In this guide
Venn diagrams use circles to show how groups are related. In RRB Group D they appear in two forms. In the picture type, you are given three words (such as mothers, women and doctors) and must pick the diagram that shows how they relate. In the number type, you are told how many people like cricket, football or both, and must find how many like only one, or neither.
Both types are quick once you know the patterns. The picture type is really a question about the real world: can one thing also be the other? The number type needs one formula and a habit of filling the overlap first.
Part 1: choosing the right diagram
For each pair of words, ask one question: can something be both?
- If every A is a B, draw A inside B.
- If some A can be B, draw A and B overlapping.
- If nothing can be both, draw them apart.
Answer that for all three pairs, and the diagram follows.
The common patterns
| Relationship | Picture | Example |
|---|---|---|
| One inside another, inside a third | Three circles nested inside each other | Mumbai, Maharashtra, India |
| Two separate groups inside a bigger one | Two separate circles inside a large circle | Dogs, Cats, Animals |
| Two overlapping groups inside a bigger one | Two overlapping circles inside a large circle | Doctors, Engineers, Humans |
| One inside another, a third overlapping both | Small circle inside a big one, a third circle crossing both | Mothers, Women, Doctors |
| A third group overlapping two separate groups | Two separate circles, a third circle crossing both | Men, Women, Teachers |
| All three overlapping | Three overlapping circles | Singers, Dancers, Students |
| All three separate | Three separate circles | Chairs, Rivers, Cars |
Worked example 1: mothers, women, doctors
- Mothers and women: every mother is a woman, so mothers inside women.
- Women and doctors: some women are doctors, so they overlap.
- Mothers and doctors: some mothers are doctors, so they overlap.
- Diagram: a small "mothers" circle inside a "women" circle, with a "doctors" circle crossing both.
Worked example 2: doctors, engineers, humans
- Every doctor and every engineer is a human, so both circles sit inside humans.
- A person can be both a doctor and an engineer, so the two inner circles overlap.
- Compare this with dogs, cats and animals, where the inner circles stay apart, because nothing is both a dog and a cat.
Part 2: number problems with two groups
For two groups A and B, let "both" be the number in the overlap.
| You want | Formula |
|---|---|
| At least one (A or B or both) | A + B − both |
| Only A | A − both |
| Only B | B − both |
| Exactly one (only A or only B) | A + B − 2 × both |
| Neither | Total − (A + B − both) |
Why subtract "both"? When you add A and B, the people in the overlap are counted twice, once in each group.
Worked example 3
In a class of 100 students, 60 like cricket, 45 like football and 20 like both.
- Overlap: 20.
- Only cricket: 60 − 20 = 40. Only football: 45 − 20 = 25.
- At least one: 40 + 25 + 20 = 85 (or 60 + 45 − 20 = 85).
- Neither: 100 − 85 = 15.
- Check: 40 + 25 + 20 + 15 = 100.
Worked example 4: finding "both"
Of 50 students, 30 study Hindi and 28 study English. Every student studies at least one of the two. How many study both?
- At least one = 50, because nobody studies neither.
- 50 = 30 + 28 − both, so both = 58 − 50 = 8.
Part 3: reading a three-circle diagram
Sometimes you are given a three-circle diagram with a number in every region and asked to read it. Suppose the regions of circles A, B and C hold these numbers:
| Region | Number |
|---|---|
| Only A | 10 |
| Only B | 8 |
| Only C | 12 |
| A and B only | 4 |
| B and C only | 5 |
| A and C only | 3 |
| All three | 2 |
Worked example 5
- Exactly two groups: 4 + 5 + 3 = 12.
- Everyone in A: 10 + 4 + 3 + 2 = 19.
- At least one group: add every region: 10 + 8 + 12 + 4 + 5 + 3 + 2 = 44.
Common mistakes
- Adding the two groups without taking away the overlap.
- Drawing separate circles when the groups can share members, such as doctors and engineers.
- Forgetting the "neither" part outside all circles.
- In three-circle questions, counting "A and B" as only the "A and B only" region when the question wants everyone in both A and B (which also includes the "all three" region).
Practice set
- Which diagram shows Fruits, Mangoes and Vegetables?
- Which diagram shows Teachers, Men and Women?
- Of 70 people, 40 drink tea, 35 drink coffee and 15 drink both. How many drink neither?
- In question 3, how many drink only tea?
- Which diagram shows Kerala, Punjab and India?
- Of 60 people, each speaks Hindi, Marathi or both. 45 speak Hindi and 30 speak Marathi. How many speak both, and how many speak only Marathi?
- Using the three-circle table above, how many are in A but not in C?
- Which diagram shows Students, Players and Boys?
Answers:
- Mangoes inside Fruits, with Vegetables separate. Every mango is a fruit, and here fruits and vegetables are treated as separate groups.
- Two separate circles (Men, Women) with a Teachers circle crossing both. Some men and some women are teachers.
- 10. At least one = 40 + 35 − 15 = 60, so neither = 70 − 60.
- 25. 40 − 15.
- Two separate circles (Kerala, Punjab) inside India. Both states are in India, and no place is in both.
- Both = 15, only Marathi = 15. 45 + 30 − 60 = 15 speak both, and 30 − 15 = 15 speak only Marathi.
- 14. Only A (10) plus A and B only (4). The regions shared with C are left out.
- Three overlapping circles. A student can be a player and a boy, and each pair can overlap.
What to do next
- Learn the pattern table, with one example of your own for each row.
- For every picture question, ask "can something be both?" for all three pairs before looking at the options.
- Solve 10 number problems and 10 picture problems this week.
- Revise syllogism, which uses the same circles, and the reasoning plan for how to fit this topic into your week.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Railway Recruitment Boards website .
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