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Step-by-step practice: Maths · Level 4Chapter 9 · Solving problems with the four operations
Level 4 · Chapter 9

Solving problems with the four operations

By the end of this chapter you will be able to

  • Choose addition, subtraction, multiplication or division for a problem
  • Solve a two-step problem
  • Work backwards to find a missing number
  • Find a fraction of an amount in a story
  • Explain why an operation fits a problem

You should already know

  • Column addition and subtraction (chapter 3)
  • Times tables and division (chapters 4 to 6)
  • Fractions of amounts (chapter 7)
  • Time to the minute (chapter 8)

Where this leads

  • Level 4 check (chapter 10)

Step-by-step practice: maths (Level 4). Small steps, repeated until fluent. Content: Choose an operation for a one-step problem with numbers to 10000 (PV6, AS4, MD3), Solve a two-step problem using addition or subtraction, then multiplication or division (Year 4 paraphrase), Solve a problem with a missing number, by working backwards (AS4), Find a fraction of an amount in a problem (F7), Choose the operation in a problem and explain the choice (Year 4 paraphrase).

Exercise P  Do you remember?

Non-calculator
1
FWhat is 3/4 of 8?
2
FWhat is 1000 more than 2500?
3
FRound 4826 to the nearest 1000.
4
FWhat time is 3 hours after 10:20?
5
FHow many minutes are there in 2 hours?
9.1

Choosing one operation

Level 4
Choose addition, subtraction, multiplication or division for a one-step problem with numbers to 10000
Video lesson: Choosing one operationComing soon

Read the problem and look for the key words. Altogether and more mean add. Left, sold and taken away mean subtract.

Equal groups mean multiply. Sharing into equal groups means divide.

Key ideas

  • Altogether, total, more: add.
  • Left, sold, taken away: subtract.
  • Equal groups or sharing equally: multiply or divide.
Worked example 9.1.1
Sam has 2450 stickers in one box and 1875 stickers in another box. How many stickers are there altogether?

Method. Find the key words, choose the operation (addition), then work out the answer.

  1. Find the key words. “altogether” means put the two amounts together.
    2450 and 1875
  2. Choose the operation. Putting amounts together is addition.
    2450 + 1875
  3. Work out the answer. Add the ones, tens, hundreds and thousands with carrying.
    2450 + 1875 = 4325
  4. Write the answer. This is the total number of stickers.
    4325

Check. Check: 4325 - 1875 = 2450.

Watch out. Check the answer with the opposite operation.

Worked example 9.1.2
There were 5000 cards in a shop. 1275 cards were sold. How many cards are left?

Method. Find the key words, choose the operation (subtraction), then work out the answer.

  1. Find the key words. “sold” and “left” mean some are taken away.
    5000 and 1275
  2. Choose the operation. Taking away is subtraction.
    5000 - 1275
  3. Work out the answer. Use exchange where a column is too small.
    5000 - 1275 = 3725
  4. Write the answer. This is how many are left.
    3725

Check. Check: 3725 + 1275 = 5000.

Watch out. Check with addition: the amount left plus the amount sold makes the start.

Worked example 9.1.3
72 pencils are shared equally between 8 children. How many pencils does each child get?

Method. Find the key words, choose the operation (division), then work out the answer.

  1. Find the key words. “shared equally” means the same number in each group.
    72 into 8 groups
  2. Choose the operation. Sharing into equal groups is division.
    72 ÷ 8
  3. Work out the answer. Ask how many 8s make 72. Use the 8 times table.
    8 × 9 = 72
  4. Write the answer. This is the number for each child.
    9

Check. Check: 9 × 8 = 72.

Watch out. Use the times table fact to check the division.

Watch out

Watch out. Read the whole question. The numbers do not tell you the operation; the story does.

Exam tip

Grown-up tip. Ask: “What is the question asking for? Which key word tells us the operation?”

