Linear equations

Simplify.

a × a × a × a × a × a × a

4b × 5b²

Solve.

6 k - 3 = 177

Solve.

8 x - 11 = 6 x + 3

Solve by factorising.

x² + 15x - 16 = 0

Each member of a tennis club is classed as a junior, an adult or a senior.

• 16% of the members are seniors.

• The ratio of the number of juniors to the number of adults is 9: 5.

• There are 95 more juniors than seniors.

Work out the total number of members of the tennis club. You must show your working.

Solve.

8 x - 11 = 6 x + 3

Solve by factorising.

x² + 15x - 16 = 0

Five adults and two children go to a theme park. The cost of an adult ticket is £6 more than the cost of a child ticket. The total cost of the seven tickets is £142.

Work out the cost of an adult ticket. You must show your working.

In this question, all lengths are in centimetres.

The diagram shows an isosceles triangle ABC.

Find the perimeter of the triangle. You must show your working.

Darcie invests £ x at a rate of 1.5% per year simple interest for 5 years. Ivan also invests £ x but at a rate of 1.1% per year simple interest for 6 years.

Darcie earns £108 more interest than Ivan.

Work out the value of x. You must show your working.

Solve.

3 x + 12 = 9 - 7 x

Here is a formula.

a = b (c + d)

Find the value of a when b = 3, c = 7 and d = 5.

Find the value of c when a = 30, b = 4 and d = 6.

The diagram shows a straight line crossing two parallel lines.

Find the value of y. You must show your working.

The diagram shows an equilateral triangle and a rectangle.

The equilateral triangle has the same perimeter as the rectangle.

Find the value of a.

In this question all measurements are in centimetres.

The shaded shape is made by cutting a square from the corner of a rectangle.

The width of the rectangle is k. The length of the rectangle is 2 k. Each side of the square is g.

Write down the relationship between the length and the width of the rectangle.

Find an expression for the area of the shaded shape. Give your answer in its simplest form.

Find an expression for the perimeter of the shaded shape. Give your answer in its simplest form.

Find the value of k when the perimeter of the shaded shape is 62.4.

The diagram shows an equilateral triangle and a rectangle.

The equilateral triangle has the same perimeter as the rectangle.

Find the value of a.

The diagram shows a cuboid with dimensions 12 cm, 5 cm and x cm.

The volume of the cuboid is 240 cm³.

Find the value of x.

A climber is in a competition.

Here are their scores after two events.

Event 3 is out of 52 points.

The climber’s overall score is found by adding the three scores together.

Find the score the climber needs in event 3 to achieve 85% of the total points.

Here are two functions.

When the input of each function is 6, the output of Function A is equal to the output of Function B.

Work out the value of p in Function B.

The bar chart shows the rainfall, in millimetres (mm), for a city in the first three months of the year.

Write down the amount of rainfall in February.

Write in its simplest form the ratio amount of rainfall in February: amount of rainfall in March.

The total amount of rainfall in January and February was the same as the total amount of rainfall in March and April.

Work out the amount of rainfall in April.

The diagram shows a rectangle with length (3 x - 4)cm and width (x + 2)cm.

The length of the rectangle is twice the width of the rectangle.

Calculate the area of the rectangle. You must show your working.

Simplify.

4 a + 2 a + a

2 x - 3 y - 3 x + 4

Solve.

p - 5 = - 4

Here are two pieces of work. Each shows a question and an incorrect solution.

For each part, describe the error made and write out a correct solution.

A correct solution is........

A correct solution is........

A motorist wants to buy a new car but does not have enough money.

The price of the car is £16 000.

The motorist sees this notice for a deal on the car they want to buy. The number of equal monthly payments is hidden.

Pay 15% of the price of the car now

Work out the number of monthly payments if the total cost of the car to the motorist is £17 400. You must show your working.

Here are two functions.

5 is input into Function A. 5 is also input into Function B. The output of Function A is 10 times the output of Function B.

Work out the value of k. You must show your working.

Finley is asked to solve the equation 5 x + 4 = 19.

Finley’s working is shown below.

Write down the error that Finley has made.

Charlie is asked to use the formula

v = u + at

to find the initial velocity, when

• the acceleration is 5 m/s²

• the final velocity is 29 m/s

• the time is 3 seconds.

Charlie’s working is shown below.

Write down the error that Charlie has made.

Solve.

2 x = - 14

Hiro and Ling work in a restaurant. Hiro is paid £9 per hour and Ling is paid £10 per hour.

