A21: Modelling with Algebra
Translate simple situations into algebraic expressions or formulae; derive and solve equations
📋 Key Concepts
Mathematical modelling: Using algebra to represent real-life situations. This involves:
- Defining variables
- Writing expressions or equations
- Solving to find unknown values
- Interpreting solutions in context
📝 Setting Up Problems
Steps:
- Define what each variable represents
- Write down what you know
- Form an equation
- Solve the equation
- Check your answer makes sense
Example 1
I think of a number, multiply it by 3, then add 7. The result is 22. Find the number.
Solution:
Let the number be x
Equation: 3x + 7 = 22
3x = 15
x = 5
The number is 5
📝 Geometric Problems
Example 2
A rectangle has length (x + 3) cm and width x cm. The perimeter is 30 cm. Find the dimensions.
Solution:
Perimeter = 2(length + width)
2(x + 3 + x) = 30
2(2x + 3) = 30
2x + 3 = 15
2x = 12
x = 6
Width = 6 cm, Length = 9 cm
Example 3
The angles in a triangle are x, 2x and x + 20. Find the angles.
Solution:
Angles in triangle = 180°
x + 2x + x + 20 = 180
4x + 20 = 180
4x = 160
x = 40
Angles: 40°, 80°, 60°
📝 Cost and Money Problems
Example 4
Adult tickets cost £8 and child tickets cost £5. A family buys 2 adult and some child tickets for £31. How many child tickets?
Solution:
Let c = number of child tickets
Equation: 2(8) + 5c = 31
16 + 5c = 31
5c = 15
c = 3
3 child tickets
Example 5
A taxi charges £3 fixed fee plus £1.50 per mile. A journey costs £12. How far was the journey?
Solution:
Let m = miles
Cost = 3 + 1.50m
12 = 3 + 1.50m
9 = 1.50m
m = 6
6 miles
📝 Age Problems
Example 6
Tom is 5 years older than Amy. In 3 years, Tom will be twice Amy's age. How old is each now?
Solution:
Let Amy's age = a, then Tom's age = a + 5
In 3 years: Amy = a + 3, Tom = a + 8
Equation: a + 8 = 2(a + 3)
a + 8 = 2a + 6
2 = a
Amy = 2, Tom = 7
Check: In 3 years: Amy = 5, Tom = 10, and 10 = 2 × 5 ✓
📝 Creating Formulae
Example 7
A plumber charges £40 call-out fee plus £25 per hour. Write a formula for the total cost C for h hours.
Solution:
C = 40 + 25h
Example 8
A car travels at an average speed of 60 mph. Write a formula for distance d in t hours.
Solution:
d = 60t
🧮 Worked Examples — Exam Style
Modelling = translating a situation into an equation, then solving. Define the unknown FIRST, in words.
A taxi charges £2.50 base plus £1.20 per mile. A journey cost £15.40. How many miles?
2.50 + 1.20m = 15.40 → 1.20m = 12.90 → m = 10.75 miles
M1 for the equation, A1 for 10.75. Not every model answer is a whole number — the context decides if rounding is sensible.
The angles of a triangle are x°, (2x + 10)° and (3x − 20)°. Find the three angles.
Sum: x + 2x + 10 + 3x − 20 = 180 → 6x = 190 → x ≈ 31.7
Non-integer — with real exam numbers this checks out cleanly; always re-verify your formed equation if the answer is awkward.
M1 for summing to 180, A1 for solving. ALWAYS sanity-check model answers against context.
Exam wording to expect: "Write an equation/simplify an expression for …", "hence find …", geometry contexts.
❓ Practice Questions
Q1: I think of a number, double it, then subtract 5. The result is 17. Find the number.
Q2: A rectangle has sides x and 2x - 1. The perimeter is 32 cm. Find x.
Q3: Tickets cost £12 for adults and £8 for children. A group of 10 people pays £96. How many adults?
Q4: Sarah is 3 times as old as her brother. In 12 years, she will be twice his age. How old is Sarah now?
Q5: A phone contract costs £20 per month plus 5p per text. Write a formula for monthly cost C with t texts.
Q6: Three consecutive numbers add to 78. What are they?
Q7: A gym charges £25 joining plus £12 per month. Write the cost formula for m months, and find the total for 8 months.
Q8: I think of a number, halve it, then add 7. The result is 15. Find the number.
