Problem-Solving Approach

AO3 (problem solving) accounts for roughly 30% of the marks across your three Edexcel papers. These questions are where many students lose marks — not because the maths is too hard, but because they do not have a structured approach. This lesson gives you a framework.

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The "What Do I Know / What Do I Need" Framework

Before writing anything, ask yourself two questions:

1. What do I know? (List all the information given in the question.)
2. What do I need to find? (What is the question actually asking for?)

Then ask: What connects them? (Which mathematical technique bridges the gap?)

Worked Example 1

Question: A cone has volume $150\pi cm^{3}$150πcm3 and height 18 cm. Find the radius.

What do I know?

• Volume $= 150\pi cm^{3}$=150πcm3
• Height = 18 cm
• Formula: V = ⅓$\pi r^{2}h ($πr2h(from formula sheet)

What do I need?

• The radius, r

What connects them?

• Substitute into the formula and solve for r.

$150\pi =$150π= ⅓ $\times \pi \times r^{2} \times 18$×π×r2×18 $150\pi = 6\pi r^{2}$150π=6πr2 $r^{2} = 150\pi \div 6\pi = 25$r2=150π÷6π=25 r = 5 cm

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Breaking Multi-Step Problems into Parts

Many exam questions require 3–5 steps. The key is to tackle them one at a time.

Strategy

1. Read the entire question first — including any follow-up parts.
2. Identify the individual steps needed.
3. Complete each step, writing down intermediate answers clearly.
4. Use each intermediate answer for the next step.
5. Check that your final answer addresses the question asked.

Worked Example

Question: A shop sells pens in packs of 12 for £4.56 and packs of 5 for £2.15. Sarah needs 50 pens. What is the cheapest way to buy at least 50 pens, and how much does it cost?

Step 1: Find the price per pen for each pack. Pack of 12: £$4.56 \div 12 =$4.56÷12= £0.38 per pen Pack of 5: £$2.15 \div 5 =$2.15÷5= £0.43 per pen The pack of 12 is better value.

Step 2: How many packs of 12? $50 \div 12 = 4$50÷12=4 remainder 2 4 packs of 12 = 48 pens. She needs 2 more.

Step 3: Consider options for the remaining 2 pens. Option A: Buy another pack of $12 \to 5 \times$12→5× £4.56 = £22.80 Option B: Buy a pack of $5 \to 4 \times$5→4× £$4.56 + 1 \times$4.56+1× £2.15 = £18.24 + £2.15 = £20.39

Step 4: Compare and answer. The cheapest way is 4 packs of 12 and 1 pack of 5, costing £20.39.

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Drawing Diagrams

For geometry, trigonometry, and many context problems, drawing a diagram is essential — even if one is not provided.

When to Draw a Diagram

• Any question involving shapes, angles, or distances.
• Questions about bearings.
• Questions about speed, distance and time (draw a simple sketch or timeline).
• Questions involving 3D objects (draw the relevant 2D cross-section).

Tips for Diagrams

• Label all known measurements and angles.
• Mark right angles with a small square.
• Mark equal sides or angles with tick marks or arcs.
• Use dotted lines for construction lines or hidden edges.
• Write "NOT drawn accurately" if your diagram is a sketch — remind yourself not to measure from it.

Worked Example 2 — Bearings

Question: A ship sails from port A on a bearing of $065°$065° for 30 km to point B, then on a bearing of $150°$150° for 40 km to point C. Find the direct distance AC.

Drawing a diagram immediately reveals:

• The angle at B between the two directions.
• The bearing $065°$065° means $65°$65° clockwise from north.
• The bearing $150°$150° means $150°$150° clockwise from north.
• Angle ABC $= 360° - (180° - 65°) - 150° = 360° - 115° - 150° = 95°$=360°−(180°−65°)−150°=360°−115°−150°=95° (Or use the simpler method: angle ABC $= 150° - 65° + 180° =$=150°−65°+180°= ... draw it and check using north lines.)

With the angle and two sides, you can use the cosine rule: AC$^{2} = 30^{2} + 40^{2} - 2(30)(40)\cos 95°$2=302+402−2(30)(40)cos95° AC$^{2} = 900 + 1600 - 2400 \times \cos 95°$2=900+1600−2400×cos95° AC$^{2} = 2500 - 2400 \times (-0.0872)$2=2500−2400×(−0.0872) AC$^{2} = 2500 + 209.3$2=2500+209.3 AC$^{2} = 2709.3$2=2709.3 AC $= \sqrt{2709.3} =$=2709.3​= 52.1 km (to 3 s.f.)

Without the diagram, finding the angle at B would be much harder.

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Working Systematically

Some problems require you to consider multiple cases or organise information carefully.

Listing Outcomes

Worked Example 3

Two fair dice are thrown. Find the probability that the product is greater than 20.

Systematically list products greater than 20:

Die 1	Die 2	Product
4	6	24
5	5	25
5	6	30
6	4	24
6	5	30
6	6	36

6 outcomes out of 36 total = 6/36 = 1/6

Using Tables

Two-way tables and frequency tables help organise data problems. Always fill in the totals column/row first, then work inwards.

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Identifying Which Topic(s) to Use

Multi-topic questions are common on Edexcel. Here are clues:

Clue in the Question	Topic to Use
Right-angled triangle with sides	Pythagoras' theorem
Right-angled triangle with an angle	Trigonometry (SOH CAH TOA)
Non-right triangle with sides and angles	Sine rule or cosine rule
"Show that... is always even/odd/multiple of"	Algebraic proof
Curved shape (cone, sphere, cylinder)	Volume/surface area formulae
"Compound interest" or "depreciation"	Repeated percentage change
"Directly/inversely proportional"	Proportion equations
A shape on a coordinate grid	Coordinate geometry (gradients, midpoints)
"Probability" with two events	Tree diagrams or two-way tables
"Frequency" and "estimate"	Histograms or grouped frequency

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Linking Topics Together

Worked Example 4

A question might combine algebra and geometry:

"The perimeter of a rectangle is 34 cm. The length is (2x + 1) cm and the width is (x + 3) cm. Find the area of the rectangle."

Step 1 (Algebra): Perimeter = 2(2x + 1) + 2(x + 3) = 34 4x + 2 + 2x + 6 = 34 6x + 8 = 34 6x = 26 x = 13/3

Step 2 (Geometry): Length = 2(13/3) + 1 = 26/3 + 3/3 = 29/3 cm Width = 13/3 + 3 = 13/3 + 9/3 = 22/3 cm

Step 3: Area $= (29/3) \times (22/3) = 638/9 \approx$=(29/3)×(22/3)=638/9≈ $70.9 cm^{2}$70.9cm2 (to 1 d.p.)

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Common Pitfalls in Problem Solving

1. Not reading the entire question. The final sentence often contains crucial information ("Give your answer in metres" when the question uses centimetres).
2. Not showing intermediate steps. Even if you can see the answer, the method marks require visible working.
3. Forgetting to answer the actual question. You might correctly find x but the question asks for the length (2x + 1).
4. Using the wrong formula. Always re-read to check which shape or scenario you are dealing with.
5. Mixing up units. Convert everything to the same unit before calculating.

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Practice Problem