One week, Hiro works a total of 30 hours. Ling earns £50 more than Hiro that week.

Work out the number of hours that Ling works.

Ryan started with 80 counters.

Ryan says

I still have enough counters to make Pattern 5 and Pattern 6.

Is Ryan correct? Show how you decide.

Complete the table below for the addition of counters in consecutive patterns.

The number of counters in Pattern k + Pattern (k + 1) is 144.

Find the value of k.

In the diagram, line AB is parallel to line ED. The points A, C and D lie on a straight line. The points B, C and E lie on a straight line.

Angle BCA = 45°, angle ABC = x ° and angle CDE = y °. The ratio x: y is 3: 2.

Work out the value of x.

The sides of this triangle are given in centimetres. The perimeter of the triangle is 80 cm.

Find the length of each side of the triangle. You must show your working.

(a)........ cm,........ cm and........ cm

Is the triangle above a right-angled triangle? Use calculations to show how you decide.

Here are two functions.

Write an algebraic expression for the output of function A when the input is x.

Here is a composite function C.

The input to function C is x. The output from function C is 2 x + 1.

Find the value of x. You must show your working.

The graph shows the velocity of a particle over the first 20 minutes of its motion.

Between 7 minutes and 15 minutes the velocity of the particle is v metres per minute. The average velocity of the particle over the 20 minutes is 11.55 metres per minute.

Find the value of v. You must show your working.

On Heidi’s bookcase, the ratio of fiction to non-fiction books is 2: 3. Heidi removes 2 fiction books from the bookcase. The ratio of fiction to non-fiction books is then 5: 8.

How many books are left on the bookcase in total?

An examination has three papers. Paper 1 is marked out of 60. Paper 2 is marked out of 40. Paper 3 is marked out of 100. The three marks are added together to form the total mark out of 200.

A student scored 65% on Paper 1 and 70% on Paper 2.

Find the mark they need to get on Paper 3 to achieve 64% of the total marks. You must show your working.

Solve.

6 x - 9 = 27 - 4 x

Solve.

8 - x = - 2

A recipe for a batch of jam needs 3 oranges, 5 lemons and 1.5 kg of sugar. A cook uses the recipe to make lots of batches of jam. They use 16 more lemons than oranges in total.

F ind how much sugar the cook should use.

An examination has three papers. Paper 1 is marked out of 60. Paper 2 is marked out of 40. Paper 3 is marked out of 100. The three marks are added together to form the total mark out of 200.

A student scored 65% on Paper 1 and 70% on Paper 2.

Find the mark they need to get on Paper 3 to achieve 64% of the total marks. You must show your working.

The diagram shows a straight line crossing a pair of parallel lines.

Find the value of y. You must show your working.

The parallelogram and the trapezium have the same area.

The ratio of x: y is 3: 5.

Find the value of x and the value of y. You must show your working.

A recipe for a batch of jam needs 3 oranges, 5 lemons and 1.5 kg of sugar. A cook uses the recipe to make lots of batches of jam. They use 16 more lemons than oranges in total.

Find how much sugar the cook should use.

This list represents four numbers.

127 x x + 1 2 x

The mean of the four numbers is 180.

Work out the numbers. You must show your working.

127

........

A town square has a fountain (F) at the centre. There is also a bell tower (B) and a statue (S).

The bearing of the statue from the fountain is x °.

The bearing of the bell tower from the fountain is 140° more than the bearing of the statue from the fountain.

Write down, in terms of x, the bearing of the bell tower from the fountain.

The bearing of the bell tower from the fountain is also three times the bearing of the statue from the fountain.

Work out the bearing of the bell tower from the fountain.

In this question, all lengths are in centimetres.

The diagram shows an equilateral triangle and a square.

The perimeter of each shape is 36 cm.

Find the value of b.

Amir, Beth and Charlie work in a cafe. Customers give spare change as tips. At the end of each week, Amir, Beth and Charlie share the total amount of tips between them in the ratio matching the number of hours they worked that week.

This week:

• Amir’s share of the tips was £25.40.

• Beth worked twice as many hours as Amir.

• Charlie worked 5 more hours than Amir.

• The total hours worked by Amir, Beth and Charlie was 85 hours.

Calculate the total amount of tips received this week. You must show your working.

Solve.

x - 14 = 30

6 y + 7 = 28

A biased five-sided spinner is numbered 1, 2, 3, 4 and 5.

