Forming & solving equations

Greg is x years old. Greg is 5 years older than Katy.

Write down an expression, in terms of x, for Katy’s age.

Carl is twice as old as Greg.

Write down an expression, in terms of x, for Carl’s age.

Solve 4y = 12

The diagram shows a solid shape made from a cylinder and a cone.

The cone has a base radius of 8 cm. The cylinder has a radius of 8 cm. The vertical height of the cone is three times the height of the cylinder. The volume of the solid shape is 640πcm³ Work out the vertical height of the cone. You must show all your working.

1 cup of tea and 1 cup of coffee cost £4.50 3 cups of tea and 1 cup of coffee cost £8.50 Work out the total cost of 4 cups of tea and 3 cups of coffee. You must show all your working.

Here is a pentagon.

Angle a = angle c Angle b = 155° Angle d is three times the size of angle c Angle e is two times the size of angle c Work out the size of angle a

Here is a pentagon.

Angle a = angle c Angle b = 155° Angle d is three times the size of angle c Angle e is two times the size of angle c Work out the size of angle a

Here are two cuboids.

All lengths are measured in centimetres. The volume of cuboid A is 142 cm³ greater than the volume of cuboid B. Work out the value of x.

The diagram shows triangle ABC.

In the diagram, all measurements are in centimetres. AC = BC The perimeter of the triangle is 72 cm. Work out the area of the triangle.

The diagram shows triangle ABC.

In the diagram, all measurements are in centimetres. AC = BC The perimeter of the triangle is 72 cm. Work out the area of the triangle.

22There are only red counters and yellow counters in a box. of the counters are red. Sophie takes at random two counters from the box. The probability that the two counters are the same colour is Work out the number of yellow counters in the box. You must show all your working.

A shop sells apples and oranges. There are 6 apples in each pack of apples. There are 7 oranges in each bag of oranges. The shop sells x packs of apples and y bags of oranges. Write an expression, in terms of x and y, for the total number of apples and oranges the shop sells.

Here is a number machine.

Work out the output when the input is 13

Work out the input when the output is 28

Show that there is a number for which the output is the same as the input.

21A solid cuboid has a volume of 300 cm³ The cuboid has a total surface area of 370 cm² The length of the cuboid is 20 cm. The width of the cuboid is greater than the height of the cuboid. Work out the height of the cuboid. You must show all your working.

ABCD is a quadrilateral.

All angles are measured in degrees. Show that ABCD is a trapezium.

Simplify m ×mm×m

In a competition, a player gets 5 points for each game they win 2 points for each game they draw 0 points for each game they lose. Amy wins x games and draws y games.

Write down an expression, in terms of x and y, for the total number of points Amy gets.

Mano has three shelves of books. There are x books on shelf A. There are (3 x + 1) books on shelf B. There are (2 x – 5) books on shelf C. There is a total of 44 books on the three shelves. All the books have the same mass. The books on shelf B have a total mass of 7500 g. Work out the total mass of the books on shelf A.

Mano has three shelves of books. There are x books on shelf A. There are (3 x + 1) books on shelf B. There are (2 x – 5) books on shelf C. There is a total of 44 books on the three shelves. All the books have the same mass. The books on shelf B have a total mass of 7500 g. Work out the total mass of the books on shelf A.

Vicky has a collection of medals. The Venn diagram gives information about the number of medals in her collection where E = {all medals} A = {English medals} B = {gold medals}

Vicky is going to take at random a medal from her collection. Given that the medal is gold, the probability that the medal is English is Work out the number of medals in Vicky’s collection.

Naomi has b bags of apples and c crates of apples. There are 5 apples in each bag. There are 28 apples in each crate. Naomi has a total of T apples. Write a formula for T in terms of b and c.

Here is a triangle and a rectangle.

All measurements are in centimetres. The area of the triangle is 10 cm² greater than the area of the rectangle. Work out the value of x.

Here is a triangle and a rectangle.