Exercise 9A  Practice: choosing one operation

Non-calculator
1
PSThere are 8 bags. Each bag has 91 books in it. How many books are there altogether?
2
PS356 cards are shared equally between 4 children. How many cards does each child get?
3
PSOmar has 2451 apples in one box and 3050 apples in another box. How many apples are there altogether?
4
PSLily has 3586 counters in one box and 1418 counters in another box. How many counters are there altogether?
5
PS156 stickers are shared equally between 6 children. How many stickers does each child get?
6
PSThere are 4 bags. Each bag has 46 stickers in it. How many stickers are there altogether?
7
PSThere are 3 bags. Each bag has 95 pencils in it. How many pencils are there altogether?
8
PSThere are 6 bags. Each bag has 47 apples in it. How many apples are there altogether?
9
PSLeo has 3945 cards in one box and 3275 cards in another box. How many cards are there altogether?
10
PS603 eggs are shared equally between 9 children. How many eggs does each child get?
11
PSThere are 8 bags. Each bag has 84 cards in it. How many cards are there altogether?
12
PS270 counters are shared equally between 3 children. How many counters does each child get?
13
PS497 eggs are shared equally between 7 children. How many eggs does each child get?
14
PSThere are 6 bags. Each bag has 41 biscuits in it. How many biscuits are there altogether?
15
PS30 stickers are shared equally between 6 children. How many stickers does each child get?
16
PSThere were 2445 apples in a shop. 1476 apples were sold. How many apples are left?
9.2

Two-step problems

Level 4
Solve a two-step problem, doing the first step, then using the answer in the second
Video lesson: Two-step problemsComing soon

A two-step problem needs two operations, in order. Find the first answer, write it down, then use it for the second step.

Look at the words that come first in the question. They often give the first step.

Key ideas

  • Do the first step, then the second.
  • Write the answer of the first step before you go on.
  • Check each step makes sense.
Worked example 9.2.1
Sam buys 4 packs of stickers. Each pack has 25 stickers. Sam also has 13 loose stickers. How many stickers does Sam have altogether?

Method. Solve in two steps: multiply, then add. Check the first step before the second.

  1. Find the first step. The packs have equal groups, so multiply.
    4 × 25 = 100
  2. Find the second step. Add the loose stickers to the total of the packs.
    100 + 13 = 113
  3. Write the answer. This is the total altogether.
    113

Check. Check: 113 - 13 = 100, and 100 is 4 packs of 25.

Watch out. Do the first step, write the answer, then use it in the second step.

Worked example 9.2.2
96 cards are shared equally between 8 bags. Then 5 more cards are put into each bag. How many cards are in each bag now?

Method. Solve in two steps: divide, then add. Check the first step before the second.

  1. Share the first amount. Equal sharing is division.
    96 ÷ 8 = 12
  2. Add the extra in each bag. Each bag gets 5 more.
    12 + 5 = 17
  3. Write the answer. This is the number in each bag now.
    17

Check. Check: 17 - 5 = 12, and 8 × 12 = 96.

Watch out. Work out the first step before you add the extra cards.

Worked example 9.2.3
There are 6 boxes with 35 buttons in each box. 48 buttons are lost. How many buttons are left?

Method. Solve in two steps: multiply, then subtract. Check the first step before the second.

  1. Find the total in the boxes. Equal groups: multiply.
    6 × 35 = 210
  2. Take away the lost ones. The question asks what is left, so subtract.
    210 - 48 = 162
  3. Write the answer. This is how many are left.
    162

Check. Check: 162 + 48 = 210.

Watch out. Multiply first, because the boxes have equal groups.

Watch out

Watch out. Do not use the total of the question for the first step if the question gives you a different number first.

Exam tip

Grown-up tip. Ask the child to draw the first step on paper, then write the answer before the second step.

