Mathematics • Year 10 • Unit 2 • Lesson 4

Factorising in the Real World

Apply common-factor factorising to recognise structure in area expressions, ticket pricing, recipe scaling and side-length problems. Then explain why "fully factorised" is the only acceptable finished form.

Apply · Real-World Maths

1. Word problems

Each problem asks you to factorise an expression that arises from a real setup, then use the factorised form to read off side lengths, group counts, or a sensible pattern.

1.1 — Side lengths of a rectangle. A rectangle has area 6x + 24 (m²).

(a) Factorise the area expression.
(b) State the two side lengths of the rectangle, in terms of x.
(c) If x = 7, what are the numerical side lengths and the area? 3 marks

Stuck? Factorise to a(b + c) and read off the two sides directly.

1.2 — Ticket revenue. A school sells x adult tickets at $12 each and x child tickets at $8 each. The total revenue is 12x + 8x.

(a) Combine like terms.
(b) Factorise the result, pulling out the common x.
(c) If x = 35 (35 adults and 35 children), what is the total revenue? 3 marks

Stuck? 12x + 8x = 20x. Factorising: this can be written as x(20). Substitute x = 35 last.

1.3 — Scaling a recipe with common factors. A cookie recipe uses 4x cups of flour, 6x tablespoons of sugar, and 2x teaspoons of salt for x batches.

(a) Write a single expression for the total quantity (in any combined unit "items").
(b) Factorise the result, taking out the common x.
(c) Use the factorised form to find the total for x = 5 batches. 3 marks

Stuck? 4x + 6x + 2x = 12x. Factorise as x(12) or just 12x. Substitute x = 5.

1.4 — Area of an L-shaped patio. An L-shaped patio is made of two rectangles: one with area 8x²y (m²) and one with area 12xy² (m²).

(a) Write a single expression for the total area.
(b) Factorise it by pulling out the HCF (a number and the common letters).
(c) If x = 2 and y = 3, calculate the total area in m². 4 marks

Stuck? Number HCF of 8 and 12 is 4. Letter HCF of x²y and xy² is xy. So HCF = 4xy.

1.5 — Splitting a bill into two parts. A bill for catering is given by 15x + 25 dollars, where x is the number of guests.

(a) Factorise the expression.
(b) Use the factorised form to give a one-sentence "story" of what the two pieces represent (think: per-guest cost, fixed fee).
(c) If x = 12, calculate the total bill. 3 marks

Stuck? HCF of 15 and 25 is 5 → 5(3x + 5). The 3x part is the per-guest, the 5 is the fixed.

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A classmate writes: "I factorised 12x²y + 18xy² and got 6(2x²y + 3xy²). Done!" Using everything from Lesson 4, explain (i) what is correct about their answer, (ii) what is still missing — i.e. why it isn't fully factorised, and (iii) what the correct fully factorised form is. Use the words "HCF" and "fully factorised" somewhere in your answer.

Stuck? Revisit lesson § "Common Pitfalls" — pulling out just the numerical factor when letters are also common is the most common slip.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Rectangle area 6x + 24

(a) HCF = 6 → 6(x + 4).
(b) Sides: 6 m and (x + 4) m.
(c) When x = 7: sides are 6 m and 11 m. Area = 6 × 11 = 66 m².

1.2 — Ticket revenue

(a) 12x + 8x = 20x.
(b) Factorised: x(20) or simply 20x.
(c) When x = 35: 20 × 35 = $700.

1.3 — Recipe scaling

(a) Total items = 4x + 6x + 2x = 12x.
(b) Factorised: x(12) or just 12x — pulling x out is mostly a comment that the quantity scales linearly with x.
(c) When x = 5: 12 × 5 = 60 items.

1.4 — L-shaped patio

(a) Total area = 8x²y + 12xy² m².
(b) Number HCF = 4, letter HCF = xy. So HCF = 4xy.
8x²y ÷ 4xy = 2x; 12xy² ÷ 4xy = 3y. Factorised: 4xy(2x + 3y).
(c) When x = 2, y = 3: 4(2)(3)(2(2) + 3(3)) = 24(4 + 9) = 24 × 13 = 312 m².

1.5 — Splitting a bill

(a) HCF = 5 → 5(3x + 5).
(b) The factor 5 outside is a multiplier; inside, 3x represents the per-guest portion (when scaled by 5 gives $15 per guest) and 5 represents a small fixed fee component (5 × 5 = $25 fixed). One way to read it: the catering bill is "5 times (3 per-guest dollars + 5 fixed dollars)".
(c) When x = 12: 5(3(12) + 5) = 5(36 + 5) = 5 × 41 = $205.

2.1 — Explain your thinking (sample response)

(i) The classmate is correct that 6 is a common factor of both terms — so 6(2x²y + 3xy²) is a valid factorising step. (ii) But the result is not fully factorised: the inner terms 2x²y and 3xy² still share the common letter group xy, so there's more to pull out. The true HCF of the original expression is 6xy, not just 6. (iii) The correct fully factorised form is 6xy(2x + 3y) — both the number HCF and the letter HCF have been extracted, and inside the bracket there are no more common factors.

Marking: 1 for crediting the correct numerical part, 1 for identifying the missing xy, 1 for the fully factorised form, 1 for using both required words.