Practice Maths

Mixed Problem Solving

Key Ideas

Key Terms

Read carefully
Identify what information is given and what you are asked to find.
Plan
Decide which Year 9 topic(s) and techniques apply. Draw a diagram if geometry is involved.
Execute
Carry out all steps, showing clear working. Set up equations before solving.
Check
Verify your answer makes sense in context. Use estimation or a different method to check.
Communicate
Write a sentence answering the question with correct units.
Hot Tip If you are stuck, try a simpler version of the problem first, draw a diagram, or work backwards from the answer to check your method.

Approaching Multi-Step Problems

Multi-step problems are the real test of mathematical understanding — they require you to choose the right technique, apply it correctly, and link steps together into a logical solution. No single procedure is given to you; you must identify what is needed. The key mindset shift is from "which formula do I use?" to "what do I know, what do I need, and how can I bridge the gap?"

General strategy: (1) Read carefully and underline key information. (2) Draw a diagram if the problem is geometric. (3) Define any unknowns with a letter. (4) Decide which Year 9 topic applies. (5) Write a clear, logical solution. (6) Check and interpret the answer.

Combining Geometry and Algebra

Many rich problems combine algebra with geometry. A common type: angles or lengths are expressed as algebraic expressions, and a geometric property (e.g. angles in a triangle sum to 180°, or Pythagoras' theorem) creates an equation to solve.

Example: In a right triangle, the two shorter sides are (x + 3) cm and (x − 1) cm. The hypotenuse is (x + 5) cm. Find x.

By Pythagoras: (x + 3)2 + (x − 1)2 = (x + 5)2. Expand: x2 + 6x + 9 + x2 − 2x + 1 = x2 + 10x + 25. Simplify: 2x2 + 4x + 10 = x2 + 10x + 25. Rearrange: x2 − 6x − 15 = 0. Solve to find x.

Working Systematically

In counting or probability problems, "working systematically" means listing or organising all cases in a structured way so none are missed. Use a table, tree diagram, or organised list. Label rows and columns clearly.

In number problems, try specific cases to spot a pattern, then generalise. For example, if asked which two consecutive integers have a product of 506, note that 20 × 25 = 500 (close), so try 22 × 23 = 506. Or set up n(n + 1) = 506 and solve.

Communicating Solutions Clearly

A correct answer with no working shown risks earning 0 marks for a multi-part problem if you made a small error. Show every step. Use equals signs correctly — never write a chain like "3 + 5 = 8 × 2 = 16" (this says 3 + 5 = 16, which is false). Each new operation belongs on a new line.

Finish with a sentence that answers the question asked, including units. "Therefore the rectangle has dimensions 8 cm × 12 cm" is far better than just writing "8 and 12."

Choosing the Right Year 9 Tool

Review this checklist when a problem feels unfamiliar: Is there a right angle? → Pythagoras or trigonometry. Are there two unknowns? → simultaneous equations. Is something growing or changing? → linear relationship or gradient. Are lengths proportional? → similarity. Is it about chance? → probability (tree diagram or Venn diagram). Is data involved? → scatter plot and line of best fit. Does the problem involve money over time? → financial maths. Once you identify the topic, apply its standard technique.

Key tip: If you are stuck on a multi-step problem, start by writing down what you know and what you are trying to find. Often this act of organising the information makes the path to the solution clear. Never stare at a blank page — write something down, even if it is just the given values and a sketch.

Mastery Practice

  1. Number and Algebra — compound interest and equations. Understanding

    1. Maya invests $4 500 at 4% per annum simple interest for 3 years. Calculate the total interest earned and the final amount.
    2. Her friend invests $4 500 at 4% per annum compound interest (compounded annually) for 3 years. Use A = P(1 + r)^n to find the final amount. Round to the nearest cent.
    3. How much more does Maya’s friend earn from compound interest compared to simple interest?
    4. Maya wants to save enough for a $6 000 laptop. Using simple interest at 4% p.a., how many years does she need to invest the original $4 500?

  2. Algebra and Geometry — finding dimensions. Understanding

    A rectangular swimming pool has a length that is 3 m more than twice its width. The perimeter of the pool is 48 m.

    1. Let the width be w metres. Write an expression for the length in terms of w.
    2. Write and solve an equation to find the width and length of the pool.
    3. Calculate the area of the pool.
    4. Tiles covering the bottom of the pool cost $75 per m². Find the total tiling cost.
    5. The pool is 1.8 m deep throughout. Find the volume of water needed to fill it, in kilolitres. (1 m³ = 1 kL)

  3. Geometry — Pythagoras and trigonometry combined. Problem Solving

    A 6 m ladder leans against a vertical wall. The base of the ladder is 2.3 m from the wall on level ground.