Exercise 9B  Practice: two-step problems

Non-calculator
1
PS230 counters are shared equally between 5 bags. Then 18 more counters are put into each bag. How many counters are in each bag now?
2
PS95 counters are shared equally between 5 bags. Then 6 more counters are put into each bag. How many counters are in each bag now?
3
PSKofi has 70 apples. Kofi gives 4 equal groups of 17 apples to friends. How many apples does Kofi have left?
4
PSRuby buys 6 packs of biscuits. Each pack has 18 biscuits. Ruby also has 46 loose biscuits. How many biscuits does Ruby have altogether?
5
PSLeo has 141 books. Leo gives 2 equal groups of 47 books to friends. How many books does Leo have left?
6
PSOmar has 42 pencils. Omar gives 2 equal groups of 10 pencils to friends. How many pencils does Omar have left?
7
PS48 counters are shared equally between 2 bags. Then 16 more counters are put into each bag. How many counters are in each bag now?
8
PSThere are 9 boxes with 33 counters in each box. 181 counters are lost. How many counters are left?
9
PSThere are 9 boxes with 26 counters in each box. 215 counters are lost. How many counters are left?
10
PSPriya buys 3 packs of cubes. Each pack has 29 cubes. Priya also has 55 loose cubes. How many cubes does Priya have altogether?
11
PSThere are 9 boxes with 26 stickers in each box. 184 stickers are lost. How many stickers are left?
12
PS75 counters are shared equally between 3 bags. Then 2 more counters are put into each bag. How many counters are in each bag now?
13
PSSam has 320 cards. Sam gives 6 equal groups of 47 cards to friends. How many cards does Sam have left?
14
PSNoah buys 5 packs of marbles. Each pack has 11 marbles. Noah also has 30 loose marbles. How many marbles does Noah have altogether?
15
PSThere are 8 boxes with 39 cubes in each box. 118 cubes are lost. How many cubes are left?
16
PS66 pencils are shared equally between 2 bags. Then 12 more pencils are put into each bag. How many pencils are in each bag now?
9.3

Missing-number problems

Level 4
Solve a problem with a missing number by working backwards
Video lesson: Missing-number problemsComing soon

In a missing-number problem, you know the answer but not the starting number. Work backwards.

Undo each step in the reverse order. Addition is undone by subtraction. Multiplication is undone by division.

Key ideas

  • Work backwards from the answer.
  • Undo each step with the opposite operation.
  • Check forwards to make sure.
Worked example 9.3.1
I think of a number. I add 350. The answer is 1200. What was my number?

Method. Work backwards: undo each step in the reverse order, then check forwards.

  1. Work backwards. Undo the addition by subtracting.
    1200 - 350 = 850
  2. Check forwards. Add the number back to check the answer.
    850 + 350 = 1200
  3. Write the answer. This was the number I started with.
    850

Check. Check: 850 + 350 = 1200.

Watch out. Undo the last step first.

Worked example 9.3.2
I think of a number. I multiply it by 6. The answer is 360. What was my number?

Method. Work backwards: undo each step in the reverse order, then check forwards.

  1. Work backwards. Undo multiplying by dividing.
    360 ÷ 6 = 60
  2. Check forwards. Multiply the number to check.
    60 × 6 = 360
  3. Write the answer. This was the number I started with.
    60

Check. Check: 60 × 6 = 360.

Watch out. Division undoes multiplication.

Worked example 9.3.3
I think of a number. I take away 40, then double the answer. The result is 220. What was my number?

Method. Work backwards: undo each step in the reverse order, then check forwards.

  1. Undo the doubling. Halve the result first.
    220 ÷ 2 = 110
  2. Undo the take away. Add back the 40 that was taken away.
    110 + 40 = 150
  3. Write the answer. This was the number I started with.
    150

Check. Check: 150 - 40 = 110, and 110 × 2 = 220.

Watch out. Work backwards, one step at a time.

Watch out

Watch out. If the last step is “double”, the first thing to undo is to halve. Undo the steps in reverse order.

Exam tip

Grown-up tip. Ask the child to follow their answer forwards through each step. Does it reach the answer in the question?

