Linear equations

Simplify 45e

Solve f 14 29

Simplify 4e × 7f

Simplify d × d × d × d × d

Simplify 3a + 2k + a – 7k

Solve x + 6 = 15

Solve 2r – 9 = 14

Aabir has some boxes and some crates. Each box has the same weight. Each crate has the same weight.

The total weight of 7 boxes and 2 crates is 27.3 kg The total weight of 4 boxes and 2 crates is 16.8 kg

Work out the weight of one crate.

Show clear algebraic working.

PTR and QTS are chords of a circle.

PT = (3y – 5) cm QT = (y + 3) cm RT = 4xcm ST = (x + 2) cm

Find an expression for y in terms of x

Simplify d + d + d + d

Simplify w w w w w

Solve 6x = 42

Simplify 7r + 9y + 3r – 4y

Simplify a + a + a + a + a

Simplify b c 7

Solve 5d 40

7 2 cupcakes and 5 doughnuts cost $4.50 3 cupcakes cost $2.25

Work out the cost of 1 doughnut.

Raymond sells 80 burgers for a total of £176 of the burgers are large. Each large burger costs £3

The remaining burgers are small.

Work out the cost of one small burger.

Factorise 15 5x

d = 5p + 7r

Work out the value of p when d = 35 and r = 2

Sweets are sold in small packets and large packets.

There are 12 sweets in a small packet. There are 25 sweets in a large packet.

Ben buys m small packets of sweets and n large packets of sweets.

Write an expression, in terms of m and n, for the total number of packets of sweets Ben buys.

Write an expression, in terms of m and n, for the total number of sweets Ben buys.

Factorise fully 18c 45cd

Show clear algebraic working.

Simplify 8d 4d 7d

Simplify m 10 p

Solve x 5 19

Simplify a a a a

k 8 and n = 12

Work out the value of 9k 4n

Simplify 10p – 4p + 9p

Simplify 94q

Solve 4r = 15

Harold works in a factory.

His normal hourly rate of pay is £14 His overtime hourly rate of pay is £21

Harold is paid the normal hourly rate of pay for 35 hours in one week. His total pay for this week is £679

Work out the number of hours of overtime he works in this week.

Simplify 10x – 7y – 6x + 4y

T = 4d – 6e

Work out the value of T when d = 13 and e = 7

Solve 5p + 11 = 28

Simplify c + c + c + c + c

Simplify 7w + 10y ‒ 9w + 2y

The nth term of a sequence is given by 7n – 4

Find the 1st term and the 5th term of the sequence.

Solve 7(x + 5) = 8 ‒ 3x Show clear algebraic working.

Here are six cards.

Five of the cards have a number written on them.

Work out the number that should be written on the last card so that the mean of the six numbers will be 11

450 students were asked how they travelled to school on Monday. Each student walked or travelled by bus or travelled by car or travelled by bicycle. Each student used just one method of travel.

One of these students is chosen at random. The table shows information about the probability of each method of travel.

Work out how many of the 450 students travelled by car.

Here is a biased spinner.

The table gives information about the probability that, when the spinner is spun once, it will land on each number.

Alexis is going to spin the spinner 400 times.

Work out an estimate for the number of times the spinner will land on an odd number.

2 450 students were asked how they travelled to school on Monday. Each student walked or travelled by bus or travelled by car or travelled by bicycle. Each student used just one method of travel.

One of these students is chosen at random. The table shows information about the probability of each method of travel.

Work out how many of the 450 students travelled by car.

Here is a biased spinner.

The table gives information about the probability that, when the spinner is spun once, it will land on each number.

Alexis is going to spin the spinner 400 times.

Work out an estimate for the number of times the spinner will land on an odd number.

Expand and simplify (m + 5) (m – 8)

Show clear algebraic working.

Simplify c × c × c

Simplify 12d × 3e

Solve 2g 3 6

Expand x(x – 4)

P 4y² w

Work out the value of P when y = −3 and w = 2

Factorise fully 12 pq 18p

There are 56 metal bars in a box. Each metal bar is gold or silver or zinc.

y metal bars are gold. (3 y + 7) metal bars are silver. (2 y − 5) metal bars are zinc.

Work out the number of metal bars that are zinc. Show clear algebraic working.

A biased spinner has three sections each of a different colour.

The table shows the probability that, when the spinner is spun once, it will land on blue or on orange or on white.

Work out the value of x

The spinner is spun 250 times.

