Word problems
A word problem hides a sum inside a short story. Here’s how to read the story, choose the right sum and spot the traps in the wrong answers.
Chapter 1
What are word problems?
A word problem is a sum told as a story. The numbers are in the story, but the sum isn’t written down. You work out which sum it is, then do it.
The story. Who has what, with the numbers in it.
The question. The last sentence. It says what you’re finding.
The sum is usually easy. Choosing it is the hard part. Rush’s stories, in Add & Subtract, Multiply & Divide and Money & Time, each need one sum. The exam’s harder ones need two or three: chapter 4 shows how to plan them.
Kai has 55 football cards. He wants 87. How many more does he need?
The options
- Option 1: 142
- Option 2: 42
- Option 3: 32
- Option 4: 33
A Maths question from Rush, level 1.
Every story is one of four kinds
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Put together: add
Two amounts join into one. Priya has 53 stamps and Leo has 32. How many altogether? 53 + 32 = 85.
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Take from, or compare: take away
Some go, and you find what’s left. Or you find the gap between two amounts. A shop has 68 kites and sells 24: 68 − 24 = 44 left.
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Equal groups, find the total: times
You know how many groups and how many in each. A table seats 6. How many can sit at 4 tables? 4 × 6 = 24.
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Equal groups, know the total: divide
18 sweets shared by 3 children: 18 ÷ 3 = 6 each. And 18 children in teams of 3 make 6 teams. Each in a group, or how many groups: both divide.
Chapter 2
Solve it in 4 steps.
Here’s the question, one step at a time.
The question
Kai has 55 football cards. He wants 87. How many more does he need?
The options
- Option 1: 142
- Option 2: 42
- Option 3: 32
- Option 4: 33
A Maths question from Rush, level 1.
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Step 1
Find the question first.
Read the last sentence before the rest. It tells you what you’re looking for: how many more cards Kai needs.
Kai has 55 football cards. He wants 87.
How many more does he need?This is what you’re finding.
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Step 2
Underline the numbers.
Next to each number, say what it counts. 55 is the cards Kai has. 87 is the cards he wants.
- 55cards Kai has
- 87cards he wants
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Step 3
Draw it, then name the sum.
Draw a bar for each. The bit sticking out is how many more he needs: a gap. A gap is take away, even though the question says more. So it’s 87 − 55.
The gap: 87 − 55.
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Step 4
Work it out, then check.
Count up from 55: 5 makes 60, then 27 more makes 87. That’s 32. Check it fits: 55 + 32 = 87, and a gap is smaller than 87.
- 55 + 5 = 60
- 60 + 27 = 87
- 5 + 27 = 32
Check: 55 + 32 = 87.
Three mistakes this question sets
| The mistake | Do this instead |
|---|---|
| Adding because it says more. 142 is 55 + 87. That’s more cards than he even wants. | “How many more” asks for the gap. Draw the two bars. |
| Counting the 55 as well. 33 counts every number from 55 to 87, and that includes the 55 he already has. | Count the jumps, not the numbers you say. |
| Slipping a ten. 42 is ten too many: 55 + 42 is 97. | Check by adding back. 55 + your answer must make 87. |
So it’s 32, option 3. The gap between 55 and 87, and 55 + 32 = 87.
Remember: Question first. Then draw it.
Chapter 3
When the words trick you.
Each card is a catch Rush’s stories set: the rule, one example, and the trap.
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2 bags of 9 apples
“Altogether” can mean times
Equal groups? Times, whatever the words say.
2 bags each have 9 apples. How many altogether? Each bag is a group of 9, so 2 × 9 = 18. Trap: 11, from adding 2 and 9. Rush often offers it.
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6 eggs, boxes of 2: 6 ÷ 2 = 3 boxes
Not sure which sum? Try tiny numbers
Tell the same story with small numbers.
72 eggs are packed in boxes of 9. How many boxes? With 6 eggs in boxes of 2, you’d fill 3 boxes: 6 ÷ 2. So it’s 72 ÷ 9 = 8. Trap: 9, the eggs in a box, copied from the story.
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10 full minibuses, and one for the last 3
Everyone must fit? Round up
The ones left over need one more.
A minibus seats 4. How many for 43 people? 43 ÷ 4 = 10 r 3. The 3 left still need a seat, so 11. Trap: 10 r 3. You can’t hire part of a minibus.
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4 ribbons, and 4 cm left over: too short for a fifth
Only full ones count? Round down
Full boxes, whole ribbons: the leftover doesn’t make one.
Ribbons must be 5 cm long. How many from 24 cm? 24 ÷ 5 = 4 r 4, so 4. But “how many each, and how many left?” wants both: 23 sweets between 5 children is 4 r 3.
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- miles per hour
- = miles ÷ hours
- 250 ÷ 5 = 50
“Per” means for each one
The units name the sum.
A van travels 250 miles in 5 hours. How many miles per hour? Miles for each hour is miles ÷ hours: 250 ÷ 5 = 50. Trap: 1250, from 250 × 5.
