Primary 5 Method and Model Drawing
Build confidence with equal parts, fractions, percentages and everyday comparisons. Learn one method, then try a fresh question together.
Choose a method
Choose the relationship you can see. These P5 examples use whole numbers, fractions and percentage parts; formal ratio and average lessons are in P6.
Read fraction and percentage wording
Identify the whole before calculating. “Of the remainder” uses a new base after the earlier action.
- 1/4 of the whole
- Split the original amount into four equal parts and take one.
- 3/4 left
- If one quarter was used, three of the original four parts remain.
- 1/2 of the remainder
- First find what remains, then halve that amount.
See the change of whole
Start with 24 counters. Use 1/4: 24 ÷ 4 = 6 counters. Now 18 remain. Half of the remainder is 18 ÷ 2 = 9, not half of 24.
Model drawing · equal beginning
What it means
Draw the two equal starting bars first. Add or remove the later changes on those bars, then compare their ending amounts.
Work through a new example
Iris and Joel had the same number of cards at first. Iris bought 12 cards while Joel gave away 6. Iris then had twice as many cards as Joel. How many did each have at first?
Read the story. Tap Next to draw the model one clue at a time.
The model will appear here as you follow the clues.
Answer: Iris and Joel each had 24 cards at first.
View the complete worked solution
- The starting amounts were equal. The difference between their ending bars is Iris’s extra 12 plus Joel’s 6 given away.1 unit = 12 + 6 = 18 cards
- Use each ending bar to work back to the equal starting amounts.Joel at first = 18 + 6 = 24 cardsIris at first = (2 × 18) − 12 = 24 cards
- Check both changes and the ending comparison.Iris after = 24 + 12 = 36 cardsJoel after = 24 − 6 = 18 cards36 = 2 × 18 ✓
Answer: Iris and Joel each had 24 cards at first.
Ask at home
“At which point were the two bars equal? Can we draw that stage before adding the changes?”
Common mistake
Drawing equal bars at the end even though equality was stated for the beginning.
Try together · Primary 5
Asha and Ben had the same number of stickers. Asha bought 10 and Ben gave away 5. Asha then had twice as many as Ben. How many did each have at first?
Reveal the worked answer
20 stickers each.
The ending gap is 10 + 5 = 15 stickers. Two ending parts minus one ending part is one part, so Ben has 15 left. At first he had 15 + 5 = 20. Check: Asha has 30 and Ben has 15.
Model drawing · part–whole
What it means
Start with one whole bar. Partition the original fraction first; subdivide the remaining part to show the second fraction’s new base.
Work through a new example
Maya spent of her money on lunch. She then spent of the remaining money on a gift and had $24 left. How much did she have at first?
Read the question. First find which amount each fraction refers to.
The four-part model will appear as you follow the question.
Answer: Maya had $48 at first.
View the complete worked solution
- Four units are left. All four original quarters have been cut in half.4 units = $241 unit = $24 ÷ 4 = $6Original money = 8 units = 8 × $6 = $48
- Check each spending step using the correct whole.Lunch = 2 × $6 = $12Gift = 2 × $6 = $12$48 − $12 − $12 = $24 left
Answer: Maya had $48 at first.
Ask at home
“What amount does the two-sixths refer to? Can we halve each of the 3 remaining parts?”
Common mistake
Treating two-sixths of the remainder as two-sixths of the starting money.
Try together · Primary 5
Nadia spent 1/3 of her money, then 1/4 of the remainder. She had $30 left. How much did she have at first?
Reveal the worked answer
$60.
She kept 3/4 of 2/3 of the original amount: 3/4 × 2/3 = 1/2. If half is $30, the whole is $60. Check: $60 − $20 = $40; $40 − $10 = $30.
Model drawing · equal ending
What it means
Read the story from its ending condition. Draw equal bars for the amounts at the end, then reverse each change to find the starting amounts.
Work through a new example
Mia and Noah had 92 cards altogether at first. Mia bought 16 cards while Noah gave away 12. They then had the same number. How many cards did Mia have at first?
Read the whole question. Find when the two amounts are equal before drawing the bars.
Start with what is known at the end: Mia and Noah have equal amounts.
Answer: Mia had 32 cards at first.
