problem solving with algebra. These include:

  • Simplifying algebraic expressions
  • Expanding brackets and factoring
  • Forming algebraic equations from word problems
  • Solving algebraic equations and inequalities
  • Substituting into expressions
  • Changing the subject of an equation
  • Working with real life graphs and straight line graphs
  • Sequences
A slide guiding pupils through the equation of a straight line from Third Space Learning's online intervention.
A slide guiding pupils through the equation of a straight line from Third Space Learning’s online intervention.


Algebra questions KS3: basic algebra

1. In this pyramid, you add two adjacent blocks to find the value of the block above.

 

basic algebra question 1a

 

What expression will be in the top box?

8a-2b
GCSE Quiz False

12a-b
GCSE Quiz True

12a+5b
GCSE Quiz False

10a+b
GCSE Quiz False

basic algebra question 1b

2.  Brian is a window cleaner. He uses the following formula to calculate the amount to charge his customers:

 

Charge = £20 + 4n

Where n is the number of windows a house has.

 

If a house has 7 windows, how much would Brian charge?

£24
GCSE Quiz False

£67
GCSE Quiz False

£48
GCSE Quiz True

£27
GCSE Quiz False

In this question, n is 7 so we can substitute 7 into the formula.

 

Charge = £20 + 4 × 7

Charge = £48

3. The area of a rectangle is 4x-6.

 

basic algebra question 3

 

Which of the following pairs could be the length and width of the rectangle?

4x and 6x

GCSE Quiz False

4 and x-6

GCSE Quiz False

2 and 2x-3

GCSE Quiz True

2x and 2x-3

GCSE Quiz False

There are two ways of attempting this question. We know that area \;of \;a \;rectangle = length × width so we could multiply each pair together to see which pair makes 4x − 6.

 

\begin{aligned} &4x × 6x \quad \quad\quad\; \;24x^2 \\ &4 × (x − 6) \quad \quad \quad4x − 24\\ &2 × (2x − 3) \quad \quad \;\;4x − 6\\ &2x × (2x − 3) \quad \quad 4x^2 − 6x\\ \end{aligned}

 

Alternatively, if we factorise 4x − 6 we get 2(2x − 3) meaning the sides could be 2 and 2x − 3.

4.  The formula for changing degrees Celsius to degrees Fahrenheit is

 

F=\frac{9C}{5}+32

 

Rearrange this formula to make C the subject.

C=\frac{5(F-32)}{9}
GCSE Quiz True

C=\frac{5F-32}{9}
GCSE Quiz False

C=\frac{5F}{9}-32
GCSE Quiz False

C=5F-\frac{32}{9}
GCSE Quiz False
\begin{aligned} F&=\frac{9C}{5}+32 \hspace{3cm} &\text{subtract 32}\\\\ F-32&=\frac{9C}{5} &\text{multiply by 5}\\\\ 5(F-32)&=9C &\text{divide by 9}\\\\ \frac{5(F-32)}{9}&=C \end{aligned}

Algebra questions KS3: forming and solving equations

5. Work out the size of the smallest angle.

algebra question ks3

20^{\circ}
GCSE Quiz False

26^{\circ}
GCSE Quiz False

8^{\circ}
GCSE Quiz False

34^{\circ}
GCSE Quiz True

The angles in a triangle add up to 180^{\circ} therefore we can write

 

4x+2x-10+3x-8=180

 

Now we have an equation we can solve.

 

\begin{aligned} 9x-18&=180 \hspace{3cm} &\text{add } 18\\ 9x&=198 &\text{divide by } 9\\ x&=22^{\circ} \end{aligned}

 

The angles are :

 

\begin{aligned} 4\times22&=88^{\circ}\\ 2\times22 -10&=34^{\circ}\\ 3\times22 -8&=58^{\circ} \end{aligned}

 

The smallest angle is 34^{\circ} .

6.  Jamie’s dad is 4 times older than Jamie. In 14 years time, Jamie’s dad will be twice the age of Jamie.

 

What is the sum of Jamie’s age now and Jamie’s dad’s age now?

70
GCSE Quiz False

42
GCSE Quiz False

22
GCSE Quiz False

35
GCSE Quiz True

To solve this we need to write an equation.

 

Let Jamie’s age now be x . Then Jamie’s dad’s age is 4x .

 

In 14 years time Jamie’s age will be x + 14 and Jamie’s dad’s age will be 4x + 14 .

