GCSE Maths Practice: venn-diagrams

Venn Diagrams and “OR” Probability

Venn diagrams are a very common GCSE Maths method for organising information about two groups that overlap. In this question, the groups are students who like History and students who like Geography. Some students like both subjects, so the two groups are not separate. If you simply add 30 and 25, you count the students who like both subjects twice, which gives an incorrect total. A Venn diagram helps you avoid this error by splitting the totals into clear regions.

How to Fill the Venn Diagram

Draw a rectangle to represent the whole class of 50 students. Inside the rectangle, draw two overlapping circles. Label one circle History and the other Geography.

Step 1: Put the overlap in first. The question says that 10 students like both subjects. Write 10 in the overlapping region of the circles.

Step 2: Find the “History only” region. 30 students like history in total, but 10 of these are already included in the overlap. Subtract the overlap:

History only = 30 − 10 = 20.

Write 20 in the history-only part of the diagram.

Step 3: Find the “Geography only” region. 25 students like geography in total, and 10 of them are in the overlap. Subtract again:

Geography only = 25 − 10 = 15.

Write 15 in the geography-only region.

Finding “History or Geography”

The probability question asks for students who like history or geography. In GCSE Maths, “or” means: history only, geography only, or both. That is everything inside the circles. Add the regions:

20 + 10 + 15 = 45.

So 45 students like at least one of the subjects.

Turning the Total into a Probability

Probability is calculated as:

Probability = favourable outcomes ÷ total outcomes

Here, favourable outcomes = 45 (students who like history or geography) and the total outcomes = 50 (students in the class). So:

\(\frac{45}{50}\)

GCSE answers should be simplified. Divide the top and bottom by 5:

\(\frac{45}{50} = \frac{9}{10}\)

Worked Examples

Example 1: In a group of 60 students, 32 like football, 28 like basketball, and 10 like both. Football only = 32 − 10 = 22. Basketball only = 28 − 10 = 18. Football or basketball = 22 + 10 + 18 = 50. Probability = \(\frac{50}{60} = \frac{5}{6}\).

Example 2: In a class of 40 students, 18 like art, 16 like music, and 6 like both. Art only = 18 − 6 = 12. Music only = 16 − 6 = 10. Art or music = 12 + 6 + 10 = 28. Probability = \(\frac{28}{40} = \frac{7}{10}\).

Common Mistakes

• Double counting: adding 30 + 25 and forgetting to subtract the 10 in the overlap.
• Not splitting totals: writing 30 in the history-only region instead of calculating history-only as 20.
• Not simplifying: leaving \(\frac{45}{50}\) instead of simplifying to \(\frac{9}{10}\).

Real-Life Applications

Overlapping groups appear in real surveys all the time. For example, a school may ask which subjects students enjoy, or a streaming platform might ask which genres users watch. Venn diagrams help organise this information clearly and avoid counting the same person twice in analysis.

FAQ

What does “or” mean in a Venn diagram?
It means everything in either circle, including the overlap.

Do I need the number who like neither subject?
Not for this question, but it can be used as a check. Inside circles = 45, so neither = 50 − 45 = 5.

What should I always write first?
Always write the overlap (both) first, then subtract to find the “only” regions.

Study Tip

If a question gives two totals and a “both”, sketch a Venn diagram immediately. It makes “OR” probability questions much easier and prevents the most common mistakes.