Exercise 9C  Practice: missing-number problems

Non-calculator
1
PSI think of a number. I add 725. The answer is 1356. What was my number?
2
PSI think of a number. I add 130. The answer is 1442. What was my number?
3
PSI think of a number. I multiply it by 4. The answer is 1264. What was my number?
4
PSI think of a number. I multiply it by 9. The answer is 6012. What was my number?
5
PSI think of a number. I take away 86, then double the answer. The result is 1986. What was my number?
6
PSI think of a number. I multiply it by 9. The answer is 5040. What was my number?
7
PSI think of a number. I add 1221. The answer is 1934. What was my number?
8
PSI think of a number. I add 2358. The answer is 6696. What was my number?
9
PSI think of a number. I multiply it by 8. The answer is 2568. What was my number?
10
PSI think of a number. I take away 112, then double the answer. The result is 3648. What was my number?
11
PSI think of a number. I add 2713. The answer is 5306. What was my number?
12
PSI think of a number. I multiply it by 3. The answer is 1122. What was my number?
13
PSI think of a number. I multiply it by 8. The answer is 6424. What was my number?
14
PSI think of a number. I take away 94, then double the answer. The result is 812. What was my number?
15
PSI think of a number. I add 468. The answer is 4277. What was my number?
16
PSI think of a number. I multiply it by 9. The answer is 1611. What was my number?
9.4

Fractions of amounts in problems

Level 4
Find a fraction of an amount in a problem, and say how many are left
Video lesson: Fractions of amounts in problemsComing soon

To find a fraction of an amount in a story, divide by the bottom number to find one equal part. Then multiply by the top number.

If the question asks what is left, take the part away from the total.

Key ideas

  • Divide by the bottom number for one equal part.
  • Multiply by the top number for the fraction.
  • Read the question: “how many are left” needs one more step.
Worked example 9.4.1
There are 60 marbles in a bag. 3/5 of them are green. How many are green?

Method. Find one equal part by dividing by the bottom number, then multiply by the top number.

  1. Find one equal part: 60 ÷ 5. The bottom number tells how many equal parts.
    60 ÷ 5 = 12
  2. Take 3 parts. The top number tells how many parts are green.
    3 × 12 = 36
  3. Write the answer. This is the number of green marbles.
    36

Check. Check: 36 green and 24 not green make 60.

Watch out. A fraction of an amount needs the division first.

Worked example 9.4.2
Priya has 48 cards. She gives away 3/8 of them. How many cards does Priya have left?

Method. Find one equal part by dividing by the bottom number, then multiply by the top number.

  1. Find one equal part: 48 ÷ 8. Divide by the bottom number.
    48 ÷ 8 = 6
  2. Find the cards given away: 3 × 6. Multiply by the top number.
    3 × 6 = 18
  3. Take it away from the total. The question asks what is left.
    48 - 18 = 30
  4. Write the answer. This is how many are left.
    30

Check. Check: 18 given and 30 left make 48.

Watch out. Find the amount given away, then take it from the total.

Worked example 9.4.3
There are 36 cards. 1/4 of them are red. How many cards are not red?

Method. Find one equal part by dividing by the bottom number. Then find the part that is not red.

  1. Find the red cards: 36 ÷ 4. A quarter of 36 is one part of four equal parts.
    36 ÷ 4 = 9
  2. Find the cards that are not red. Take the red cards away from the total.
    36 - 9 = 27
  3. Write the answer. This is the number of cards that are not red.
    27

Check. Check: 9 red and 27 not red make 36.

Watch out. The part that is not red is the total minus the red part.

Watch out

Watch out. 3/8 of 48 is 18, not 3/8 of 8. Divide the total by the bottom number, not by the top.

Exam tip

Grown-up tip. Ask: “How many equal parts are there? What is one part? How many parts do we need?”