Work out an estimate for the number of times the spinner will land on blue.

A bag contains only 10 cent coins and 20 cent coins.

Josip takes at random a coin from the bag, records its value and replaces it in the bag. He then takes at random a second coin from the bag, records its value and replaces it in the bag.

Josip finds the mean value of the two coins. The probability that the two coins have a mean value of 10 cents is Work out the probability that the two coins have a mean value of 15 cents.

Barney went for 4 walks on Tuesday.

The lengths of the walks were

800 metres 2 kilometres 1.7 kilometres x metres

The total length of the 4 walks was 6250 metres.

Work out the value of x

Show clear algebraic working.

Simplify h¹⁵ h³

Simplify fully 2g³k^{5 4}

work out the value of n

The diagram shows a hexagon ABCDEF

AB = 25 cm BC = (x + 2) cm CD = 8 cm EF = 7 cm AF = (x + 6) cm

The area of hexagon ABCDEF is 258 cm²

Work out the value of x

Show clear algebraic working.

Simplify p + p + p + p

Simplify 5e + 6f + 7e – 2f

Solve 13 – x = 7

Solve 4y + 7 = 43

Simplify 6a × 2c

Simplify 4d + 3e + d – 5e

Solve 4x – 7 = 23

Carla wraps some parcels with ribbon. She wraps 2 large parcels and 3 small parcels.

For each large parcel, Carla uses 185 centimetres of ribbon. For each small parcel, she uses p centimetres of ribbon.

In total, Carla uses exactly 7 metres of ribbon to wrap the parcels.

Find the value of p

There are 30 counters in a bag.

13 of the counters are purple. 11 of the counters are white. The rest of the counters are red.

Suha takes at random a counter from the bag.

Write down the probability that the counter is purple.

Work out the probability that the counter is red.

The counter is put back into the bag.

Clive now puts 10 more counters into the bag. When a counter is taken at random from the bag, the probability that it is white is now

How many of the 10 counters that Clive puts into the bag are white?

Pablo buys some tickets to go to the theatre.

3 of the tickets are for adults. The remaining tickets are for children.

Each adult ticket costs 12 euros. The children’s tickets each cost 30% less than an adult ticket.

The total amount of money that Pablo pays for all the tickets is 94.80 euros.

Find the number of children’s tickets Pablo buys.

Roland, Seiso and Tim share the total cost of buying a plot of land.

Roland and Seiso share some of the cost in the ratio 2: 5 Roland’s share of the cost is $1700

Tim’s share of the cost is $2150 more than Roland’s share.

Work out the total cost of buying the plot of land.

Team A and Team B take part in a quiz league.

After 11 rounds, Team A has a mean score per round of 17 After 9 rounds, Team B has a mean score per round of 18

Both teams take part in a further round. After this round, both teams have a mean score per round of 18.5

In the further round, Team A scored more points than Team B.

How many more?

Express in the form pxⁿ where p and n are integers.

Show clear algebraic working.

Simplify 10y – y

Simplify 3p4p

Solve 7x = 42

Solve n + 6 = 5

Simplify 8c + 5d – 2c – 3d

Simplify 5p × 9k

Simplify 3e + 2f + 8e – 7f

Solve 2d + 7 = 16

Solve 3(2 – 4 x) = 5 – 8 x Show clear algebraic working.

The diagram shows a pentagon.

Work out the value of x

Some members of a library were asked to name the type of book that they each liked to read the best.

One of the members is chosen at random. The table shows information about the probability of the type of book that this member answered.

48 members answered comedy books.

Work out how many of the members answered mystery books.

Solve 3(2 – 4x) = 5 – 8x Show clear algebraic working.

Some members of a library were asked to name the type of book that they each liked to read the best.

One of the members is chosen at random. The table shows information about the probability of the type of book that this member answered.

48 members answered comedy books.

Work out how many of the members answered mystery books.

22The first term of an arithmetic series is (2t + 1) where t 0 The n th term of this arithmetic series is (14 t – 5)

The common difference of the series is 3

The sum of the first n terms of the series can be written as p(qt – 1)^r where p, q and r are integers.

Find the value of p, the value of q and the value of r Show clear algebraic working.

p =_........ q =_........ r =_........

Pat has 4 parcels A, B, C and D

The weight of parcel A is 400 grams.

The weight of parcel B is 350 grams more than the weight of parcel A The weight of parcel C is twice the weight of parcel A

The total weight of the 4 parcels is 2.5 kilograms.