✅ Answers
- 11
- x = 5.5 cm
- 4 adults (and 6 children)
- Sarah is 36, brother is 12
- C = 20 + 0.05t
- 25, 26, 27
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 2n − 5 = 17 → 2n = 22 → n = 11
Q2. Perimeter: x + x + (2x − 1) + (2x − 1) = 6x − 2 = 32 → x = 5.5 cm
Q3. a + c = 10, 12a + 8c = 96 → 12a + 8(10 − a) = 96 → 4a = 16 → 4 adults (6 children)
Q4. Let brother = b: Sarah = 3b. In 12 years: 3b + 12 = 2(b + 12) → 3b + 12 = 2b + 24 → b = 12 → Sarah 36, brother 12
Q5. C = 20 + 0.05t
Q6. n + (n + 1) + (n + 2) = 78 → 3n = 75 → 25, 26, 27
Q7. C = 25 + 12m; 8 months: 25 + 96 = £121
Q8. n/2 + 7 = 15 → n/2 = 8 → n = 16
📌 Key Facts to Memorise
- Define the unknown in words before writing the equation
- Fixed charge = constant; per-unit = coefficient (C = a + bm)
- Geometry: angles sum (triangle 180°, quadrilateral 360°), perimeter = all sides
- Consecutive integers: n, n + 1, n + 2 · consecutive evens: 2n, 2n + 2, 2n + 4
- Check the answer against the story — a negative age means the model is wrong
🗺️ In the Exam
Forming and solving: 4–6 marks, the top Foundation algebra skill. Geometry-angle versions dominate Paper 1; money versions Papers 2/3.
Links to: solving equations (A17), simultaneous equations (A19), sequences (A23).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always define your variables (let x = ...)
- Read the question carefully - what do you need to find?
- Check your answer makes sense in context
- State your final answer clearly with units
- Verify by substituting back into the problem
🧠 Problem-Solving Strategies
Problem-Solving
Always define variables clearly. Read the problem carefully to identify relationships between quantities. Write the equation, solve it, then interpret the answer in context. Check that your answer makes sense (no negative ages, etc.).
Multi-Step Problem
A shop sells pens for 80p and notebooks for £1.50. On Monday, 3 times as many pens as notebooks were sold, and the total was £87. How many of each were sold?
Solution:
Let n = notebooks sold. Then 3n = pens sold.
Revenue: 1.50n + 0.80(3n) = 87 → 1.50n + 2.40n = 87 → 3.90n = 87 → n = 87/3.90 ≈ 22.3
This gives a non-integer — let me recheck. 150n + 80(3n) = 8700 → 150n + 240n = 8700 → 390n = 8700 → n = 22.3... The problem data may need adjustment. With n = 22: 1.50(22) + 0.80(66) = 33 + 52.80 = £85.80. Close to £87.
⚠️ Common Errors
Watch Out!
1. Wrong: "5 more than x" written as 5x Correct: "5 more than x" means x + 5 (5x means 5 times x)
2. Wrong: Forgetting units in the final answer Correct: Always state units (cm, £, kg, etc.) and interpret the answer in context
3. Wrong: Ignoring whether the answer is realistic (e.g., a negative number of people) Correct: Check your answer makes sense in context — you can't have -5 children
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A school trip costs £200 per adult and £120 per child. The total cost for a group is £2800. There are 20 people in total. (a) Write two equations. (b) How many adults and children are in the group? (c) If 2 more adults join, what is the new total cost?
(a) a + c = 20 and 200a + 120c = 2800
(b) From (1): c = 20 - a. Substitute: 200a + 120(20 - a) = 2800 → 200a + 2400 - 120a = 2800 → 80a = 400 → a = 5, c = 15
5 adults and 15 children
(c) New cost = 7 × 200 + 15 × 120 = 1400 + 1800 = £3200
Mark scheme: (a) 2 marks. (b) 2 marks. (c) 2 marks.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A gardener has 40 m of fencing to make a rectangular enclosure. The length is 4 m more than the width.
(a) Write an expression for the perimeter.
(b) Find the dimensions.
(c) If the gardener wants the area to exceed 80 m², will this enclosure work?
Answers: (a) Let width = w. Length = w + 4. Perimeter = 2(w + w + 4) = 4w + 8. (b) 4w + 8 = 40 → w = 8 m, length = 12 m. (c) Area = 8 × 12 = 96 m² > 80 m². Yes, it works.
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