Factors: Real-Life Problems
A diagnostic ladder from Year 4 to Year 7+. Word problems using factors.
This booklet helps you use factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. to solve real-life word problems, step by step.
When is a real-life problem secretly a factor problem?
Look for these words and ideas:
- “Share equally” or “split into equal groups” — for it to work with no leftover, the group size must be a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the total.
- “Pack into boxes of N” — N must be a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the total.
- “Arrange in a rectangle” — the sides are a factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. of the area.
- “Largest equal share of two different things” — that’s the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
A useful method: RUCSACS
When you read a word problem, follow these steps:
- Read the question twice.
- Underline the important numbers and the question word (how many? what’s the largest? …).
- Choose your operation. (Sharing → divide. Largest equal share → HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..)
- Solve.
- Answer in a sentence (so you check it’s the right kind of thing).
- Check the answer fits the question (is it sensible?).
- Sense check: does the answer make real-life sense?
What you’ll learn
We start with sharing things into equal groups. Then we choose group sizes that fit a range. Then we look at rectangles and arrays. Finally we work out the LARGEST equal share when there are TWO different things (which is HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. in disguise).
How to use the booklet
There are four levels. Start at Level 1 and work through in order.
If you get stuck, that’s OK — stop there. The level you stop at tells your tutor exactly where you need help.
TUTOR — MATERIALS TO HAVE READY
For pupils who struggle with word problems:
- Counters or beads — for Level 1, lay out the numbers physically. “24 chocolates, 6 children” becomes 24 beads, 6 piles. Sharing becomes visible.
- Squared paper — for Level 3, drawing the rectangle from the dimensions makes the “area = side × side” rule concrete.
- Two colours of counter or sticker — for Level 4, when the question says “30 red and 45 blue”, lay them out. The HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. becomes the number of identical piles you can make.
- A highlighter — for any level, ask the pupil to highlight the important numbers and the question word BEFORE doing any arithmetic. Half of word-problem errors are reading errors, not maths errors.
The ladder:
· Level 1 / Section A — Year 4–5 (KS2). Sharing equally and equal groups.
· Level 2 / Section B — Year 5–6 (KS2). Choosing group sizes that fit a range.
· Level 3 / Section C — Year 6–7. Rectangles, arrays, and area.
· Level 4 / Section D — Year 7+. Greatest equal share (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. in real-life clothes).
· Level 1 / Section A — Year 4–5 (KS2). Sharing equally and equal groups.
· Level 2 / Section B — Year 5–6 (KS2). Choosing group sizes that fit a range.
· Level 3 / Section C — Year 6–7. Rectangles, arrays, and area.
· Level 4 / Section D — Year 7+. Greatest equal share (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. in real-life clothes).
LEVEL 1 · YEAR 4–5 (KS2) · LESSON 1
Sharing equally
Aim: Recognise when a word problem is asking you to divide. Get the answer right when the share works exactly.
Rule: “Share X things between Y people equally” = X ÷ Y.
For the share to work exactly (no remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1.), Y must be a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of X.
For the share to work exactly (no remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1.), Y must be a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of X.
Worked example
A teacher has 20 cookies. She shares them equally between 5 children. How many does each child get?
- 20 cookies, 5 children. “Equally” means divide.
- 20 ÷ 5 = 4.
- Each child gets 4 cookies. ✓
Watch out
If a question asks whether the share works exactly and you get a remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1., the answer is NO. Don’t round up — that would mean making up extra things to share.
LEVEL 2 · YEAR 5–6 (KS2) · LESSON 2
Group sizes that fit a range
Aim: When a question puts a limit on group size (“at least 5 and at most 10”), list the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. and pick the right one.
Method:
1. List ALL the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the total.
2. Keep only the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. that are inside the range.
3. Read the question again: do you want the LARGEST, SMALLEST, or HOW MANY?
1. List ALL the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the total.
2. Keep only the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. that are inside the range.
3. Read the question again: do you want the LARGEST, SMALLEST, or HOW MANY?
Worked example
A coach has 36 players. He wants equal teams, each with at least 3 and at most 9 players. What is the LARGEST team size that works?
- FactorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
- In the range 3 to 9: 3, 4, 6, 9.
- Largest of these = 9. ✓
LEVEL 3 · YEAR 6–7 · LESSON 3
Rectangles, arrays, and area
Aim: Use the area-side-side relationship to find a missing side.
Rule: Area of a rectangle = side 1 × side 2.
Knowing the area and one side, the other side = area ÷ known side.
For whole-number sides, both sides must be factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the area.
Knowing the area and one side, the other side = area ÷ known side.
For whole-number sides, both sides must be factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the area.
Worked example
A rectangular garden has area 40 m². One side is 5 m. What’s the other side?
- Area = 40, one side = 5.
- Other side = 40 ÷ 5 = 8 m. ✓
LEVEL 4 · YEAR 7+ · LESSON 4
Greatest equal share (HCF in real life)
Aim: Recognise when a word problem is secretly an HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. problem, and solve it.
Signal: You have TWO different quantities, and you want the LARGEST number of identical packs/groups/bags using ALL of both.
That largest number = HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. (Highest Common FactorHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.) of the two quantities.
That largest number = HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. (Highest Common FactorHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.) of the two quantities.
Worked example
A florist has 20 roses and 30 daisies. She makes identical bouquets using all the flowers. What’s the largest number of bouquets she can make?
- The number of bouquets must divide BOTH 20 and 30.
- FactorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 20: 1, 2, 4, 5, 10, 20.
- FactorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 30: 1, 2, 3, 5, 6, 10, 15, 30.
- Largest in both lists = 10.