Work out the weight, in grams, of parcel D

Simplify c × c × c c c

Solve 5 + x = 12

Expand 5(2 + 3h)

Factorise g² + 7g

Here is a number machine.

Work out the output when the input is 9

Here is a different number machine.

When the input is 30 the output is 18

Find a suitable way to complete the number machine.

The following rule is used to work out the total cost, in euros, of hiring a cement mixer.

James hires a cement mixer for 3 days.

Work out the cost to James of hiring the cement mixer.

The cost to Sophia of hiring a cement mixer is 61 euros.

For how many days does Sophia hire the cement mixer?

There are 380 students in a Sixth Form. The students are either in the Upper Sixth or in the Lower Sixth.

The number of students in the Upper Sixth is 20 fewer than the number of students in the Lower Sixth. of the Upper Sixth students study mathematics. 32% of the Lower Sixth students study mathematics.

Work out the total number of students in the Sixth Form who study mathematics.

Expand x (10 – x)

Factorise 6y + 27

Make m the subject of the formula h = + 4

Solve 7g + 3 = 2g – 5 Show clear algebraic working.

In a bag, there are only red counters, blue counters, green counters and yellow counters.

The total number of counters in the bag is 80

In the bag

the number of red counters is x + 7 the number of blue counters is x – 11 the number of green counters is 3x

Jude takes at random a counter from the bag. The probability that he takes a red counter is

Work out the probability that Jude takes a yellow counter.

The diagram shows rectangle ABCD

Work out the perimeter of the rectangle. Show your working clearly.

In a bag, there are only red counters, blue counters, green counters and yellow counters.

The total number of counters in the bag is 80

In the bag

the number of red counters is x + 7 the number of blue counters is x – 11 the number of green counters is 3x

Jude takes at random a counter from the bag. The probability that he takes a red counter is

Work out the probability that Jude takes a yellow counter.

Show clear algebraic working.

Simplify 12a + 3a – 7a

Simplify 8 3b

Sandeep buys some flowers. He has 5000 rupees to spend.

He buys 6 carnations at 220 rupees each. He also buys some roses at 295 rupees each.

Sandeep should receive 140 rupees in change from his 5000 rupees.

Work out how many roses Sandeep buys.

Simplify 12g − 8e − 5g + 6e

A = 3b − 5c

Work out the value of A when b = 12 and c = 4

Solve 4p + 9 = 24

Expand and simplify (n − 6)(n + 4)

Show clear algebraic working.

Here are five cards.

Each card has a number written on it.

The mean of the five numbers is 12

Work out the value of x

Expand and simplify (n − 6)(n + 4)

Show clear algebraic working.

The diagram shows a quadrilateral ABCD

In the diagram, ABC and DAC are right‑angled triangles.

BC = 6 cm AC = 7.5 cm

The area of quadrilateral ABCD is 31.5 cm²

Work out the length of AD

Find the value of

14³

Write a number on each dotted line to make the calculation correct.

10 –........ 2 = 4

(5 +........)3 = 36

6

Show clear algebraic working.

Simplify a⁰where a 0

Factorise fully 10c³d² + 15cd⁴

Solve 5x = 30

Solve y – 7 = 12

Simplify h + h + h + h + h

Simplify 5a + 7f – 2a + 4f

Simplify 3c + 5d – c + 2d

Simplify 8e 5f

Solve 5r – 3 = 8

Thabisa is organising a trip to the theatre.

The cost of a ticket for each adult is £11.75 The total cost of the tickets for 12 adults and 5 children is £181

Work out the cost of a ticket for each child.

Solve 5 c = 15

Expand x(8 – x)

T = 5m – 6n

Work out the value of T when m = 4.2 and n = –2.5

Make g the subject of k = 2g + t

Solve 4y + 5 12

Show clear algebraic working.

Solve 4y + 5 12

Show clear algebraic working.

The frequency table gives information about the number of points scored by a player.

The mean number of points scored is 2

Work out the value of x

Solve 5x = 20

Simplify 3a 8b

Simplify 8w – 4y + w – 3y

Factorise fully 16 + 12t

Yasmin has some identical rectangular tiles. Each tile is Lcm by Wcm.

Diagram NOT L cm accurately drawn

W cm

Using 9 of her tiles, Yasmin makes rectangle ABCD shown in the diagram below.

The area of ABCD is 1620 cm²

Work out the value of L and the value of W.

L =_........ W =_........

Yasmin has some identical rectangular tiles. Each tile is Lcm by Wcm.