All measurements are in centimetres. The area of the triangle is 10 cm² greater than the area of the rectangle. Work out the value of x.

Gabriel thinks of a number. He multiplies his number by 5 and then adds 7 His answer is 72 What number did Gabriel think of?

Here is a quadrilateral ABCD.

All the measurements are in centimetres. The perimeter of ABCD is 52 centimetres. Work out the length of DC.

Simplify 4 c + 7 d + 3 c − d

Solve 5(2m − 6) = 40

There are x sweets in a box. There are y sweets in a packet.

Write an expression, in terms of x and y, for the total number of sweets in 3 boxes and 2 packets.

A first aid test has two parts, a theory test and a practical test. The probability of passing the theory test is 0.75 The probability of passing only one of the two parts is 0.36 The two events are independent. Work out the probability of passing the practical test.

The ratio of Marta’s hourly pay to Khalid’s hourly pay is 6: 5 Both Marta and Khalid get an increase of £1.50 in their hourly pay. The ratio of Marta’s hourly pay to Khalid’s hourly pay after this increase is 13: 11 Work out the hourly pay before the increase for Marta and for Khalid.

Rick, Selma and Tony are playing a game with counters. Rick has some counters. Selma has twice as many counters as Rick. Tony has 6 counters less than Selma. In total they have 54 counters. the number of counters Rick has: the number of counters Tony has = 1: p Work out the value of p.

Rick, Selma and Tony are playing a game with counters. Rick has some counters. Selma has twice as many counters as Rick. Tony has 6 counters less than Selma. In total they have 54 counters. the number of counters Rick has: the number of counters Tony has = 1: p Work out the value of p.

The diagram shows rectangle STUV. TQU and SRV are straight lines. All measurements are in cm.

The area of trapezium QUVR is Acm² Show that A = 2x² + 20x

ABCD is a kite.

AB = (4x – 2) cm Jasper says that x could be 0.5

Explain why Jasper cannot be correct.

AD = 3AB The kite has a perimeter of 64 cm.

Find the value of x.

26The diagram shows a cube with edges of length xcm and a sphere of radius 3 cm.

The surface area of the cube is equal to the surface area of the sphere.

The diagram shows a cube with edges of length xcm and a sphere of radius 3 cm.

The surface area of the cube is equal to the surface area of the sphere.

The diagram shows two rectangles, A and B.

All measurements are in centimetres. The area of rectangle A is equal to the area of rectangle B. Find an expression for y in terms of w.

The first four terms of a Fibonacci sequence are a 2 a 3 a 5 a The sum of the first five terms of this sequence is 228 Work out the value of a.

The diagram shows two shaded shapes, A and B. Shape A is formed by removing a sector of a circle with radius (3x – 1) cm from a sector of the circle with radius (5 x – 1) cm. Shape B is a circle of diameter (2 – 2 x) cm.

The area of shape A is equal to the area of shape B. Find the value of x. You must show all your working.

Here are two squares, A and B.

The length of each side of square B is 4 cm greater than the length of each side of square A. The area of square B is 70 cm² greater than the area of square A. Find the area of square B. Give your answer correct to 3 significant figures. You must show all your working.

There are only red sweets and yellow sweets in a bag. There are n red sweets in the bag. There are 8 yellow sweets in the bag. Sajid is going to take at random a sweet from the bag and eat it. He says that the probability that the sweet will be red is

Show why the probability cannot be

After Sajid has taken the first sweet from the bag and eaten it, he is going to take at random a second sweet from the bag. Given that the probability that both the sweets he takes will be red is

work out the number of red sweets in the bag. You must show all your working.

There are 60 people in a choir. Half of the people in the choir are women.

The number of women in the choir is 3 times the number of men in the choir. The rest of the people in the choir are children.

the number of children in the choir: the number of men in the choir = n: 1

Work out the value of n. You must show how you get your answer.

The number of days, d, that it will take to build a house is given by

where n is the number of workers used each day.

Ali’s company will take 40 days to build the house. Hayley’s company will take 30 days to build the house.

