Mathematics • Year 7 • Unit 2 • Lesson 5

Adding and Subtracting Expressions in the Real World

Use bracket-removal and like-term collection to combine real-world expressions — comparing two phone plans, calculating profit (income minus costs), and tidying multi-part party budgets.

Apply · Real-World Maths

1. Word problems

For each, write the expression in brackets, then expand, then simplify by collecting like terms.

1.1 — Two card collections. Sara has (2x + 5) cards. Mia has (3x + 4) cards. How many cards do they have altogether?

(a) Write the total as a sum of two expressions in brackets.
(b) Simplify your expression. 2 marks

Stuck? (2x + 5) + (3x + 4). Drop the brackets, then collect like terms.

1.2 — Profit calculation. A lemonade stand's income for n cups sold is (3n + 5) dollars (including the $5 tip jar). Its costs are (n + 2) dollars (cups + lemons). Profit = income − costs.

(a) Write a profit expression using brackets.
(b) Expand and simplify.
(c) Use your simplified expression to find the profit if n = 10 cups are sold. 3 marks

Stuck? Profit = (3n + 5) − (n + 2). Subtracting a bracket flips ALL signs inside.

1.3 — Comparing phone plans. Plan A costs (2x + 10) dollars and Plan B costs (x + 25) dollars, where x is the number of GB used.

(a) Write an expression for "how much MORE Plan A costs than Plan B" (i.e. Plan A − Plan B).
(b) Expand and simplify.
(c) For what value of x do the two plans cost the same? 3 marks

Stuck on (c)? "Cost the same" means the difference = 0. Set your simplified expression equal to 0 and solve.

1.4 — Party budget. Three things at the party each cost an amount depending on the number of guests g: drinks cost (2g + 3) dollars, food costs (5g + 10) dollars, and decorations cost a flat 8 dollars.

(a) Write a single expression for the total cost of the party.
(b) Expand and simplify.
(c) Find the total cost when there are 12 guests. 3 marks

Stuck? Add all three expressions. Decorations is just "8" — no variable in it.

1.5 — Bus trip and refunds. A school trip costs 3(p + 2) dollars for p students (each pays $3 and there's a flat $6 added in). Then the bus company gives a refund of (p + 1) dollars per group.

(a) Write an expression for the total cost AFTER the refund.
(b) Expand 3(p + 2) and then subtract the refund. Simplify fully. 3 marks

2. Explain your thinking

This question is about why subtraction flips the signs. Use full sentences. 4 marks

2.1 Drew writes: "(4x + 3) − (2x − 1) = 4x + 3 − 2x − 1 = 2x + 2." Drew has made the same mistake twice in a row. (i) Identify the mistake, (ii) explain in words WHY a minus sign in front of a bracket flips every sign inside, (iii) give a numerical example (one sentence) to show that NOT flipping gives the wrong answer, and (iv) write Drew's corrected final answer.

Stuck? Try x = 1. Original: (7) − (1) = 6. Drew's "answer" 2(1) + 2 = 4. Doesn't match — so Drew is wrong.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Card collections

(a) Total = (2x + 5) + (3x + 4).
(b) Drop both brackets: 2x + 5 + 3x + 4. Collect: x-terms 2x + 3x = 5x; constants 5 + 4 = 9. Answer: 5x + 9 cards.

1.2 — Profit calculation

(a) Profit = (3n + 5) − (n + 2).
(b) Drop first: 3n + 5. Flip second: −n − 2. Together: 3n + 5 − n − 2 = 2n + 3 dollars.
(c) With n = 10: 2(10) + 3 = $23 profit.

1.3 — Comparing phone plans

(a) Difference = (2x + 10) − (x + 25).
(b) Drop first: 2x + 10. Flip second: −x − 25. Together: 2x + 10 − x − 25 = x − 15.
(c) Plans cost the same when x − 15 = 0, i.e. x = 15 GB.

1.4 — Party budget

(a) Total = (2g + 3) + (5g + 10) + 8.
(b) Drop brackets: 2g + 3 + 5g + 10 + 8. Collect: g-terms 2g + 5g = 7g; constants 3 + 10 + 8 = 21. Answer: 7g + 21 dollars.
(c) With g = 12: 7(12) + 21 = 84 + 21 = $105.

1.5 — Bus trip and refunds

(a) Total = 3(p + 2) − (p + 1).
(b) Distribute: 3(p + 2) = 3p + 6. Flip second bracket: −p − 1. Together: 3p + 6 − p − 1 = 2p + 5 dollars.

2.1 — Drew's error (sample response)

(i) Drew dropped the second bracket without flipping the signs of the terms inside. The −(2x − 1) should become −2x + 1, not −2x − 1.
(ii) A minus sign in front of a bracket means "subtract the WHOLE thing inside". Subtracting +2x is the same as adding −2x; subtracting −1 is the same as adding +1. So every term's sign flips.
(iii) Numerical check: let x = 1. Original = (4 + 3) − (2 − 1) = 7 − 1 = 6. Drew's wrong answer = 2(1) + 2 = 4. Different — so Drew's working is wrong.
(iv) Drew's corrected final answer: 4x + 3 − 2x + 1 = 2x + 4.

Marking: 1 for spotting the sign error; 1 for the verbal explanation; 1 for a clear numerical check; 1 for the corrected final answer.