Diagram NOT L cm accurately drawn

W cm

Using 9 of her tiles, Yasmin makes rectangle ABCD, shown in the diagram below.

The area of ABCD is 1620 cm²

Work out the value of L and the value of W.

L =_........ W =_........

Simplify 10x + 4y + 3x – 6y

Solve 2n + 5 = 16

Expand 5(3 a + 4)

Factorise 4c – 14

Solve 5x – 11 = x + 6 Show clear algebraic working.

The diagram shows a cuboid and a cylinder.

The dimensions of the cuboid are xcm by 12 cm by 5 cm. The volume of the cuboid is 270 cm³

The radius of the cylinder is xcm. The height of the cylinder is 2 x cm.

Work out the volume of the cylinder. Give your answer correct to the nearest whole number.

Change 1 m³ to cm³

Factorise fully 15y⁴ + 20uy³

Show clear algebraic working.

Factorise fully 15y⁴ + 20uy³

Show clear algebraic working.

The diagram shows a cuboid and a cylinder.

The dimensions of the cuboid are xcm by 12 cm by 5 cm. The volume of the cuboid is 270 cm³

The radius of the cylinder is xcm. The height of the cylinder is 2 x cm.

Work out the volume of the cylinder. Give your answer correct to the nearest whole number.

Change 1 m³ to cm³

Here is a number machine.

Work out the output when the input is 7

Work out the input when the output is 160

When the input is n, the output is P.

Find a formula for P in terms of n.

Simplify p + p + p + p + p + p

Simplify 5y² + 6y² – 3y²

Simplify eeeee

Simplify 5c4d

Solve x – 7 = 19

18² + 15² – 5³ = 4n

Work out the value of n.

Factorise 9t – 6

The diagram shows a right-angled triangle.

The perimeter of the triangle is 126 cm.

Work out the area of the triangle.

A mathematics teacher at a school asked a group of students how far, in kilometres, each student had travelled to get to school that day.

The table gives information about their answers.

Distance travelled (dkm)	Number of students
0 d 2	x
2 d 4	11
4 d 6	8
6 d 8	6
8 d 10	5

The teacher calculated that an estimate for the mean distance travelled by the whole group of students was 4.25 km.

Work out the value of x. Show your working clearly.

The diagram shows four parcels. The total weight of the four parcels is 8.3 kg.

The weight of the parcel labelled B is 3.2 kg. Each of the three parcels labelled A have the same weight.

Work out the weight of each of the parcels labelled A.

Here are another three parcels.

The total weight of the three parcels is 9.45 kg.

Each of the two parcels labelled D have the same weight.

The weight of each parcel labelled D is 3 the weight of the parcel labelled C.

Work out the weight of the parcel labelled C.

Simplify 5c d

Solve x + 5 = 12

Solve 9y = 36

Simplify 8k + 5m – 2k + 6m

Expand 4(3g + 1)

Here is a list of six numbers written in order of size.

4 7 x 10 y y

The numbers have

a median of 9 a mean of 11

Find the value of x and the value of y.

Simplify h⁷h²

G = c² – 4c

Find the value of G when c = –5

Show clear algebraic working.

Solve 5(4 – x) = 7 – 3 x Show clear algebraic working.

Factorise fully 16m³g³ + 24m²g⁵

Factorise y² – 2y – 48

Hence, solve y² – 2y – 48 = 0

Here is a 10-sided polygon.

Work out the value of x.

Simplify h⁷h²

G = c² – 4c

Find the value of G when c = –5

Show clear algebraic working.

Solve 5(4 – x) = 7 – 3x Show clear algebraic working.

Factorise fully 16m³g³ + 24m²g⁵

Factorise y² – 2y – 48

Hence, solve y² – 2y – 48 = 0

The following rule is used to work out the total cost, in euros, of hiring a room.

Paolo hires the room for 5 hours.

Work out the total cost.

Agathe also hires the room. The total cost is 164 euros.

For how many hours does Agathe hire the room?

The total cost of hiring the room for n hours is T euros.

Write down a formula for T in terms of n.

Expand 4(m + 2)

Solve 2x + 5 = –19

Show clear algebraic working.

Simplify 3r ×5t

Solve 4x+ 5 = 27

P= 7w– 5y

Find the value of P when w= 2 and y= 4

Q= 2u²− 5

Find the value of Q when u= −3

Simplify 6m− 2k+ 5m−k

P= 2a+ 3b

Work out the value of P when a= 5 and b= 8

P= 2a+ 3b

Work out the value of a when P= 16 and b= 20

Kamal sells 240 ice creams for a total of $640 of the ice creams he sells are large.