View the complete worked solution
- Mia added 16 cards and Noah removed 12, so the total rose by 4 cards.Ending total = 92 + 16 − 12 = 96 cards
- The two ending bars are equal.Each at the end = 96 ÷ 2 = 48 cards
- Work backwards from 48 cards for each person.Mia at first = 48 − 16 = 32 cardsNoah at first = 48 + 12 = 60 cards
- Check the original total and the equal ending.32 + 60 = 92 cardsMia after buying: 32 + 16 = 48 cardsNoah after giving away: 60 − 12 = 48 cards
Answer: Mia had 32 cards at first.
Ask at home
“When were the amounts equal? Which change must we reverse first to get back to the start?”
Common mistake
Splitting the starting total of 92 into two equal amounts.
Try together · Primary 5
Kai and Mei had 80 cards altogether. Kai bought 12 and Mei gave away 8. They then had the same number. How many did Kai have at first?
Reveal the worked answer
30 cards.
The new total is 80 + 12 − 8 = 84. Each then has 84 ÷ 2 = 42. Kai started with 42 − 12 = 30. Mei started with 50; 50 − 8 = 42.
Sets concept
What it means
Treat one recurring combination as a single set. Find the set’s total value, then count how many complete sets fit.
Work through a new example
A school buys 3 notebooks at $4 each and 2 pens at $2 each in every stationery set. It spends $64 on complete sets. How many notebooks does it buy?
Start by finding the cost of one set: 3 notebooks at $4 each and 2 pens at $2 each.
Build one complete stationery set from the clues.
Answer: The school buys 12 notebooks.
View the complete worked solution
- Find the cost of the three notebooks in one set.3 × $4 = $12
- Find the cost of the two pens in the same set.2 × $2 = $4
- Add both costs to price one complete set.$12 + $4 = $16 per set
- Use the total spending to find how many complete sets were bought.$64 ÷ $16 = 4 sets
- Count the notebooks in all the sets.4 × 3 = 12 notebooks
- Check the notebook and pen costs against the $64 total.12 × $4 = $48 for notebooks4 × 2 = 8 pens; 8 × $2 = $16$48 + $16 = $64 ✓
Answer: The school buys 12 notebooks.
Ask at home
“How much does one set of 3 notebooks and 2 pens cost?”
Common mistake
Dividing $64 by the cost of one notebook and ignoring the pens that are bought with it.
Try together · Primary 5
Each stationery set contains 2 notebooks at $3 each and 1 pen at $2. A parent buys complete sets for $40. How many notebooks does the parent buy?
Reveal the worked answer
10 notebooks.
One set costs 2 × $3 + $2 = $8. There are $40 ÷ $8 = 5 sets. Each has 2 notebooks, so 5 × 2 = 10 notebooks.
Groups concept
What it means
Count whole bundles and any individual items separately. For a least-cost question, compare the whole-number combinations that give the required item count.
Work through a new example
Maya needs 26 badges. A pack of 3 badges costs $5, while one badge bought separately costs $2. What is the least amount Maya needs to pay?
Maya needs exactly 26 badges. Find the least cost using whole packs and single badges.
First compare a whole pack with a single badge.
Answer: The least Maya needs to pay is $44.
View the complete worked solution
- Make as many whole packs of three as possible from 26 badges.26 ÷ 3 = 8 packs, remainder 2 badges8 × 3 = 24 badges
- Buy the two badges left over separately and add the costs.8 × $5 = $402 × $2 = $4$40 + $4 = $44
- Check whether fewer packs with more single badges could cost less.3 singles cost 3 × $2 = $6, which is $1 more than a $5 pack7 packs + 5 singles = $35 + $10 = $456 packs + 8 singles = $30 + $16 = $460 packs + 26 singles = $52
- Check the chosen purchase gives exactly 26 badges for $44.8 × 3 + 2 = 26 badges8 × $5 + 2 × $2 = $44 ✓
Answer: The least Maya needs to pay is $44.
Ask at home
“How many whole packs of three fit into 26? What happens to the two badges left over? Is a pack or three singles cheaper?”
Common mistake
Rounding 26 ÷ 3 up to 9 packs, or forgetting to add the cost of the two single badges.
Try together · Primary 5
A pack of 4 badges costs $7. A single badge costs $2. What is the least cost of buying exactly 18 badges?
Reveal the worked answer
$32.
A pack saves $1 compared with four singles. Use as many complete packs as possible: 4 packs give 16 badges for $28. Add 2 singles for $4. Total: $32.
Make a supposition (alternative to guess and check for two totals)
What it means
Pretend every item is the cheaper or smaller kind. The difference between that imagined total and the real total shows how many items must be switched.
