The four operations

Write a number in each box to make each statement true.

Calculate.

Give your answer correct to 3 significant figures.

Here is a list of numbers.

15 17 21 28 39 49

From this list, write down

the prime number,

the square number,

two numbers with a difference of 18.

Work out.

10 - 2 × 3

-1 + 3²

Insert one pair of brackets to make this calculation correct.

6 × 10 ÷ 2 - 4 = 6

Bottle A contains 1.5 litres of water and bottle B contains 100 millilitres of water.

1.5 litres 100 millilitres

Water is poured from bottle A into bottle B until both bottles contain the same amount.

How many millilitres of water are poured from bottle A into bottle B ?

Sundip has some £5, £10, £20 and £50 bank notes. She buys 30 litres of fuel for her car. One litre of fuel costs £1.50.

Sundip pays the exact amount of money for the fuel using three bank notes.

Work out the three bank notes that she uses to pay for the fuel.

(a) £........, £........ and £........

On one journey, Sundip’s car uses fuel at a rate of one litre every 16 kilometres. At the start of the journey, her car has 50 litres of fuel. At the end of the journey, her car has used of this fuel. Use the conversion 5 miles = 8 kilometres to work out the number of miles Sundip travels on this journey. You must show your working.

Work out.

- 3.5 × - 4

0.1 ÷ 0.4

Write 5 × 5 × 5 × 5 × 5 × 5 as a power of 5.

Write down the value of 1¹⁰.

Work out.

4 × 4 × 4 - 3³

A shop assistant works for 7 hours each day for 5 days. The shop assistant is paid the same amount for each hour they work. In the 5 days, the shop assistant earns £448.

How much does the shop assistant earn per hour?

A student thinks of a number. The student takes 16 from the number and then divides by 4. The answer is 8.

What number is the student thinking of?

A teacher thinks of a number. 150% of the number is 36.

What number is the teacher thinking of?

This is a function machine.

Find the output when the input is 10.

Find the input when the output is 24.

This is a different function machine.

a and b are both positive integers. When a is 7 the value of b is 26.

Find another pair of values for a and b.

Work out.

- 3.5 × - 4

0.1 ÷ 0.4

Insert two of these symbols +, -, × or ÷ to make this calculation correct.

5........ (3........ 1) = 20

Calculate

Give your answer correct to 3 significant figures.

The cost of sending a child to nursery is £5.80 per hour on each weekday. On Saturday the cost is times the weekday hourly rate.

A child goes to nursery for:

• 6 hours on Wednesday

• 5 hours on Thursday

• 4 hours on Saturday.

The total cost for the three days is reduced by 5% as a special offer.

Work out the cost for the three days after the 5% reduction. You must show your working,

Work out.

7 + - 5

26 × 6

1648 ÷ 8

4.78 + 5.3

The vertical line chart shows the rainfall, in millimetres (mm), in March and April.

Write down the rainfall in March.

In May there was 55 mm of rainfall.

Complete the vertical line chart for May.

Work out the total rainfall for March, April and May.

A sequence is generated using the rule:

• multiply the previous term by 3

• then subtract 1.

The 2nd term of the sequence is 20.

Find the 3rd term of the sequence.

Find the 1st term of the sequence.

Here are the first four terms of a different sequence.

5 10 15 20

Find the n th term of the sequence.

80 students each chose two activities, one from Option 1 and one from Option 2.

This frequency tree shows the number of students choosing some of the activities.

How many more students chose hockey and multigym rather than football and multigym?

Ten more students chose football rather than hockey.

Complete the frequency tree.

Anika has a shelf 79.6 cm long. She has many books, each of width 3.4 cm. Anika puts two paperweights, each of width 5 cm, and the maximum possible number of books on the shelf.

Work out the amount of space on the shelf that is left over. You must show your working.

Ben thinks of a number.

Ben says,

When I square root my number and divide the result by 10 the answer is 1.3.

Find Ben’s number.

These are the ingredients for making some scones.

10 ml of milk weighs 10.4 g.

Work out the total weight of all the ingredients.

This is a function machine.

Find the output when the input is 9.

Find the input when the output is 36.

Complete this function machine to show the equation y = 8 + 3x.

Machine A makes enough lollipops to fill 300 packs.

There are 8 lollipops in each pack.

