GCSE Maths Problem Solving: Venn Diagram and Two-Way Table Exam Question Walkthrough

Exam Board: Suitable for AQA, Edexcel, OCR GCSE Maths
Topic: Probability & Statistics / Problem Solving
Difficulty: Higher Tier Foundation

If you've ever looked at a wordy GCSE maths problem and felt overwhelmed by all the numbers and categories, you're not alone. Today, we're walking through a classic type of question that tests your ability to organise information systematically. Let's break down the problem and learn a reliable method using a two-way table.

The Question

Here's the problem exactly as it might appear in your exam:

> 200 students chose one language to study.
> Each student chose one language from French or Spanish or German.
> Of the 200 students, 90 are boys and the rest of the students are girls.
> 70 chose Spanish.
> 60 of the 104 students who chose French are boys.
> 18 girls chose German.
> Work out how many boys chose Spanish.

Why This Question Type Matters

This isn't just about finding one number—it's about organising data logically. These skills are essential for:

Probability questions (finding probabilities of combined events)

Statistics problems

Real-world data analysis

• Many problem-solving contexts

Step 1: Understanding What We Know

Let's extract the key facts:

Total students: 200

Gender breakdown: 90 boys, so girls = 200 - 90 = 110 girls

Language totals: Spanish = 70 students (we don't know boys/girls split yet)

French specifics: Total French = 104 students. Of these, 60 are boys, so French girls = 104 - 60 = 44

German specifics: 18 girls chose German

1. Important constraint: Every student chooses exactly one language (French, Spanish, or German)

Step 2: Choosing Our Method – The Two-Way Table

While you might think of a Venn diagram first, for this type of problem (with two categories: gender and language), a two-way table is often clearer. Here's why:

It visually organises all possibilities

It helps spot missing values

• It prevents double-counting or missing students

Here's our table structure:

Step 3: Filling In What We Know

Let's add the information we have directly:

How we got those numbers:

Boys French: Given as 60

Girls French: 104 total French - 60 boys = 44 girls

Girls German: Given as 18

• Girl total: 200 total - 90 boys = 110 girls

Step 4: Finding Missing Values

Now we can use the totals to find missing pieces. Remember: rows must add across, columns must add down.

First, let's find total German students:

We know total students = 200
Total = French + Spanish + German
200 = 104 + 70 + German
200 = 174 + German
German = 200 - 174 = 26

Update our table:

Now find Boys German:

Total German = 26
Girls German = 18
Boys German = 26 - 18 = 8

Update:

Find Boys Spanish using the boys row:

Boys total = 90
Boys French = 60
Boys German = 8
Boys Spanish = 90 - 60 - 8 = 22

Wait! We could stop here—we've found our answer! But let's complete the table to check our work.

Step 5: Completing and Checking the Table

Find Girls Spanish:

Total Spanish = 70
Boys Spanish = 22
Girls Spanish = 70 - 22 = 48

Final check with girls row:

Girls total should be: French (44) + Spanish (48) + German (18) = 44 + 48 + 18 = 110 ✓

Complete table:

flowchart TD
    A["Total Students: 200"] --> B["Boys: 90"]
    A --> C["Girls: 110"]
    
    B --> D["French Boys: 60"]
    B --> E["Spanish Boys: 22"]
    B --> F["German Boys: 8"]
    
    C --> G["French Girls: 44"]
    C --> H["Spanish Girls: 48"]
    C --> I["German Girls: 18"]
    
    D --> J["Total French: 104"]
    G --> J
    E --> K["Total Spanish: 70"]
    H --> K
    F --> L["Total German: 26"]
    I --> L

Step 6: Answer and Presentation

Answer: 22 boys chose Spanish.

In the exam, you might present this as:
> Boys choosing Spanish = 90 (total boys) - 60 (boys French) - 8 (boys German) = 22

Or show your table with workings.

Common Mistakes to Avoid

Forgetting the constraint: Each student chooses exactly one language—no one chooses two or none.

Misunderstanding "of the": "60 of the 104 students who chose French are boys" means there are 104 French students total, and 60 of them are boys.

Not using totals to check: Always check row and column totals match.

1. Jumping to conclusions: Don't assume equal distributions—calculate carefully.

Why This Method Works

The two-way table approach:

Organises information visually

Reveals relationships between categories

Provides checks through row/column totals

1. Minimises errors from mental calculations

Practice This Technique

Try similar problems by:

Drawing the table framework first

Filling in what you know directly

Using totals to find missing values

1. Checking all rows and columns add up correctly

Exam Tips

Show your working: Even if you make a calculation error, method marks could save you.

Label clearly: Write "Boys Spanish = " before your answer.

Check feasibility: Does your answer make sense? (22 is less than 70 total Spanish and less than 90 total boys—so it's plausible.)

• Time management: This is a 3-4 mark question—don't spend 15 minutes on it!

Final answer: 22 boys chose Spanish.

Remember: Systematic organisation is key to solving complex-looking problems. The two-way table turns a confusing word problem into a clear, step-by-step puzzle. Master this technique, and you'll be ready for any similar question that comes your way in the GCSE maths exam!

• Need more practice? Look for questions involving "two-way tables," "contingency tables," or "organising data" in your revision guide.