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- 85p = £0.85
- £4.40 + £0.85
- = £5.25
Pounds and pence: one unit first
Write both in pounds, or both in pence.
A drink costs 85p and a sandwich £4.40. How much altogether? 85p is £0.85, so £5.25. Trap: £89.40, from adding 85 to 4.40 as if both were pounds.
Words that point the way, and the size to expect
The words point to a sum, but the picture decides: “altogether” can mean times, and “more” can mean take away. Then check the answer’s size fits.
| Sum | Words | The answer is |
|---|---|---|
| + Add | altogether, total, in all, more join | bigger than both numbers |
| − Take away | left, empty, not sold, difference, how many more | smaller than the biggest number |
| × Times | lots of, rows of, packs of, every | bigger than both, when both are whole numbers above 1 |
| ÷ Divide | shared, split, how many each, groups of, how many fit | smaller than the total, when dividing by a whole number above 1 |
Chapter 4: In the exam
Two steps or more.
Rush asks one step at a time. The exam joins them up. We counted the steps in GL’s two free maths familiarisation papers: the last third of the questions average between two and three each.
The questionA school takes 4 classes of 28 children to the zoo. Each coach seats 50. What is the smallest number of coaches needed?
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Step 1
Find the question, then plan.
It asks for coaches. To fill coaches you need the number of children, and the story doesn’t give it. So plan two parts: children first, then coaches.
- Part 1: how many children?
- Part 2: how many coaches?
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Step 2
Work out part 1.
4 classes of 28 are equal groups: 4 × 28 = 112 children. Don’t stop here: 112 isn’t a number of coaches.
4 × 28 = 112 children.
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Step 3
Feed it into part 2.
112 ÷ 50 = 2 r 12. Two full coaches take 100, and 12 children are left. They still need a seat, so round up: 3 coaches.
Coach 3 takes the last 12.
Three mistakes this question sets
| The mistake | Do this instead |
|---|---|
| Stopping after part 1. 112 is the number of children. The question asks for coaches. | Before you choose, check your answer counts what the question asks for. |
| Rounding down. 2 coaches seat only 100, so 12 children stay behind. | Everyone has to go, so round up. |
| A coach for each class. 4 coaches would work, but 3 seat 150, which is plenty. | Read the last words again: the smallest number. |
So it’s 3 coaches. Part 1 gives 112 children, and part 2 puts them in coaches of 50, rounding up.
Remember: Plan it. Each answer feeds the next.
Other ways the exam asks it
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Zara is
10. She buys 3 packs of 8 stickers and gives away 5.Numbers you don’t need
Some stories add a number that isn’t part of the sum, like Zara’s age. Cross it out. How many stickers now? 3 × 8 = 24, then 24 − 5 = 19.
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a dozen = 12a pair = 2a week = 7 daysdouble = × 2
Numbers hidden in words
Eggs come in boxes of a dozen. How many eggs in 5 boxes? A dozen is 12, so 5 × 12 = 60. Trap: missing a number because it’s written as a word.
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Sam has 4 lots of Jo’s 6
More than, or times as many?
Sam has 24 marbles and Jo has 6. How many more has Sam? The gap: 24 − 6 = 18. How many times as many? How many 6s make 24: 24 ÷ 6 = 4.
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7 less than 20
20 − 7 = 13
“Less than” reads backwards
What is 7 less than 20? Start from 20 and take 7: 13. The number you read first isn’t always first in the sum. “5 fewer than 30” is 30 − 5 too.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Find the question first.
- Underline the numbers.
- Draw it, then name the sum.
- Work it out, then check.
Remember: Question first. Then draw it.
For parents.
What parents ask us about word problems.
Questions parents ask
How do I help my child with 11+ word problems?
Read the last sentence first, together, and ask what it wants: a total, a gap, a share or a number of groups. Let your child say which operation that is before working anything out. Saying it out loud catches the commonest slip, adding when the story wants a take-away.
Which words tell you which operation to use?
'In all', 'total' and 'altogether' point to adding, unless the story is about equal groups, when 'altogether' means multiply. 'Left', 'difference', 'how many more' and 'fewer' mean take away. 'Rows of' and 'lots of' usually mean multiply. 'Shared', 'split', 'in groups of' and 'how many fit' mean divide. 'Each' and 'per' can go either way, so draw it: equal groups with the total unknown is multiply, and with the total known it's divide. They don't always work, so check the answer fits the story.
What is a bar model?
It's a drawing that shows the numbers as bars, with the whole on top and the parts underneath. It shows whether you're finding a part (take away) or the whole (add), and the same drawing works later for fractions and ratio.
Rush: 11+ Practice & KS2 SATsPractise word problems in Rush.
- Add & Subtract, Multiply & Divide and Money & Time as 100-second games, easy at first and harder as your child improves
- Every miss kept with the right answer and why
- Free to start, with a new family game every day
On the App Store for iPhone and iPad
Add & Subtract and Multiply & Divide are free, and four more games too, plus a new family game every day. Money & Time is in Premium: all 41 games are £24.99 a year after a free week1, or £3.99 a month.