Show that 2400 lollipops are made by machine A.

Machine B makes 3600 lollipops in the same time it took by machine A to make 2400 lollipops.

Machine B makes lollipops one at a time and at a constant rate.

What fraction of the time needed to fill 300 packs is saved if machine B is used rather than machine A? Give your answer in its simplest form.

Shop A and shop B have special offers on the same cupcakes.

Show that the special offer cost of 6 cupcakes from Shop A is £6.25.

Gabi wants 25 cupcakes for a party.

Which shop will be cheapest and by how much? Show how you decide.

(b) Shop........ by........ p

Some sequences are defined using this term-to-term rule.

u_n+1 = 5u_n - 8.

If u₃ = 22, show that u₄ = 102.

If u₃ = 22, work out u₂.

If u₁ = 2, write down the value of u₅₀. Give a reason for your answer.

Anika has a shelf 79.6 cm long. She has many books, each of width 3.4 cm. Anika puts two paperweights, each of width 5 cm, and the maximum possible number of books on the shelf.

Work out the amount of space on the shelf that is left over. You must show your working.

A teacher writes down a number. They subtract 6 from the number and then divide by 8. Their answer is 81.

What number did the teacher write down?

Work out.

6.3 + 4.7

£4.50 - £0.27

Work out.

-7 - 4

- 42 ÷ 6

(-4)²

A bathroom shower uses water at a rate of 15 litres per minute.

How much water is used in a bathroom shower that lasts 12 minutes?

Compost is used to grow plants. Ivan has a sack containing 50 litres of compost. He uses this compost to fill pots for his plants.

Ivan fills six large pots each holding 7.5 litres.

Work out how much compost is left in the sack.

Ivan uses the remaining compost to fill small pots each holding 400 ml.

Work out the maximum number of small pots Ivan can fill with the remaining compost.

Work out how much compost will then be left in the sack.

Work out.

1.2 ÷ 0.03

Kai buys some keyrings. Each keyring costs £1.19. Kai pays with a £20 note and buys the largest number of keyrings possible.

How much change should Kai receive?

Work out.

1.2 ÷ 0.03

Find the number that is halfway between 4.2 and 6.

The costs to hire a motorhome are shown below.

Finley hired a motorhome and travelled 560 miles. The total cost of hiring the motorhome was £916.

Calculate how many days Finley hired the motorhome for. You must show your working.

Riley and Sam are conducting surveys. They are both given the same list of 12 463 people from which to select their sample.

Riley selects every 56th person. Sam selects every 64th person. They both start counting from the first name in the list.

Work out how many people will be selected to be in both surveys. You must show your working.

Work out.

45 × 100

37 ÷ 100

2.36 + 1.79

0.82 - 0.36

Complete the three missing values on this multiplication grid.

Complete each statement.

........ × 4 = 12.08

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.

What type of numbers are 2, 3 and 5?

Circle one answer from the list.

cube numbers even numbers odd numbers prime numbers square numbers

These are five tiles.

Arrange the five tiles to make a calculation with the answer 3.

Write down a multiple of 8 that can be made using two of the five tiles.

A family buys a television for £599. They pay a deposit of £119. They then pay the rest of the cost in 12 equal payments.

How much is each payment?

Insert one pair of brackets into each calculation to make it correct.

15 ÷ 7 - 2 = 3

5 × 2 + 3 × 2 = 26

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.

Work out.

0.8 ÷ 0.004

A carpenter has a plank of wood of length w metres. The carpenter cuts of the plank of wood into 20 equal pieces. Each piece has length 0.06 metres.

Work out the value of w, the original length of the plank of wood. 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.

Cookies are made using these ingredients.

How many eggs are needed to make 48 cookies?

How much sugar is needed to make 6 cookies?

Ashley has 520 g of cranberries and plenty of the other ingredients. Ashley thinks this is enough to make at least 80 cookies.

Is Ashley correct? Show working to support your answer.

Darcie makes 100 cookies. They are put into packets, each holding 6 cookies. Each packet of 6 cookies is sold for £1.35. Darcie sells all of these packets.

Work out how much money Darcie receives.

Work out.

20 - 16 ÷ 2

Insert one pair of brackets to make this calculation correct.

2 + 7 - 3 × 8 = 34

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.