- So 10 bouquets. Each: 2 roses, 3 daisies. ✓
Watch out
If the question asks “how many of each in a bag?” instead of “how many bags?”, divide each total by the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.. In the example: 20 ÷ 10 = 2 roses, 30 ÷ 10 = 3 daisies per bouquet.
VOCABULARY
- Factor
- A number that fits exactly into another number, with no remainder.
- Share equally
- Divide so that each part is the same size, with nothing left over.
- HCF (Highest Common Factor)
- The biggest number that divides BOTH of two numbers exactly. Used for “largest equal share”.
- Array
- An arrangement in equal rows and columns. The total count is the number of rows × number of columns.
- Remainder
- What’s left over after a division that doesn’t come out exactly.
Recap
- Share equally: divide. Group size must be a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
- Fits a range: list factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., pick the ones in range.
- Rectangles: sides are a factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. of the area.
- Largest equal share of two things: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
Part Two
Now try the practice questions
45 questions across three sections, 30 minutes when you’re ready. The step-by-step walkthroughs unlock after you mark.
Instructions
- Type each numeric answer in the box. For sign questions, use a minus sign (e.g.
-7). For radio questions, tap your choice. - You can start the 30-minute timer or work untimed.
- When finished, click Mark My Work. Your answers will then be locked, the correct answers will appear under each wrong question with the working, and you will get 5 similar practice questions inline for each one you missed.
Timed practice · 30 minutes
When you’re ready, start the timer
This practice is designed to take 30 minutes. Your tutor will see how long you took. You can work untimed if you prefer.
Section A · Level 1 · Year 4–5 (KS2)
Sharing equally and equal groups.
Q1.
A teacher has 24 chocolates and 6 children. She shares them equally. How many chocolates does each child get?
Your answer:
(2 marks)
Q2.
A baker has 30 cookies. Can she pack them into boxes of 7 with none left over? Type 1 for yes, 0 for no.
Your answer:
(2 marks)
Q3.
A gardener has 45 flowers. She plants them in 9 equal rows. How many flowers are in each row?
Your answer:
(2 marks)
Q4.
I have 36 sweets. I want to give 4 sweets to each friend. How many friends can I share with?
Your answer:
(2 marks)
Q5.
A teacher has 28 books. Can she stack them in equal piles of 6? Type 1 for yes, 0 for no.
Your answer:
(2 marks)
Q6.
48 children visit a park in equal groups of 8. How many groups are there?
Your answer:
(2 marks)
Section B · Level 2 · Year 5–6 (KS2)
Group sizes that fit a range.
Q7.
A coach has 24 players. He puts them into equal teams of 4 players. How many teams does he make?
Your answer:
(2 marks)
Q8.
A coach has 30 players. He wants equal teams with at least 5 and at most 10 players each. What is the LARGEST team size that works?
Your answer:
(3 marks)
Q9.
A teacher has 40 stickers to share equally between her 8 pupils. How many stickers does each pupil get?
Your answer:
(2 marks)
Q10.
A baker has 60 cookies. She packs them into equal boxes. She wants at least 5 and at most 15 boxes. What is the SMALLEST number of boxes that works?
Your answer:
(3 marks)
Q11.
A florist has 48 flowers. She makes identical bouquets. Each bouquet must have at least 4 and at most 12 flowers. How many DIFFERENT bouquet sizes are possible?
Your answer:
(4 marks)
Q12.
A teacher arranges 42 chairs in equal rows. Each row has 6 chairs. How many rows?
Your answer:
(2 marks)
Q13.
56 children take part in a competition in equal teams. Each team has at least 4 and at most 14 members. What is the LARGEST team size that works?
Your answer:
(3 marks)
Section C · Level 3 · Year 6–7
Rectangles, arrays, and area.
Q14.
A rectangle has area 36 cm². One side is 4 cm. What is the other side, in cm?
Your answer:
(2 marks)
Q15.
A garden bed has area 60 m² and is rectangular. One side is 5 m. What is the other side, in m?
Your answer:
(2 marks)
Q16.
A teacher arranges 100 stickers in a rectangular array on the wall. One side has 20 stickers. How many stickers are on the OTHER side?
Your answer:
(3 marks)
Q17.
A rectangular floor has area 72 m² with whole-number sides. One side is 9 m. What is the other side, in m?
Your answer:
(3 marks)
Q18.
A teacher splits 84 worksheets into equal piles. She wants at least 10 and at most 15 piles. What is the SMALLEST number of piles that works?
Your answer:
(4 marks)
Q19.
A baker has 96 muffins. She packs them into equal boxes. Boxes can hold at least 6 and at most 16 muffins. How many DIFFERENT box sizes can she use?
Your answer:
(4 marks)
Section D · Level 4 · Year 7+
Greatest equal share (HCF in real-life clothes).
Q20.
I have 30 red sweets and 45 blue sweets. I want to make identical party bags using ALL of the sweets (no sweets left over). What is the LARGEST number of party bags I can make?
Your answer:
(4 marks)
Q21.
A florist has 24 roses and 36 lilies. She makes identical bouquets using ALL the flowers. What is the LARGEST number of bouquets she can make?
Your answer:
(4 marks)
Q22.
I have 18 red pencils and 27 blue pencils. I make identical pencil packs using ALL of them. What is the LARGEST number of packs I can make?
Your answer:
(4 marks)
Q23.
A teacher has 40 pens and 60 erasers. She makes identical stationery packs for pupils, using everything. What is the LARGEST number of packs?
Your answer:
(4 marks)
Q24.
I have 72 stickers and 96 sweets. I want to give the same number of each to each child, using everything. What is the LARGEST number of children I can give to?
Your answer:
(5 marks)
Q25.
A school has 42 boys and 63 girls in Year 7. They are split into equal-sized mixed groups, each with the same number of boys and the same number of girls. What is the LARGEST number of groups possible?
Your answer:
(5 marks)