 

Since we know Jamie’s dad’s age will be two times Jamie’s age, we can write

 

4x+14=2(x+14)

 

Now we have an equation we can solve.

 

\begin{aligned} 4x+14&=2(x+14) \hspace{3cm} &\text{expand the brackets}\\ 4x+14&=2x+28 \hspace{3cm} &\text{subtract 2x}\\ 2x+14&=28 \hspace{3cm} &\text{subtract 14}\\ 2x&=14 \hspace{3cm} &\text{expand the brackets}\\ x&=7 \hspace{3cm} \end{aligned}

 

Jamie is currently 7 years old meaning his dad is 28 years old. The sum of their ages is 35 .

Algebra questions KS3: graphs

7.  Which of the following lines passes through the point (2, 5)?

y=2x-1
GCSE Quiz False

y=2x+1
GCSE Quiz True

y=4x-2
GCSE Quiz False

y=2x+5
GCSE Quiz False

At the point (2, 5), x is 2 and y is 5. We can check which equation works when we substitute in these values:

 

\begin{aligned} y&=2x−1 \quad \quad \quad 5=2×2−1 \quad \quad \text{False}\\ y&=2x+1 \quad \quad 5=2×2+1 \quad \quad \text{True}\\ y&=4x−2 \quad \quad \quad 5=4×2−2 \quad \quad \text{False}\\ y&=2x+5 \quad \quad 5=2×2+5 \quad \quad \text{False} \end{aligned}


Algebra questions KS4

Algebra is studied extensively in the GCSE and IGCSE curriculum.

In KS4 we build on the techniques learnt in KS3. Topics include:

  • Expanding and factorising polynomials
  • Solving quadratic equations
  • Solving simultaneous equations
  • Inequalities
  • Algebraic fractions
  • Further work on graphs
  • Functions
One of the GCSE algebra slides on Third Space Learning's online intervention, guiding students through factorising quadratics.
One of the GCSE algebra slides on Third Space Learning’s online intervention, guiding students through factorising quadratics.


Algebra questions KS4: algebraic manipulation

8. Which of the following expressions has the smallest value when a=5 and b=-3?

\frac{1}{2}(a-b)
GCSE Quiz False

ab
GCSE Quiz False

b^2
GCSE Quiz False

b − 4a
GCSE Quiz True
\begin{aligned} &\frac{1}{2}(a−b) \quad \quad \quad 4 \\\\ &ab \quad \quad \quad \quad-15 \\\\ &b^2 \quad \quad \quad \quad \quad 9 \\\\ &b−4a \quad \quad \quad -23 \end{aligned}

9. Find an expression in terms of x for the volume of this cuboid

 

algebra question ks4 1

30x^3-49x^2+4x+3
GCSE Quiz True

30x^3-46x^2-8x-3
GCSE Quiz False

30x^3-3
GCSE Quiz False

30x^3+3
GCSE Quiz False
Volume of a cuboid = length × width × height

 

Volume = (5x+1)(2x-3)(3x-1)

 

Volume = (10x^2+2x-15x-3)(3x-1)

 

Volume = (10x^2-13x-3)(3x-1)

 

Volume = 30x^3-39x^2-9x-10x^2+13x+3

 

Volume = 30x^3-49x^2+4x+3

Algebra questions KS4: forming and solving equations

10. The area of this triangle is 24cm^2 .

 

algebra question ks4 2

 

Work out the perimeter of the triangle.

16cm
GCSE Quiz False

24cm
GCSE Quiz True

12cm
GCSE Quiz False

9cm
GCSE Quiz False

The area of a triangle is area = \frac{1}{2} × b × h.

 

If we fill in what we know we get:

 

\begin{aligned} 24&=\frac{1}{2}\times 6 \times (3x-1) \hspace{3cm} &\text{simplify}\\\\ 24&=3(3x-1) &\text{multiply out the brackets}\\\\ 24&=9x-3 &\text{add 3}\\\\ 27&=9x &\text{divide by 9}\\\\ x&=3 \end{aligned}

 

Since x = 3 , the side lengths are 6m, 8cm and 10cm .

 

The perimeter is 6 + 8 + 10 = 24cm .

11.  Solve the equation x+2-\frac{15}{x}=0

x=-2 or x=15

GCSE Quiz False

x=-3 or x=5

GCSE Quiz False

x=-5 or x=3

GCSE Quiz True

x=-15 or x=2

GCSE Quiz False

We can make this a bit easier by getting rid of the fraction involving x. We do this by multiplying each term by x.

