Year 4: Word problems with bar models
Part of the Year 4 short multiplication and short division chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Word problems with bar models
Real problems often take more than one step. Use these five steps every time.
- Read the problem. What do you know? What do you need to find out?
- Draw a bar model.
- Choose which operations to use.
- Work it out, one step at a time, with a written method.
- Check your answer. Does it make sense? Does it answer the question?
A bar model is a picture of the numbers. For equal groups, draw equal boxes: the problem is a multiplication or a division. For a whole made of parts, draw the parts side by side: add or subtract. To compare two amounts, draw two bars, one above the other.
Some problems use two operations. A baker makes 6 trays of 24 buns, then sells 105. First multiply (6 × 24 = 144 buns baked). Then subtract (144 − 105 = 39 buns left). Always check the question at the end: the answer to the first step is not always the answer to the question.
Key words. A bar model is a drawing of boxes that shows the numbers in a problem. A multi-step problem needs more than one calculation.
Key ideas
- Read, draw, choose, work, check.
- Equal groups: multiply to find the total, divide to find the size of a group or the number of groups.
- Whole made of parts: add the parts, or subtract a part from the whole.
- Write the working for each step and the unit with each answer.
- Check: does the answer fit the story and answer the question?
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. The whole is the buns baked. 105 are sold. We want the part that is left.
- Step 1: Find how many buns the baker makes. 6 trays with 24 buns on each, so multiply.24 × 6 = 144
H T O 2 4 × 6 1 4 4 1 2 - Step 2: Find how many buns are left. She sells 105 buns, so take them away from the total.144 − 105 = 39
H T O 1 3 414 4− 1 0 5 3 9 - Write the answer. Include the unit and make sure it answers the question.39 buns left
Check. Add back: 39 + 105 = 144, and 144 ÷ 6 = 24. ✓
Watch out. The answer to step 1 (144) is not the answer to the question. Keep going to step 2.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. 4 equal tickets share the £36. We find one ticket first.
- Step 1: Find the cost of 1 ticket. 4 tickets cost £36 altogether, so share £36 into 4 equal parts.£36 ÷ 4 = £9
0 9 4 3 36 - Step 2: Find the cost of 7 tickets. 7 tickets cost 7 times as much as 1 ticket.£9 × 7 = £63
- Write the answer. Include the unit and make sure it answers the question.£63
Check. Another way: 7 tickets is 4 tickets plus 3 tickets: £36 + 3 × £9 = £36 + £27 = £63. ✓
Watch out. Find the cost of ONE first. Then multiply by the number of tickets you need.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. All 252 stickers are shared into 6 equal packs.
- Step 1: Find how many stickers Jack and Mia have altogether. Put the two amounts together: add.138 + 114 = 252
H T O 1 3 8 + 1 1 4 2 5 2 1 - Step 2: Share the stickers equally between 6 packs. Equal sharing means divide.252 ÷ 6 = 42
0 4 2 6 2 25 12 - Write the answer. Include the unit and make sure it answers the question.42 stickers in each pack
Check. Multiply back: 42 × 6 = 252, and 138 + 114 = 252. ✓
Watch out. Add first, then divide. If you divide 138 by 6 you forget Mia’s stickers.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. The whole is the £5 note (500p). Part of it pays for the pens. The rest is the change.
- Step 1: Find the cost of 5 pens. 5 pens at 85p each, so multiply.85p × 5 = 425p
H T O 8 5 × 5 4 2 5 4 2 - Step 2: Find the change. The change is the money paid take away the cost.500p − 425p = 75p
H T O 4 59 010 0− 4 2 5 7 5 - Write the answer. Include the unit and make sure it answers the question.75p change
Check. Add back: 425p + 75p = 500p. ✓
Watch out. Keep the units the same: £5 is 500p. Work in pence all the way through.
Method. First find how many minibuses are needed (divide, and think about the remainder). Then multiply by the cost of one minibus.
- Draw a bar model. The 52 pupils are put into groups of 8, with some left over.
- Step 1: divide the pupils by 8. Each minibus holds 8 pupils.52 ÷ 8 = 6 remainder 4
0 6 r 4 8 5 52 - Decide what to do with the remainder. 4 pupils are left over, but they still need to travel. We need one more minibus: round up.6 + 1 = 7 minibuses
- Step 2: find the cost. Each minibus costs £35 to hire, so multiply 7 by 35.£35 × 7 = £245
H T O 3 5 × 7 2 4 5 2 3 - Write the answer. £245
Check. 7 × 8 = 56, which is enough for 52 pupils, but 6 × 8 = 48 is not enough. ✓
Watch out. Everyone must have a seat, so round the number of minibuses UP.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. Bea has one box. Amir has three boxes the same size. Together they have 84.