The cost of each large ice cream he sells is $3.80

All the other ice creams he sells are small. He sells each small ice cream for the same cost.

Work out the cost of each small ice cream.

The diagram shows a shape.

The shape has area 129 cm²

Work out the value of x.

Expand and simplify (m − 8)(m + 5)

Factorise fully 5y+ 20y²

Simplify (p²+ 3)⁰

Show clear algebraic working.

The diagram shows a regular hexagon, ABCDEF, and an isosceles triangle, GHJ.

The perimeter of the hexagon is equal to the perimeter of the triangle.

Find the length of each side of the hexagon. Show clear algebraic working.

The diagram shows a shape.

The shape has area 129 cm²

Work out the value of x.

Expand and simplify (m− 8)(m+ 5)

Factorise fully 5y+ 20y²

Simplify (p²+ 3)⁰

Show clear algebraic working.

The diagram shows a regular hexagon, ABCDEF, and an isosceles triangle, GHJ.

The perimeter of the hexagon is equal to the perimeter of the triangle.

Find the length of each side of the hexagon. Show clear algebraic working.

Simplify 10ab

Solve n + 3 = 7

In the diagram, A, B, C and D are points on a straight line.

AB = 30 cm AC = 96 cm BC = 3CD

Work out the length of AD.

In a shop,

3 bottles of juice cost $5.25

2 bottles of juice and 5 bars of chocolate cost $9.75

Work out the cost of 5 bottles of juice and 3 bars of chocolate.

Betsy was given $75 for her birthday.

She saved some of the money and spent the rest on a T-shirt and a bag.

Betsy saved 40% of the $75 She spent $12 more on the bag than she spent on the T-shirt.

Work out how much Betsy spent on the bag.

Three tins, A, B and C, each contain buttons.

Tin A contains x buttons. Tin B contains 4 times the number of buttons that tin A contains. Tin C contains 7 fewer buttons than tin A.

The total number of buttons in the three tins is 137

Work out the number of buttons in tin C.

The diagram shows a triangle.

Work out the value of x.

Three tins, A, B and C, each contain buttons.

Tin A contains x buttons. Tin B contains 4 times the number of buttons that tin A contains. Tin C contains 7 fewer buttons than tin A.

The total number of buttons in the three tins is 137

Work out the number of buttons in tin C.

Show clear algebraic working.

Solve the inequality 5y² – 17y 40

Simplify 6e 2f

Simplify 5m + 7km + k

Solve 5 y + 3 = 14

Becky has a biased 6-sided dice.

The table gives information about the probability that, when the dice is thrown, it will land on each number.

Becky is going to throw the dice 200 times.

Work out an estimate for the number of times that the dice will land on an even number.

Factorise 2x²x + 6

Show clear algebraic working.

Simplify 6x + 8xx

Simplify 4 e × 5 f

Solve 8 p = 24

Solve k

Simplify 10t + 4dt + 2d

7 3 kg of potatoes and 2 kg of apples cost a total of £7.33 4 kg of potatoes cost £3.80

Work out the cost of 1 kg of apples.

Simplify y⁵ × y⁹

Simplify (2m³)⁴

Solve 5(x + 3) = 3x Show clear algebraic working.

Factorise x² + 2x

Hence, solve x² + 2x

Simplify y⁵y⁹

Simplify (2m³)⁴

Solve 5(x + 3) = 3 x – 4 Show clear algebraic working.

Factorise x² + 2x – 24

Hence, solve x² + 2x – 24 = 0

3 kg of carrots cost £1.35 5 kg of carrots and 2 kg of potatoes cost a total of £4.15

Find the cost of 1 kg of potatoes.

Solve 8 – 2 p = 15

Show clear algebraic working.

x, 10 and y are three integers written in order of size, starting with the smallest integer.

The mean of x, 10 and y is 11 The range of x, 10 and y is 7

Work out the value of x and the value of y.

x, 10 and y are three integers written in order of size, starting with the smallest integer.

The mean of x, 10 and y is 11 The range of x, 10 and y is 7

Work out the value of x and the value of y.

Expand and simplify 3(c – 7) + 2(3 c + 4)

Expand and simplify (x + 7)(x – 2)

Factorise fully 28y² – 21y

Show clear algebraic working.