Exercise 9D  Practice: fractions of amounts in problems

Non-calculator
1
PSThere are 16 counters in a box. 1/2 of them are blue. How many are blue?
2
PSNoah has 155 books. Noah gives away 1/5 of them. How many does Noah have left?
3
PSThere are 78 biscuits in a box. 1/2 of them are blue. How many are blue?
4
PSThere are 57 marbles in a box. 1/3 of them are blue. How many are blue?
5
PSKofi has 69 eggs. Kofi gives away 1/3 of them. How many does Kofi have left?
6
PSLily has 132 marbles. Lily gives away 3/4 of them. How many does Lily have left?
7
PSKofi has 184 pencils. Kofi gives away 7/8 of them. How many does Kofi have left?
8
PSThere are 200 counters in a box. 1/5 of them are blue. How many are blue?
9
PSThere are 28 biscuits in a box. 1/2 of them are blue. How many are blue?
10
PSThere are 20 cards in a box. 4/5 of them are blue. How many are blue?
11
PSThere are 46 pencils in a box. 1/2 of them are blue. How many are blue?
12
PSThere are 120 stickers in a box. 2/3 of them are blue. How many are blue?
13
PSPriya has 32 apples. Priya gives away 1/2 of them. How many does Priya have left?
14
PSSam has 93 stickers. Sam gives away 2/3 of them. How many does Sam have left?
15
PSPriya has 48 counters. Priya gives away 1/4 of them. How many does Priya have left?
16
PSThere are 280 books in a box. 2/8 of them are blue. How many are blue?
9.5

Choosing the operation and explaining why

Level 4
Choose the operation for a problem and explain the choice in one sentence
Video lesson: Choosing the operation and explaining whyComing soon

Sometimes a question asks which operation to use and why. A good answer names the operation and gives a reason about the story.

Your reason can say what the numbers mean, such as equal groups, putting together, or taking away.

Key ideas

  • Name the operation.
  • Give a reason that uses the story.
  • Check the operation would give the right answer.
Worked example 9.5.1
Lily has 4 bags, with 9 apples in each bag. Which operation would you use to find the total? Explain why in one sentence.

Method. Pick the operation from the key words, then explain why. A good answer names the operation and gives a reason.

  1. Find the key words. The bags each hold the same number: equal groups.
    4 groups of 9
  2. Choose the operation. Equal groups of the same size use multiplication.
    4 × 9 = 36
  3. Give the reason. A good reason mentions the equal groups.
    equal groups of the same size
  4. Write the answer. Accept any answer that names multiplication and gives a reason about equal groups.
    Accept any answer that says multiplication and gives a reason about equal groups.

Check. Check: 4 × 9 = 36 is the total.

Watch out. The reason matters as much as the operation.

Worked example 9.5.2
Ruby has 380 stickers and gets 245 more. Which operation would you use to find how many there are now? Explain why in one sentence.

Method. Pick the operation from the key words, then explain why. A good answer names the operation and gives a reason.

  1. Find the key words. The word more means add.
    380 and 245
  2. Choose the operation. Putting two amounts together is addition.
    380 + 245 = 625
  3. Give the reason. A good reason mentions putting the amounts together.
    putting the two amounts together
  4. Write the answer. Accept any answer that names addition and gives a reason about putting amounts together.
    Accept any answer that says addition and gives a reason about putting amounts together.

Check. Check: 625 - 245 = 380.

Watch out. The reason matters as much as the operation.

Worked example 9.5.3
Omar has 520 pencils. 185 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.

Method. Pick the operation from the key words, then explain why. A good answer names the operation and gives a reason.

  1. Find the key words. Taken away means subtract.
    520 take away 185
  2. Choose the operation. Taking away is subtraction.
    520 - 185 = 335
  3. Give the reason. A good reason mentions taking away or what is left.
    some are taken away, so the rest is found by subtracting
  4. Write the answer. Accept any answer that names subtraction and gives a reason about taking away.
    Accept any answer that says subtraction and gives a reason about taking away.

Check. Check: 335 + 185 = 520.

Watch out. The reason matters as much as the operation.

Watch out

Watch out. Saying “multiply” with no reason is not enough. Say what the groups are.

Exam tip

Grown-up tip. Accept any sensible reason that uses the story. Ask: “Why would you use that operation here?”