Work through a new example
A show sold 30 tickets. Adult tickets cost $8 and child tickets cost $5. The total collected was $192. How many adult tickets were sold?
Read the ticket count, both prices and the actual collection. We will make a deliberate assumption, then correct it.
Start by imagining all tickets are the cheaper kind.
Answer: The show sold 14 adult tickets.
View the complete worked solution
- Suppose all 30 tickets were the cheaper child tickets.30 × $5 = $150 assumed collection
- Compare this imagined collection with the actual $192.$192 − $150 = $42 shortfall
- Replace one imagined child ticket with an adult ticket and find the extra money it brings.$8 − $5 = $3 extra per switch
- Divide the $42 shortfall by the extra $3 per switch to find the adult tickets.$42 ÷ $3 = 14 adult tickets30 − 14 = 16 child tickets
- Check both the ticket count and the money collected.14 + 16 = 30 tickets14 × $8 + 16 × $5 = $112 + $80 = $192 ✓
Answer: The show sold 14 adult tickets.
Ask at home
“If we pretend they are all the cheaper kind, how much does one swap change the total?”
Common mistake
Dividing the $42 difference by $8 instead of by the $3 difference between ticket prices.
Try together · Primary 5
A show sold 20 tickets. An adult ticket costs $9 and a child ticket costs $5. The total collected was $132. How many adult tickets were sold?
Reveal the worked answer
8 adult tickets.
If all were child tickets, the total would be 20 × $5 = $100. The extra $32 comes from adult tickets. Each adult ticket adds $9 − $5 = $4, so $32 ÷ $4 = 8.
Guess and check
What it means
Record a sensible trial in a table, check every condition, and use whether the result is too high or low to improve the next trial.
Work through a new example
A rack holds 18 bicycles and tricycles. Bicycles have 2 wheels and tricycles have 3 wheels. There are 43 wheels altogether. How many tricycles are there?
Read both totals. Each guess must satisfy the vehicle count and then be checked against the wheel count.
The two conditions and a record of guesses will appear here.
Answer: There are 7 tricycles and 11 bicycles.
View the complete worked solution
Swipe the table sideways to see all checks →
| Bicycles | Tricycles | Bicycle wheels | Tricycle wheels | Total wheels | Check |
|---|---|---|---|---|---|
| 9 | 9 | 9 × 2 = 18 | 9 × 3 = 27 | 18 + 27 = 45 | ✗ Too many wheels |
| 10 | 8 | 10 × 2 = 20 | 8 × 3 = 24 | 20 + 24 = 44 | ✗ Still too many |
| 11 | 7 | 11 × 2 = 22 | 7 × 3 = 21 | 22 + 21 = 43 | ✓ Both conditions met |
- Start with 9 bicycles and 9 tricycles. The wheel count is too high.9 + 9 = 18 vehicles9 × 2 + 9 × 3 = 18 + 27 = 45 wheels → too many
- Keep 18 vehicles: try one more bicycle and one fewer tricycle. There are still too many wheels.10 + 8 = 18 vehicles10 × 2 + 8 × 3 = 20 + 24 = 44 wheels → still too many
- Adjust once more to 11 bicycles and 7 tricycles. This time the wheel count matches.11 + 7 = 18 vehicles11 × 2 + 7 × 3 = 22 + 21 = 43 wheels ✓
- Check both totals and state the number of tricycles.11 bicycles + 7 tricycles = 18 vehicles22 bicycle wheels + 21 tricycle wheels = 43 wheels ✓
Answer: There are 7 tricycles and 11 bicycles.
Ask at home
“Which condition must every row keep fixed? Is this trial too high or too low?”
Common mistake
Changing both counts independently, so a trial no longer has exactly 18 vehicles.
Try together · Primary 5
There are 12 bicycles and tricycles with 29 wheels altogether. How many tricycles are there?
Reveal the worked answer
5 tricycles.
Start with 12 bicycles: 24 wheels. Replacing one bicycle with one tricycle adds one wheel. We need 29 − 24 = 5 extra wheels, so there are 5 tricycles and 7 bicycles. Check: 5 × 3 + 7 × 2 = 29.
Look for a pattern
What it means
Compare consecutive stages, state a rule in words, and test it on more than one given stage before using it for a distant stage.
Work through a new example
A fruit seller arranges apples in rows. The first row has 5 apples. Each new row has 3 more apples than the row before it. How many apples are in the 10th row?