The table shows the probability of the spinner landing on 1, 2 and 4.

The spinner is four times more likely to land on 5 than on 3.

Complete the table.

The diagram shows a straight line crossing two parallel lines.

Find the value of y. You must show your working.

Alex, Blake and Charlie play a computer game.

Alex goes first and scores n points.

• Blake scores 8 points less than 3 times the number of points scored by Alex.

• Charlie scores 25 more points than Blake.

• The three people score a total of 618 points.

Work out how many points they each score. You must show your working.

The diagram shows a square.

(4 x – 10) cm
Not to scale

(11 – 2 x) cm

By setting up and solving an equation, show that the perimeter of the square is numerically equal to the area of the square.

5 (2 x + 1) + c (x + d) = 12 x - 1

Work out the value of c and the value of d.

A rectangle is three times as long as it is wide. It has a perimeter of 44 cm.

Find the length of the rectangle.

The area of the triangle is equal to the area of the trapezium.

Calculate the height, h cm, of the trapezium.

In a survey, 60 students were asked whether they have a bank account (B) and whether they have a part-time job (J).

The number of students who had neither a bank account nor a part-time job was x. The Venn diagram shows the results in terms of x.

One of the 60 students is chosen at random.

Find the probability that they have a bank account. Show your working.

Solve.

4 x + 3 = 13

Multiply out and simplify.

5 (2 x + 3) + 2 (x - 4)

Solve.

Here are two functions.

Jo chooses a number, x. She inputs x into each function. The two outputs are equal.

Work out the value of x.

Explain why there is no other input that gives two outputs that are equal.

Here is function C.

Kai chooses values for p and q so that if he inputs any number into both function A and function C, he will always get two outputs that are equal.

Find the value of p and the value of q.

The base of a cone is fixed to the top of a cylinder to make a decoration.

The radius of the base of the cone and of the cylinder is r cm. The cone’s height is 5 r cm. The total height of the decoration is 6 r cm. The total volume of the decoration is 225 cm³.

Calculate the value of r. Show your working.

Mr and Mrs Jones buy cinema tickets for themselves and their three children. The cost of an adult ticket is £6 more than a child ticket. The total cost of the five tickets is £45.

Work out the cost of an adult ticket.

An adult ticket costs £........

The area of the parallelogram is four times the area of the triangle.

Calculate the length, L, of the parallelogram.

Dan thinks of a number. He adds 3 and divides the result by 2. His answer is 16.

What number is Dan thinking of?

Amrit’s income is 32% more than Bethan’s income. Amrit and Bethan’s combined income is £54 868.

Calculate Amrit’s income.

The diagram shows two rectangles, A and B.

Rectangle A has a width of 25 cm and a height of 12 cm. The width of rectangle B is three times the height of rectangle B.

The area of rectangle A is equal to the area of rectangle B.

Find the perimeter of rectangle B.

Kay invests £1500 in an account paying 3% compound interest per year. Neil invests £1500 in an account paying r % simple interest per year.

At the end of the 5th year, Kay and Neil’s accounts both contain the same amount of money.

Calculate r. Give your answer correct to 1 decimal place.

In this question, all lengths are in centimetres.

Here is a square.

Find the length of one side of the square when b = 4.

Solve.

6 x - 10 = 4 x + 1

Amrit’s income is 32% more than Bethan’s income. Amrit and Bethan’s combined income is £54 868.

Calculate Amrit’s income.

The diagram shows two rectangles, A and B.

Rectangle A has a width of 25 cm and a height of 12 cm. The width of rectangle B is three times the height of rectangle B.

The area of rectangle A is equal to the area of rectangle B.

Find the perimeter of rectangle B.

Kay invests £1500 in an account paying 3% compound interest per year. Neil invests £1500 in an account paying r % simple interest per year.

At the end of the 5th year, Kay and Neil’s accounts both contain the same amount of money.

Calculate r. Give your answer correct to 1 decimal place.

Bennie is 7 years older than Ayesha. Chloe is twice as old as Bennie. The sum of their three ages is 57.

Work out the ages of Ayesha, Bennie and Chloe.

Darren has these 20 crayons in a box:

• 8 blue

• 4 red

• 5 black

• 3 green.

He chooses a crayon at random from the box.

Which arrow shows the probability that this crayon is

blue,

yellow,

not black.

Darren buys 16 more crayons that are either blue or red. He puts these in the box with the 20 crayons he already has.

He now picks a crayon at random from the box. The probability that he picks a blue crayon is evens.