 

\begin{aligned} x+2-\frac{15}{x}&=0 \hspace{3cm} &\text{multiply by x}\\\\ x^{2}+2x-15&=0 &\text{factorise}\\\\ (x+5)(x-3)&=0 &\text{solve}\\\\ &x=-5 \text{ or } x=3 \end{aligned}

12. At a theme park the Jones family purchased 2 adult tickets and 3 child tickets for £48 .  The Evans family purchased 3 adult tickets and 1 child ticket for £44

 

Calculate the cost of one child ticket.

£10
GCSE Quiz False

£8
GCSE Quiz True

£12
GCSE Quiz False

£6
GCSE Quiz False

We can write simultaneous equations to solve this.

 

2a+3c=48 (Equation 1)
3a+c=44 (Equation 2)

 

Multiply equation 2 by 3 to make the coefficients of c equal: 9a+3c=132 (Equation 3)

 

Subtract equation 1 from equation 3:
7a= 84
a=12

 

Substitute a into equation 3:
3×12+c=44
36+c=44
c=8

 

The cost of an adult ticket is £12 and a child ticket is £8 .

Algebra questions KS4: graphs

13.  Which of these lines is parallel to the line 2y = x + 7

3y=x+7
GCSE Quiz False

y=2x-2
GCSE Quiz False

y=\frac{1}{2}x-9
GCSE Quiz True

2y=3x+8
GCSE Quiz False

For two lines to be parallel, their gradient must be equal.

 

If we rearrange 2y=x+7 to make y the subject we get y=\frac{1}{2}x+\frac{7}{2}.

 

The gradient is \frac{1}{2}

14. Find the minimum value of the function f(x) = x^2+4x+5 .

1
GCSE Quiz True

-2
GCSE Quiz False

5
GCSE Quiz False

0
GCSE Quiz False

To find the minimum value we need to complete the square.

 

\begin{aligned} f(x)&=x^2+4x+5\\ f(x)&=(x+2)^2-4+5\\ f(x)&=(x+2)^2+1 \end{aligned}

 

The minimum value is 1. This occurs when (x+2) is 0.

15. The diagram shows the circle x^2+y^2=25 . The line is a tangent to the circle at the point (3,4) . Work out the equation of the line.

 

algebra question ks4 3

y=5x+5
GCSE Quiz False

y=3x+4
GCSE Quiz False

y=\frac{-3}{4}x+3
GCSE Quiz False

y=\frac{-3}{4}x+\frac{21}{4}
GCSE Quiz True

To work out the gradient of the line we need to work out the gradient of the normal.

 

We know that the normal goes through the points (0, 0) and (3, 4) so we can calculate the gradient: \frac{4-0}{3-0}=\frac{4}{3}.

 

The gradient of the tangent will be \frac{-3}{4}.

 

We can now use y=mx+c . We know the tangent goes through the point (3, 4) and that it’s gradient is \frac{-3}{4} .

 

Therefore

 

\begin{aligned} 4&=\frac{-3}{4} \times 3 +c\\\\ 4&=\frac{-9}{4}+c\\\\ 4+\frac{9}{4}&=c\\\\ \frac{25}{4}&=c \end{aligned}

 

y=\frac{-3}{4}x+\frac{25}{4} \text{ or } 4y=-3x+25

Looking for more algebra questions and resources?

Third Space Learning’s free GCSE maths resource library contains detailed lessons with step-by-step instructions on how to solve algebra problems, as well as worksheets with algebra practice questions and more GCSE exam questions.

Take a look at the Algebra lessons today – more are added every week.

Looking for more KS3 and KS4 maths questions?

Try these:

Read more: Question Level Analysis Of Edexcel Maths Past Papers (Foundation)

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Beki Christian
Beki Christian
Maths Teacher and Author
Beki has 11 years experience teaching secondary Maths. She is currently working as a Maths tutor as well as writing resources and blog posts for Third Space Learning.
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Downloadable resource

Set of 25 printable GCSE maths questions

Try these challenging GCSE maths questions with your students in class. They all include answers and are organised by difficulty!

Download Free Now!

Set of 25 printable GCSE maths questions

Downloadable resource

Try these challenging GCSE maths questions with your students in class. They all include answers and are organised by difficulty!

Download Free Now!