- Step 1: Find how many equal boxes there are altogether. Bea has 1 box and Amir has 3 boxes the same size.1 + 3 = 4
- Step 2: Find the size of one box. The 4 equal boxes share the 84 cards, so divide.84 ÷ 4 = 21
2 1 4 8 4 - Step 3: Find how many cards Amir has. Amir has 3 boxes.21 × 3 = 63
T O 2 1 × 3 6 3 - Write the answer. Include the unit and make sure it answers the question.Bea 21 cards, Amir 63 cards
Check. Add: 21 + 63 = 84. And 63 is 3 times 21. ✓
Watch out. Amir has 3 times as many as Bea, so there are 4 boxes in the whole bar (1 + 3), not 3.
Watch out
The answer to the first step is not always the answer to the question. Look back at what the question asks for. Also keep your units the same: pounds with pounds, pence with pence. And do not add when the problem says “times as many”.Exercise 4G Word problems with bar models
Non-calculatorMethod. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. The whole is the pencils in the boxes. 95 are sold.
- Step 1: Find how many pencils there are in the boxes. 8 boxes with 36 pencils in each, so multiply.36 × 8 = 288
H T O 3 6 × 8 2 8 8 2 4 - Step 2: Find how many are left. 95 are sold, so subtract.288 − 95 = 193
H T O 1 218 88 − 9 5 1 9 3 - Write the answer. Include the unit and make sure it answers the question.193 pencils left
Check. Add back: 193 + 95 = 288. ✓
Watch out. The question asks how many are LEFT, so do not stop at step 1.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. Two amounts are spent. Add them to find the total.
- Step 1: Find the cost of the tickets. 5 tickets at £14 each.£14 × 5 = £70
T O 1 4 × 5 7 0 2 - Step 2: Find the cost of the drinks. 2 drinks at £3 each.£3 × 2 = £6
- Step 3: Add the two costs. Both are spent, so add.£70 + £6 = £76
- Write the answer. Include the unit and make sure it answers the question.£76
Check. Estimate: 5 × £15 = £75, and the drinks cost a few pounds, so about £80. ✓
Watch out. There are two different things to buy. Work out both before you add.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. The whole is 500p. Part of it pays for the comics.
- Step 1: Find the cost of 4 comics. 4 comics at 85p each, so multiply.85p × 4 = 340p
H T O 8 5 × 4 3 4 0 3 2 - Step 2: Find the change. Take the cost away from the £5 note (500p).500p − 340p = 160p
H T O 4 510 00 − 3 4 0 1 6 0 - Write the answer. Include the unit and make sure it answers the question.160p change
Check. Add back: 340p + 160p = 500p. ✓
Watch out. Change to pence first: £5 = 500p.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. 5 equal notebooks cost £45 in total.
- Step 1: Find the cost of 1 notebook. 5 notebooks cost £45, so share £45 into 5 equal parts.£45 ÷ 5 = £9
0 9 5 4 45 - Step 2: Find the cost of 8 notebooks. 8 notebooks cost 8 times as much as 1.£9 × 8 = £72
- Write the answer. Include the unit and make sure it answers the question.£72
Check. Another way: 8 notebooks is 5 notebooks and 3 more: £45 + £27 = £72. ✓
Watch out. Find the cost of one notebook first.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. All 282 loaves are shared into 6 equal crates.
- Step 1: Add the loaves. Loaves sold on both days are put together.138 + 144 = 282
H T O 1 3 8 + 1 4 4 2 8 2 1 - Step 2: Share the loaves equally. Equal sharing between 6 crates means divide.282 ÷ 6 = 47
0 4 7 6 2 28 42 - Write the answer. Include the unit and make sure it answers the question.47 loaves in each crate
Check. Multiply back: 47 × 6 = 282. ✓
Watch out. Add the two days first. Then divide.
Method. Find how many minibuses are needed. Divide, then round up because everyone needs a seat. Then multiply by the cost of one minibus.
- Step 1: divide the pupils by 9. Each minibus holds 9 pupils.85 ÷ 9 = 9 remainder 4
0 9 r 4 9 8 85 - Decide what to do with the remainder. 4 pupils are left over, but they still need a minibus. Round up.9 + 1 = 10 minibuses
- Step 2: find the cost. Each minibus costs £30, so multiply by 10.£30 × 10 = £300
- Write the answer. £300
Check. 9 × 9 = 81, which is not enough for 85 pupils. 10 × 9 = 90 is enough. ✓
Watch out. Round the number of minibuses UP. 9 minibuses would leave 4 pupils behind.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. Sue’s bar is 3 times as long as Tom’s. The dashed part is the difference.
- Step 1: Find how long Sue’s rope is. Sue’s rope is 3 times as long as Tom’s, so multiply.125 × 3 = 375
H T O 1 2 5 × 3 3 7 5 1 - Step 2: Find how much longer it is. The difference is the bigger length take away the smaller.375 − 125 = 250
H T O 3 7 5 − 1 2 5 2 5 0 - Write the answer. Include the unit and make sure it answers the question.250 cm longer
Check. The difference is 2 more boxes of 125: 2 × 125 = 250. ✓
Watch out. Step 1 gives Sue’s length, not the difference. Subtract Tom’s length to answer the question.