    1. Draw a diagram and find the height the ladder reaches up the wall, correct to 2 decimal places.
    2. Find the angle the ladder makes with the ground, to the nearest degree.
    3. Safety guidelines say the base should be no more than 1/4 of the ladder length from the wall. Is this ladder placed safely?
    4. If the base is moved so the angle with the ground is 75°, how far is the base from the wall? (Answer to 2 decimal places.)
    5. At 75°, how far up the wall does the ladder reach? (Answer to 2 decimal places.)

  4. Algebra — simultaneous linear equations in context. Problem Solving

    At a school fete, adult tickets cost $8 and child tickets cost $3. A total of 240 tickets were sold and $1 260 was collected.

    1. Define variables and write two equations representing this situation.
    2. Solve the system of equations to find the number of adult and child tickets sold.
    3. The fete organisers planned to sell at least 60 adult tickets. Was this target met?
    4. If the child ticket price is raised to $4, and all other conditions remain the same, write new equations and find the new ticket numbers. Comment on the result.

  5. Statistics — analysing and comparing data sets. Understanding

    Class A scored the following on a maths test (out of 50): 32, 45, 38, 27, 41, 36, 29, 44, 38, 31, 43, 35.
    Class B scored: 40, 42, 28, 39, 41, 30, 44, 38, 29, 43, 41, 37.

    1. Find the mean score for each class.
    2. Find the median score for each class.
    3. Find the range for each class.
    4. Which class performed more consistently? Use the range to justify your answer.
    5. A new student joins Class A with a score of 48. Recalculate the mean for Class A. What is the effect?

  6. Measurement — composite shapes and surface area. Problem Solving

    A storage shed has a rectangular floor (10 m × 6 m) and a triangular roof. The triangular cross-section has a base of 6 m and a vertical height of 2.5 m. The shed is 10 m long.

    1. Find the area of the triangular cross-section of the roof.
    2. Find the slant height of each side of the roof (i.e. from apex to base edge), to 2 decimal places.
    3. Find the total surface area of the two triangular end walls of the shed.
    4. Find the surface area of the two sloping roof panels (each is a rectangle with width = slant height and length = 10 m).
    5. The four walls of the shed are vertical rectangles. Find the total wall area (excluding roof and floor).
    6. Roofing iron costs $18.50 per m². Find the total cost to clad the two roof panels only.

  7. Algebra and Number — equations involving surds and indices. Problem Solving

    1. Simplify √(3x²) when x > 0.
    2. Solve: 2x² = 72
    3. A square has an area of 98 cm². Find the exact side length in simplest surd form and calculate the perimeter.
    4. Evaluate: 5² × 5³ ÷ 5&sup4; using index laws.
    5. Simplify: (2a³b)² ÷ (4a²b)

  8. Probability and Statistics — combined context. Problem Solving

    A local football team played 20 games last season. They won 13, drew 2, and lost 5.

    1. Find the experimental probability of the team winning a randomly chosen game.
    2. Find the experimental probability of the team NOT losing.
    3. If they play 30 games next season and their results are consistent with last season, predict the number of wins.
    4. The mean goals scored per game last season was 2.3 with a range of 6. What does this tell you about their scoring performance?
    5. This season, in their first 5 games, they scored: 4, 1, 3, 5, 2. Is their mean so far higher or lower than last season?

  9. Financial mathematics — income, tax, and budgeting. Problem Solving

    Zara earns $1 350 per fortnight (2 weeks). She pays 19% of her income in tax. She spends 35% of her after-tax income on rent.

    1. Calculate Zara’s fortnightly tax payment.
    2. Calculate her after-tax (take-home) income per fortnight.
    3. Calculate her fortnightly rent payment.
    4. Zara wants to save $3 600 for a holiday in 9 months. She saves the same amount each fortnight. How much must she save per fortnight? (There are approximately 2.17 fortnights per month.)
    5. After rent and savings, how much is left per fortnight for other expenses?

  10. Trigonometry — angles of elevation and depression. Problem Solving

    From the top of a cliff 80 m above sea level, a lighthouse keeper observes two boats. Boat A has an angle of depression of 22° and Boat B has an angle of depression of 35°. Both boats are in the same direction from the cliff base.

    1. Draw a labelled diagram showing the cliff, the lighthouse keeper’s position, and the two boats.
    2. Find the horizontal distance from the base of the cliff to Boat A, to the nearest metre.
    3. Find the horizontal distance from the base of the cliff to Boat B, to the nearest metre.
    4. How far apart are the two boats (horizontal distance), to the nearest metre?
    5. The lighthouse keeper receives a distress signal from a third boat located 200 m from the cliff base. What is the angle of depression to this boat? (Answer to 1 decimal place.)
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