A student is buying some gifts for their friends. The gifts are shown below with the prices.

Key ring
Badge
Wallet £3.50 £2.99 £7.50

The student has £50 to spend. They first buy 6 key rings and 2 wallets. They then buy badges with the remainder of the money.

Work out the maximum number of badges that the student can buy. You must show your working.

(a) Number of badges........

Work out the amount of money they have left over.

(b) Amount left over £........

Work out.

3.8 ÷ 0.02

Morgan makes 15 cakes. Each cake needs a piece of ribbon 18 cm long for decoration.

Ribbon is only sold in rolls of 1.2 metres, which can then be cut into 18 cm pieces. One roll of ribbon costs 92p.

Calculate the cost of the ribbon that Morgan must buy to decorate the 15 cakes. You must show your working.

Write a number in each box to make each statement true.

Work out.

0.35 + 6.2

4.8 ÷ 8

Ashley has £7 to spend on fruit. The table shows the prices.

Ashley buys 2 pineapples and 3 kilograms of bananas. Ashley spends the remaining money on strawberries.

Work out the mass, in grams, of strawberries that Ashley buys. You must show your working.

Write these numbers in order of size, starting with the smallest.

- 20 10 0.351

smallest

Find the difference between the largest and the smallest of these numbers.

34 304 3.04 300.4

Insert brackets to make this calculation correct.

5 - 5 × 5 = 0

Insert two of these symbols +, -, × or ÷ to make this calculation correct.

20 …. 5(1 …. 3) = 0

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.

Jamie has some empty boxes. Each box can hold 73 pencils. Jamie has 590 pencils.

Jamie says that eight boxes are needed to hold all of the pencils.

Is Jamie correct? You must show your working.

Here are the ticket prices for a zoo when bought at the gate.

Mr and Mrs Khan take their four children to the zoo. They buy their tickets at the gate.

How much do Mr and Mrs Khan save by buying a family ticket?

All ticket prices are reduced by 15% if bought online rather than at the gate. Mr and Mrs Morris take their one child to the same zoo. They buy their tickets online.

What is the lowest possible total cost of their tickets?

Here is a rule to work out the time, in minutes, needed to cook a turkey.

Ling’s turkey takes 150 minutes to cook.

Use the rule to work out the weight of Ling’s turkey.

James cooks a different turkey. His turkey weighs 6 kg. James wants to take his turkey out of the oven at 1:15 pm.

Use the rule to work out at what time James should put his turkey in the oven. You must show your working.

Work out.

4 – 5

2 ×-3

Write down the largest prime factor of 30.

Bananas cost 25p each.

How many bananas can be bought for £2?

Work out.

3.08 + 0.82

7.7 ÷ 11

Work out.

Give your answer as a decimal.

A local theatre is putting on a show. 50 child tickets are sold. The ratio of the number of child tickets sold to the number of adult tickets sold is 5: 2.

The cost of a child ticket is £2.50. The cost of an adult ticket is £5.00.

Work out the total amount paid for the tickets.

The diagram shows Kai’s garden. It is in the shape of a right-angled triangle.

Kai is going to spread grass seed on the garden.

• A bag of grass seed covers an area of 35 m².

• Each bag of grass seed costs £8.99.

Kai can only buy whole bags of grass seed. Kai buys the least number of bags needed for the garden.

Calculate the cost of buying the bags of grass seed that Kai needs. You must show your working.

Frankie goes on holiday. They change £375 into euros (€) at a rate of £1 = €1.15. They spend €217.49 of this money. After the holiday, Frankie changes the remaining euros back into pounds at a rate of £1 = €1.28.

Work out how many pounds Frankie gets back.

Calculate.

A student thinks of a number. They square it and then add 6. Their answer is 295.

What number is the student thinking of?

Work out.

- 7 + 10

4 × -2

Work out.

9.06 ÷ 3

15 × 0.6

Work out.

4 + 16 ÷ 2

Insert one pair of brackets to make the calculation correct.

5 × 7 + 1 ÷ 9 = 4

Here is a function.

Find the output when the input is 2.

Find the input when the output is 63.

A shopper buys 4 apples costing 60 p each and 3 peaches. They pay with a £5 note and receive 44 p in change. Each peach costs the same amount.