Exercise 9E  Practice: choosing the operation and explaining

Non-calculator
1
PS424 cards are shared equally between 8 children. How many cards does each child get?
2
PS147 stickers are shared equally between 3 children. How many stickers does each child get?
3
RMei has 561 books. 451 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.
4
RKofi has 567 apples. 101 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.
5
PS84 marbles are shared equally between 6 children. How many marbles does each child get?
6
PSThere were 7434 stickers in a shop. 5576 stickers were sold. How many stickers are left?
7
RLily has 607 apples. 283 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.
8
RNoah has 842 stickers. 434 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.
9
PSThere are 6 bags. Each bag has 6 biscuits in it. How many biscuits are there altogether?
10
PSThere are 8 bags. Each bag has 81 books in it. How many books are there altogether?
11
RLeo has 8 bags, with 8 biscuits in each bag. Which operation would you use to find the total? Explain why in one sentence.
12
RPriya has 6 bags, with 7 pencils in each bag. Which operation would you use to find the total? Explain why in one sentence.
13
RSam has 4 bags, with 8 marbles in each bag. Which operation would you use to find the total? Explain why in one sentence.
14
RNoah has 621 marbles and gets 669 more. Which operation would you use to find how many there are now? Explain why in one sentence.
15
ROmar has 236 marbles and gets 542 more. Which operation would you use to find how many there are now? Explain why in one sentence.
16
RRuby has 672 marbles and gets 348 more. Which operation would you use to find how many there are now? Explain why in one sentence.

Chapter summary

SkillMethod
Choosing one operationKey words: altogether = add; left = subtract; equal groups = multiply; shared = divide
Two-step problemsDo the first step, write the answer, then use it in the second step
Missing numbersWork backwards: undo each step in reverse, with the opposite operation
Fractions of amountsDivide by the bottom number, multiply by the top; take away if asked what is left
Explaining the choiceName the operation and give a reason that uses the story

Can you…?

  • I can choose the operation for a one-step problem with numbers to 10000
  • I can solve a two-step problem, one step at a time
  • I can work backwards to find a missing number
  • I can find a fraction of an amount in a problem
  • I can name an operation and explain why it fits a problem

Key terms

operation
a way of combining numbers: add, subtract, multiply or divide
two-step problem
a problem that needs two operations, done in order
work backwards
undo each step, in the reverse order, to find the start
key words
words in a problem that tell you the operation, such as altogether or left

Exercise 9Z  Check-up

Non-calculator
1
Mei has 812 apples and gets 365 more. Which operation would you use to find how many there are now? Explain why in one sentence.
2
Priya has 340 eggs. Priya gives 9 equal groups of 34 eggs to friends. How many eggs does Priya have left?
3
There are 130 cards in a box. 6/10 of them are blue. How many are blue?
4
148 stickers are shared equally between 4 bags. Then 15 more stickers are put into each bag. How many stickers are in each bag now?
5
Leo has 881 cards. 582 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.
6
Priya has 270 apples. Priya gives away 8/10 of them. How many does Priya have left?
7
Mei has 6 bags, with 7 stickers in each bag. Which operation would you use to find the total? Explain why in one sentence.
8
Mei has 4404 eggs in one box and 3627 eggs in another box. How many eggs are there altogether?
9
Mei has 1967 biscuits in one box and 737 biscuits in another box. How many biscuits are there altogether?
10
I think of a number. I take away 247, then double the answer. The result is 334. What was my number?

Total 19 marks. Allow about 19 minutes.

Exercise 9X  Practice test questions

Non-calculator
1
PSThere are 4 bags. Each bag has 52 pencils in it. How many pencils are there altogether?[1 mark]
2
RAmira has 257 apples and gets 672 more. Which operation would you use to find how many there are now? Explain why in one sentence.[1 mark]
3
PSThere are 9 bags. Each bag has 50 cards in it. How many cards are there altogether?[1 mark]
4
PS264 books are shared equally between 8 bags. Then 20 more books are put into each bag. How many books are in each bag now?[1 mark]
5
RRuby has 611 marbles. 121 are taken away. Which operation would you use to find how many are left? Explain why in one sentence.[2 marks]
6
PSThere are 64 biscuits in a box. 4/8 of them are blue. How many are blue?[2 marks]
7
PS35 marbles are shared equally between 7 children. How many marbles does each child get?[2 marks]
8
PSI think of a number. I multiply it by 4. The answer is 2672. What was my number?[3 marks]
9
PSThere were 5081 counters in a shop. 3384 counters were sold. How many counters are left?[3 marks]
10
PS170 cards are shared equally between 5 children. How many cards does each child get?[3 marks]
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