Key words. You can multiply three numbers in any order and the answer is the same.
Method. You can multiply three numbers in any order. Look for two numbers that make an easy answer (like a pair that makes 10 or 100) and multiply those first.
- Look for a pair that is easy to multiply. 6 × 5 = 30, which we can work out easily.6 × 5 = 30
- Multiply by the last number, 8. Multiply the answer, 30, by the last number, 8.3 × 8 = 24, so 3 tens × 8 = 24 tens
30 × 8 = 240 - Write the answer. 6 × 8 × 5 = 240
Check. Try a different order: 40 × 6 = 240. ✓
Watch out. You can multiply in any order. Pick the pair that gives an easy answer first.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. The whole rope is 245 m. 35 m is used. The rest is cut into 7 equal pieces.
- Step 1: Find how much rope is left. 35 m is used, so take it away from 245 m.245 − 35 = 210
H T O 2 4 5 − 3 5 2 1 0 - Step 2: Share the rest into 7 equal pieces. Equal pieces: divide.210 ÷ 7 = 30
0 3 0 7 2 21 0 - Write the answer. Include the unit and make sure it answers the question.30 m long
Check. Multiply back: 30 × 7 = 210, and 210 + 35 = 245. ✓
Watch out. Take the 35 m away FIRST. Then divide what is left.
Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.
- Draw a bar model. The whole is the £50. Part of it pays for the tickets.
- Step 1: Find the cost of the adult tickets. 2 adults at £8 each.£8 × 2 = £16
- Step 2: Find the cost of the child tickets. 3 children at £5 each.£5 × 3 = £15
- Step 3: Find the total cost. Add the two costs.£16 + £15 = £31
- Step 4: Find the change. Take the total cost away from the £50 paid.£50 − £31 = £19
- Write the answer. Include the unit and make sure it answers the question.£19 change
Check. Add back: £31 + £19 = £50. ✓
Watch out. There are two kinds of ticket. Work out both costs before you take away.
Method. Leo has added, but the tickets are equal groups. Work out the cost of 1 ticket first.
- Find the cost of 1 ticket. Share £36 between 4 tickets.£36 ÷ 4 = £9
- Find the cost of 8 tickets. Each ticket costs £9, so multiply by 8.8 × £9 = £72
- Say what went wrong. Leo added £8, but 4 more tickets cost 4 × £9 = £36 more, not £8 more.The cost goes up in equal steps. This is multiplication, not adding the number of tickets.
- Write the correct answer. £72
Check. 8 tickets is double 4 tickets, so double £36 = £72. ✓
Watch out. If the number of tickets doubles, the cost doubles. It does not go up by the number of tickets.
Method. Draw a bar model. Count the boxes. Then divide to find the size of one box.
- Draw the bar model. Priya has 1 box. Dev has 4 boxes the same size. The total is 5 boxes.
- Find how many boxes there are altogether. 1 box for Priya and 4 boxes for Dev.1 + 4 = 5
- Find the size of one box. The 5 equal boxes make 75.75 ÷ 5 = 15
- Find how many Dev has. Dev has 4 boxes.4 × 15 = 60
Check. 15 + 60 = 75, and 60 = 4 × 15. ✓
Watch out. There are 5 boxes in the whole bar (1 + 4). Do not divide by 4.
Method. Try some numbers that divide exactly by 6. Check whether they also divide exactly by 3. Then explain why.
- Try some numbers in the 6 times table. Divide each by 3.12 ÷ 6 = 2 and 12 ÷ 3 = 4
18 ÷ 6 = 3 and 18 ÷ 3 = 6
30 ÷ 6 = 5 and 30 ÷ 3 = 10
102 ÷ 6 = 17 and 102 ÷ 3 = 34 - Explain why. A number that divides by 6 is made of equal groups of 6. Each group of 6 is two groups of 3, so it divides by 3 too.6 = 3 × 2
- Decide. Always
Check. Example: 42 = 7 × 6 = 14 × 3. ✓
Watch out. The rule only goes one way. A number that divides by 3 does not always divide by 6 (for example 9).
Method. Work out both sides. Then explain why they are equal.
- Work out 6 × 24. Partition 24 into 20 + 4.6 × 20 = 120 6 × 4 = 24 120 + 24 = 144
- Work out 3 × 48. Partition 48 into 40 + 8.3 × 40 = 120 3 × 8 = 24 120 + 24 = 144
- Compare. Both make 144.6 × 24 = 144 = 3 × 48
- Explain why. 6 is double 3 and 24 is half of 48. Doubling one number and halving the other keeps the answer the same.6 = 3 × 2 and 48 = 24 × 2
Check. 144 ÷ 6 = 24 and 144 ÷ 3 = 48. ✓
Watch out. Doubling one number and halving the other leaves the answer unchanged.