How many red crayons did he buy?

The diagram shows two intersecting straight lines.

Find the value of y.

Solve.

4 x + 5 = 35

Bennie is 7 years older than Ayesha. Chloe is twice as old as Bennie. The sum of their three ages is 57.

Work out the ages of Ayesha, Bennie and Chloe.

Simplify fully.

Solve.

Solve.

x - 6 = 4

The perimeter of the pentagon is equal to the perimeter of the square.

Not to scale

Find an expression for the length of one side of the square. Give your answer in terms of x in its simplest form.

Marcin buys 7 rulers and 15 crayons for £7. A ruler costs 12p more than a crayon.

Find the cost of one crayon.

Multiply out.

Give your answer in its simplest form.

32x + d + cx + 5 = 10x + 17

Work out the value of c and the value of d.

Solve by factorising.

The depth of water in a garden pond is 57.8 cm. The depth decreases by 0.3 cm per day.

Assume the depth continues to decrease at the same rate.

After how many days will the depth reach 54.2 cm?

If the depth of water decreases at a slower rate, what effect will this have on your answer to part (a) ?

Gill uses the formula

h = 2 fg

Find the value of h when f = 1 and g = 3.

Find the value of g when h = 18 and f = 6.

Marcin buys 7 rulers and 15 crayons for £7. A ruler costs 12p more than a crayon.

Find the cost of one crayon.

In the diagram, the square and the trapezium share a common side of length x cm.

The area of the square is equal to the area of the trapezium.

Work out the value of x.

Solve.

4 x = 56

8 x - 6 = 46

Solve by factorising.

Aditi, Becky and Calli collect coins. Aditi has 6 more coins than Becky. Calli has 1 less coin than Aditi. Altogether they have 71 coins.

How many coins do they each have? Show all your working.

Aditi has........ coins

Becky has........ coins

Calli has........ coins

The diagram shows a straight line that passes through points A and B, and a curve that passes through points P and Q.

Find the equation of the straight line.

The equation of the curve is y = x² + kx + 8.

Find the value of k.

Diann draws line BQ. She says

Triangle ABQ is isosceles.

Is Diann correct? You must show all your working.

The output of function A is x.

Write an algebraic expression, in terms of x, for the input of function A.

A number, k, is put into function A. The output is also k.

Find the value of k.

Katy buys x cakes. Gugu buys 3 times as many cakes as Katy. Deanna buys 2 more cakes than Katy.

Each cake costs 85p. The total cost of the cakes is £52.70.

How many cakes did each girl buy?

Halina cycled from A to B at an average speed of 26 km per hour. She then cycled from B to C at an average speed of 20 km per hour.

She left A at 10.00 am, did not stop at B and arrived at C at 3.00 pm.

It took Halina x hours to cycle from A to B.

Explain why the distance from A to B, in kilometres, is 26 x.

Write down an expression, in terms of x, for the time taken to cycle from B to C.

Hence show that the distance from B to C, in kilometres, is 100 – 20 x.

The total distance cycled by Halina from A to C is 118 km.

Find the distance from A to B.

Calculate the volume of this cuboid.

In this cuboid all lengths are in centimetres.

The cuboid has a volume of 320 cm³.

Find the value of k.

This rectangle has length (4 x – 5) cm and width (x + 3) cm.

The perimeter of the rectangle is 46 cm.

Calculate the area of the rectangle.

The diagram shows the positions of two towns, Amton and Bisham.

The bearing of Bisham from Amton is b °. The bearing of Amton from Bisham is 6 b °.

Calculate the 3-figure bearing of Amton from Bisham.

Mr and Mrs Thomas buy tickets for themselves and their four children. The cost of an adult ticket is £7 more than the cost of a child ticket. The total cost of the six tickets is £86.

Work out the cost of an adult ticket.

Multiply out.

3 x (x + 2 y)

Solve.

7 x = 28

A square is divided into three rectangles, A, B and C.

Rectangle A has length n cm and a width of 2 cm. Rectangle C has length 4 cm.

Write down an algebraic expression for the width of rectangle C.

Write down an algebraic expression for the area of rectangle A.

The three rectangles all have the same area.

Work out the value of n.

Hardeep buys 11 identical shirts and 24 identical ties for £403.51. The cost of a shirt is £15.65.

Find the cost of a tie.

Solve the inequality.

3 x - 2 > 10

Solve.

6 x + 2 = 5 - 4 x