Hayley’s company will have to use more workers each day than Ali’s company.

How many more?

The diagram shows a parallelogram.

The area of the parallelogram is greater than 15 cm²

Find the range of possible values of x.

Here is a rectangle.

Coby has to find the perimeter of this rectangle.

He writes,

What mistake has Coby made?

Here is a triangle.

Iram solves a problem about this triangle to find the value of x.

Her answer is

Explain why Iram’s answer must be wrong.

There are only red counters and g green counters in a bag. A counter is taken at random from the bag. The probability that the counter is green is The counter is put back in the bag. 2 more red counters and 3 more green counters are put in the bag. A counter is taken at random from the bag. The probability that the counter is green is Find the number of red counters and the number of green counters that were in the bag originally.

There are y boats on a lake. There are 7 people in each boat. Write an expression, in terms of y, for the total number of people in the boats.

The diagram shows a rectangle, ABDE, and two congruent triangles, AFE and BCD.

area of rectangle ABDE area of triangle AFE + area of triangle BCD AB: AE 1: 3 Work out the length of AE.

There are only green pens and blue pens in a box.

There are three more blue pens than green pens in the box. There are more than 12 pens in the box.

Simon is going to take at random two pens from the box. The probability that Simon will take two pens of the same colour is Work out the number of green pens in the box.

p and q are two numbers such that p q

When you subtract 5 from p and subtract 5 from q the answers are in the ratio 5: 1 When you add 20 to p and add 20 to q the answers are in the ratio 5: 2

Find the ratio p: q Give your answer in its simplest form.

The diagram shows a pentagon. The pentagon has one line of symmetry.

AE = 4 x AB = 2 x + 1 BC = x + 2

All these measurements are given in centimetres.

The perimeter of the pentagon is 18 cm.

Show that 10 x + 6 = 18

Find the value of x.

A pattern is made from four identical squares.

The sides of the squares are parallel to the axes.

Point A has coordinates (6, 7) Point B has coordinates (38, 36) Point C is marked on the diagram.

Work out the coordinates of C.

Here are five straight rods.

All measurements are in centimetres.

The total length of the five rods is L cm.

Find a formula for L in terms of a. Write your formula as simply as possible.

Kiaria is 7 years older than Jay. Martha is twice as old as Kiaria. The sum of their three ages is 77

Find the ratio of Jay’s age to Kiaria’s age to Martha’s age.

Here is a rectangle.

All measurements are in centimetres.

The area of the rectangle is 48 cm².

Show that y = 3

Kiaria is 7 years older than Jay. Martha is twice as old as Kiaria. The sum of their three ages is 77

Find the ratio of Jay’s age to Kiaria’s age to Martha’s age.

Here is a rectangle.

All measurements are in centimetres.

The area of the rectangle is 48 cm².

Show that y = 3

3 teas and 2 coffees have a total cost of £7.80 5 teas and 4 coffees have a total cost of £14.20

Work out the cost of one tea and the cost of one coffee.

Here is a rectangle.

The length of the rectangle is 7 cm longer than the width of the rectangle.

4 of these rectangles are used to make this 8-sided shape.

The perimeter of the 8-sided shape is 70 cm.

Work out the area of the 8-sided shape.

Here is a rectangle.

The length of the rectangle is 7 cm longer than the width of the rectangle.

4 of these rectangles are used to make this 8-sided shape.

The perimeter of the 8-sided shape is 70 cm.

Work out the area of the 8-sided shape.

The area of square ABCD is 10 cm².

Show that x² + 6x = 1

Cups are sold in packs and in boxes.

There are 12 cups in each pack. There are 18 cups in each box.

Alison buys p packs of cups and b boxes of cups.

Write down an expression, in terms of p and b, for the total number of cups Alison buys.

The size of the largest angle in a triangle is 4 times the size of the smallest angle. The other angle is 27° less than the largest angle.

Work out, in degrees, the size of each angle in the triangle. You must show your working.