Work out the cost of one peach. You must show your working.

A car mechanic has a tin containing 5 litres of engine oil. Each week they use 450 millilitres of this oil for their vehicles.

The car mechanic says

After 9 weeks I will have used over 80% of the oil in this tin.

Are they correct? Show how you decide.

There are 150 coins in a jar. 20% of the coins are 10p coins. of the coins are 20p coins. The rest of the coins are 50p coins.

Work out the total value, in £, of the 150 coins. You must show your working.

Ellis has 28 m of ribbon. They cut the ribbon into lengths of 60 cm.

What is the least length of ribbon, in cm, that can be left over? You must show your working.

Calculate.

(6² + 5)³

Write your answer correct to 3 significant figures.

A car mechanic has a tin containing 5 litres of engine oil. Each week they use 450 millilitres of this oil for their vehicles.

The car mechanic says

After 9 weeks I will have used over 80% of the oil in this tin.

Are they correct? Show how you decide.

Yoghurts are packed in trays. Each tray holds 12 yoghurts.

What is the smallest number of trays needed to pack 460 yoghurts?

Nadia thinks of a number. She finds the square root and then divides by 5. Her answer is 20.

What number is she thinking of?

Work out.

-1 + 6

7 - - 3

Write down two prime numbers between 10 and 20.

Complete each statement by writing the missing value in the box.

Work out.

0.8 ÷ 2

1.7 × 2

Charlie and Jasmine share cartons of apple juice. Charlie drinks of a carton every day. Jasmine drinks of a carton every day.

Any apple juice left in a carton at the end of the day is used the following day.

The cost of a carton is 70p. Charlie and Jasmine buy just enough cartons to last them for 10 days.

How much do they spend in total for these cartons? Give your answer in £. Show your working.

Alex has a number game. He must put down tiles to make two calculations with the same answer.

Here is what Alex put down.

Is he correct? Show how you decide.

Alex is........ because........

Lily buys and sells microwaves.

She buys each one for £32 and sells it for £60. She also pays £7 for the delivery of each microwave she sells.

If she sells a microwave that is faulty then Lily must pay for its repair and redelivery. This costs her another £25 for each faulty microwave.

Last month, 6 out of the 80 microwaves Lily sold were faulty.

This month she has orders for 133 microwaves.

Calculate her expected percentage profit on this month’s order. Showing your working in the boxes below may help you present your work.

Plaza United are playing a football match away from home.

379 supporters are going to the match by coach. Each coach seats 45 people.

What is the smallest number of coaches that will be needed?

In their last 50 matches, Plaza United have drawn 10 matches, lost 5 and won the rest.

Sam claims

The probability that Plaza United will win this match is 0.7.

Show calculations to support Sam’s claim.

Give one reason why Sam’s claim may not be reliable.

A number that does not divide exactly by 2........

A number that has no factors except itself and 1........

Write down all the multiples of 4 between 21 and 29.

Write down a common multiple of 4 and 6.

Insert brackets to make this calculation correct.

4 - 1 × 2 = 6

Write 7% as a fraction.

Work out.

1.52 kg + 80 g

Give your answer in kilograms.

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

What number is Dan thinking of?

Martina has answered some questions on algebra. In each question, she has made an error.

Describe her error and give the correct answer to each problem.

Question 1 Simplify. 2a × a × a

Martina’s answer 4a

Martina’s error is........

Martina’s error is........

s = 225

Martina’s error is........

Kim is paid £9.40 per hour for the first 35 hours she works each week. After 35 hours she is paid at one and a quarter times the hourly rate.

One week Kim works 42 hours.

Calculate how much she is paid for that week.

Work out.

£ 4.25 + £ 5.18

-8 + 11

-6 × -9

Use one of these symbols <, > or = to make each statement true.

4.5........ 4.34

Work out.

0.7 × 0.3

0.48 ÷ 6

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.

A sunflower grows at a rate of 4 cm each day.

How many days does it take to grow from a height of 80 cm to more than 1.06 m?

If the sunflower grows at a faster rate, how would this affect your answer to part (a) ?

Insert brackets to make each of these calculations correct.

5 × 3 - 1 = 10

3 + 6 - 2 ÷ 2 = 3.5

Joan makes cups of tea and coffee at a lunch club. Each cup requires 250 ml of boiling water. She has a kettle that boils up to 1.7 litres of water each time.