Read the apple-row question. Build the picture and table from the first row.
The first apple rows and a pattern table will appear here.
Answer: The 10th row has 32 apples.
View the complete worked solution
| Row | Number of apples | Working |
|---|---|---|
| 1 | 5 | Given |
| 2 | 8 | 5 + 3 |
| 3 | 11 | 8 + 3 |
| 10 | 32 | 9 × 3 = 27 more 5 + 27 = 32 |
Check: row 2 is 5 + 3 = 8; row 3 is 8 + 3 = 11. The +3 rule fits both. ✓
- Record the 5 apples in row 1.Row 1 = 5 apples
- Add 3 apples to make row 2.Row 2 = 5 + 3 = 8 apples
- Add another 3 to make row 3. The pattern is +3 for each new row.Row 3 = 8 + 3 = 11 apples
- Count the jumps from row 1 to row 10.10 − 1 = 9 jumps
- Find how many apples the 9 increases add.9 × 3 = 27 more apples
- Add those apples to the 5 apples in row 1.Row 10 = 5 + 27 = 32 apples
- Check the rule against the known second and third rows.5 + 3 = 8; 8 + 3 = 11 ✓
Answer: The 10th row has 32 apples.
Ask at home
“How many jumps are there from the first row to the tenth?”
Common mistake
Multiplying 3 by 10 rather than by the 9 gaps between row numbers.
Try together · Primary 5
The first row has 4 chairs. Each next row has 3 more chairs than the row before. How many chairs are in the 8th row?
Reveal the worked answer
25 chairs.
There are 8 − 1 = 7 increases after the first row. Add 7 × 3 = 21 to the first 4 chairs: 4 + 21 = 25.
Count the gaps
What it means
Distance belongs to the spaces between consecutive objects. A straight row of n objects has n − 1 gaps from the first to the last.
Work through a new example
Five trees are planted in a straight row, 5 m apart. What is the distance from the first tree to the last tree?
Read the tree question. Count the spaces between trees, not the trees themselves.
The tree row and its gaps will appear here.
Answer: The distance is 20 m.
View the complete worked solution
5 trees − 1 = 4 gaps
Distance: 4 gaps × 5 m = 20 m
Check: 3 trees have 2 gaps. The same rule gives 4 gaps for 5 trees. ✓
- Draw three trees to see that gaps sit between neighbouring trees.3 trees have 2 gaps
- Mark the 5 m distance in each gap.Each gap = 5 m
- Continue the same straight row until there are 5 trees.Total trees = 5
- Count the gaps from the first tree to the last tree.5 − 1 = 4 gaps
- Multiply the number of gaps by 5 m.4 × 5 m = 20 m
- Check the small case and make sure there is no extra gap after Tree 5.3 − 1 = 2 gaps ✓5 − 1 = 4 gaps ✓
Answer: The distance is 20 m.
Ask at home
“Can you point to the gap after the last tree?”
Common mistake
Using 5 × 5 m. There is no extra gap beyond the last tree.
Try together · Primary 5
Six lamp posts are equally spaced along a straight path. The distance from the first to the last is 40 m. How far apart are neighbouring posts?
Reveal the worked answer
8 m.
Six posts make 6 − 1 = 5 gaps. Each gap is 40 ÷ 5 = 8 m. Do not divide by the number of posts.
Work backwards through two steps
What it means
Begin with the final amount and undo the actions in reverse order. Undo taking half by doubling; undo spending by adding back.
Use this when the final amount is given and the earlier actions are clear. Write the actions in order before reversing them.
Work through an example
Mei spent $18, then spent half of her remaining money. She had $24 left. How much money did she have at first?
The $24 left is half of the amount after the first purchase.
View the complete worked solution
Mei spent $18, then spent half of her remaining money. She had $24 left. How much money did she have at first?
Replay the story from $66 to check the answer.
Answer: $66
Try together · Primary 5
Mei spent $14, then spent half of the money remaining. She had $20 left. How much did she have at first?
Reveal the worked answer
$54.
Before she spent half, she had $20 × 2 = $40. Add back the first $14: $40 + $14 = $54. Check: $54 − $14 = $40; half of $40 is $20.
Put the method into practice
- Start with one clear model. Ask your child to label the known whole and the part to find.
- Practise the same idea with new numbers. Use the P5 “Try together” question before moving on.
- Check forwards. Replay each purchase, grouping or change in the original story.
For topic practice, revisit P5 fractions or P5 percentage.