Forming and solving equations

Heidi is x years old. Finley is 5 years younger than Heidi. Zayn is 4 times Finley’s age.

Find an expression in terms of x for the total age of Heidi, Finley and Zayn. Write your answer as simply as possible.

An electrician charges £30 per visit plus £22 per hour.

Write an expression for the cost, in £, charged by the electrician for one visit lasting n hours.

An electrician charges £30 per visit plus £22 per hour.

Write an expression for the cost, in £, charged by the electrician for one visit lasting n hours.

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 lengths are in centimetres.

The area of the rectangle is 70 cm².

Show that x² + 5x - 84 = 0.

Solve by factorising.

(b)(i) x =........ or x =........

Find the length of the longer side of the rectangle.

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.

Eight consecutive numbers are written in ascending order in this grid, starting from the top and working left to right.

Kareem writes the numbers 5 to 12 in the grid.

Show that for Kareem’s grid, the sum of the numbers in the top half of the grid is 16 less

than the sum of the numbers in the bottom half of the grid.

Use algebra to prove that for any set of eight consecutive numbers written in this grid in the same way, the sum of the numbers in the top half of the grid is 16 less than the sum of the

numbers in the bottom half of the grid.

Write an expression for the weight, in grams, of an object weighing x kilograms.

Write an expression for the area, in m², of a garden of area ycm².

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.

Here is a function.

Find the input when the output is 87.

The input is x and the output is y.

Write an equation for y in terms of x.

The diagram shows a quadrilateral and an equilateral triangle. The perimeter of the quadrilateral is equal to the perimeter of the equilateral triangle.

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

The diagram shows a prism of length 10 cm.

The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volume of the prism is 240 cm³.

A student is explaining how they worked out the value of b.

They say

b is 6 because that means h is 4 and 6 × 4 × 10 = 240.

Describe the student’s error and find the correct value of b.

Here is a function machine.

Find the output when the input is 10.

Find the input when the output is 17.

The input is x and the output is y.

Write an equation for y in terms of x.

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.

Charlie is making some wooden frames. Charlie has a strip of wood 1.6 m long and 2 cm wide.

Not to scale

Each frame will be made from four pieces of wood cut from the strip to form a rectangle, as shown below.

The width of each frame is x cm. The length of each frame is 2 x cm. The area enclosed by each frame must be 96 cm².

Work out the maximum number of frames Charlie can make from the 1.6 m length of wood. You must show your working.

In this question, all measurements are in centimetres.

The square and the rectangle have the same area.

Show that x² - 8x - 20 = 0.

Solve x² - 8x - 20 = 0.

Explain why one of the answers in part (b) is not possible in the context of the question.

Write down the following.

The area of the square.

The length of the rectangle.

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.

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

Find the length of the rectangle.

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 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.

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.

Here is a function machine.

Find the output when the input is 7.

Find the input when the output is 42.

The input is x and the output is y.

Write an equation for y in terms of x.

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.

A cone has radius r cm and height h cm.

The height is three times the radius. The volume of the cone is 2100 cm³.

Calculate the radius of the cone.

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.

A circle, with centre O and radius 6 cm, contains another circle, with centre O and radius x cm.

Write down an expression, in terms of π and x, for the shaded area in cm².

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.

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

In this question, assume all dimensions are in centimetres.

Jess joins three tiles together to make a larger rectangle, as shown.

Write an expression for the perimeter of her rectangle. Give your answer in its simplest form.

An expression for the area of her rectangle is 6ab + 6b².

Factorise this expression fully.

Pete joins some tiles together to make a different rectangle. The area of his rectangle is 8ab + 8b².

Draw a possible arrangement of tiles for Pete’s rectangle. Write down expressions for the length and for the width of the rectangle.

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.

n is an integer.

Explain why 2 n + 1 is an odd number.

Write down an algebraic expression for the next odd number after 2 n + 1.

Use algebra to show that the sum of two consecutive odd numbers will always be a

multiple of 4.

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.

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.

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.

Each week Dan drives two routes, route X and route Y.

One week he drives route X three times and route Y twice. He drives a total of 134 miles that week.

Another week he drives route X twice and route Y five times. He drives a total of 203 miles that week.

Find the length of each route.

(a) route X =........ miles

State an assumption that has been made in answering part (a).