She boils 10 litres of water in an urn. She then uses the kettle to boil the rest of the water she needs.

Find the least number of times that Joan needs to boil the kettle to make 56 cups. Show how you decide.

A sunflower grows at a rate of 4 cm each day.

How many days does it take to grow from a height of 80 cm to more than 1.06 m?

If the sunflower grows at a faster rate, how would this affect your answer to part (a) ?

Sophie is organising a raffle.

• Each raffle ticket costs 50p.

• She sells 400 tickets.

• The probability that a ticket, chosen at random, wins a prize is 0.1.

• Each winning ticket receives a prize worth £3.

Sophie says

I expect the raffle to make over £100 profit.

Show that Sophie is wrong.

A water company charged the following in 2017.

£2.00 for each m³ of water used plus a fixed charge of £45

In 2017 Jenny used 110 m³ of water. For the 12 months of 2017 she paid £20 per month to the water company.

How much more money does Jenny need to pay to the water company?

Work out.

8 ÷ 0.4

By writing each number correct to 1 significant figure, find an estimate for this calculation.

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) ?

Work out.

5 × (2 + 4)

Tia thinks of a number. She finds the square root and subtracts 4. Her answer is 1.

What number is she thinking of?

This table shows the ticket price for each person to visit a zoo in winter.

A family of two adults, one child aged 8 and one child aged 1 visit the zoo in winter.

What is the total cost of the tickets?

In summer, the zoo increases the prices.

An Adult ticket increases by 20%. A Child ticket increases by 15%.

How much more does it cost the family to visit in the summer than in the winter?

Shari buys a box of 60 candles for £125. She sells the candles for £2.25 each.

Calculate her percentage profit.

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) ?

Patrick writes down a number.

He says

If I find the square root of that number and then add 15, I get 27.

What number did Patrick write down?

Reuben hires a car. It costs £150, plus 85p for each mile he travels.

When Reuben hires the car, its mileage is 27 612 miles. When Reuben returns the car, its mileage is 28 361 miles.

How much did Reuben pay to hire the car?

Write these fractions as decimals.

A plank of wood 2.4 m long is cut into 6 pieces of equal length.

How long is each piece?

Work out.

10³

9(8 – 3 × 2)

Put brackets into this sum so that the answer is correct.

1 + 2 × 3 + 5 = 17

Liam is 0.83 metres tall. William is 1.31 metres tall. Jacob is taller than Liam by half the difference between Liam’s height and William’s height.

How tall is Jacob?

Here is the floor plan of a rectangular room.

Tim buys carpet tiles for this room.

Each tile is a square measuring 50 cm by 50 cm. The tiles are only sold in packs of ten. Each pack costs £20. Tim pays for fitting at a rate of £7.50 per square metre, with any fraction of a square metre rounded up.

Work out the total cost of the tiles and fitting.

Calculate.

Work out.

1 + 4 ÷ 2

2 + 5 × (8 - 4)

Evaluate.

2⁵

Estimate the value of

This is a rule to find the time, in minutes, needed to roast lamb.

Use the rule to work out the time needed to roast a piece of lamb which weighs 4 pounds.

A different piece of lamb takes 95 minutes to roast.

Use the rule to work out the weight of this piece of lamb.

Fill in each missing number.

24 –........ = 36

The length of a line is 10.4 cm, correct to 1 decimal place.

Write down the shortest possible length of the line.

To find the number in a square, multiply the numbers in the two circles connected to it.

Fill in the missing numbers.

Factorise.

x² – 43²

Calculate.

57² – 43²

320 people go on a coach trip. Each coach holds 53 people.

Gary says 6 coaches are needed.

Is Gary correct? You must show your working.

Nathan works out 23 × 12.4 without a calculator.

This is Nathan’s working.

Nathan’s working is incorrect.

Explain the error that Nathan has made and work out the correct answer.

Four friends buy cinema tickets using this offer.

Cinema tickets

Buy 3 tickets and get a ticket free

They each pay £6.45.

How much does a ticket cost?

The shape below is formed from a rectangle measuring 12 cm by 15 cm from which a rectangle of length 7.4 cm has been removed.

Work out the perimeter of the shape.