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Venn Diagram Questions with Solutions

Venn Diagram questions here are about counting overlapping groups correctly — the core idea is "inclusion-exclusion": add the groups, then subtract whatever got counted twice. Here’s the concept, four fully worked examples, and a few to try yourself.

The concept, quickly

  • If some people belong to group A, some to group B, and some to BOTH, the number who belong to A or B is A + B − (both) — not simply A + B, which double-counts the overlap.
  • To find how many belong to EXACTLY one group (not both), subtract the "both" count from each group’s total: only A = A − both, only B = B − both.
  • For three overlapping groups, every pairwise "both" count given usually includes people in all three — the formula is: total in at least one = A + B + C − (A∩B) − (B∩C) − (A∩C) + (A∩B∩C), adding the all-three group back once since it got subtracted three times in the pairwise terms.
  • "Neither" (people in none of the groups) = Total people − (number in at least one group).

Worked examples

1. In a class of 60 students, 35 like Mathematics, 30 like Science, and 15 like both subjects. How many students like at least one of the two subjects? How many like neither?

Solution

  1. At least one = Mathematics + Science − Both = 35 + 30 − 15 = 50.
  2. Neither = Total − At least one = 60 − 50 = 10.

Answer: At least one = 50, Neither = 10

2. In a survey of 100 people, 60 read Newspaper A, 40 read Newspaper B, and 25 read both. How many read only Newspaper A? How many read only Newspaper B?

Solution

  1. Only A = A − Both = 60 − 25 = 35.
  2. Only B = B − Both = 40 − 25 = 15.

Answer: Only A = 35, Only B = 15

3. In a group of 50 people, 20 speak Hindi, 25 speak English, and 10 speak both languages. How many speak neither Hindi nor English?

Solution

  1. At least one = 20 + 25 − 10 = 35.
  2. Neither = 50 − 35 = 15.

Answer: 15

4. In a class of 100 students, 50 play Cricket, 40 play Football, 30 play Hockey. 15 play both Cricket and Football, 12 play both Football and Hockey, 10 play both Cricket and Hockey, and 5 play all three games. How many students play at least one of the three games?

Solution

  1. Apply the three-set formula: 50 + 40 + 30 − 15 − 12 − 10 + 5.
  2. = 120 − 37 + 5 = 88.

Answer: 88

Try these yourself

Apply inclusion-exclusion the same way, then check against the answer.

1. In a class of 50 students, 28 like Painting, 22 like Dancing, and 10 like both. How many like at least one activity? How many like neither?

Answer: At least one = 40, Neither = 10

2. In a survey of 80 people, 45 drink Tea, 35 drink Coffee, and 15 drink both. How many drink only Tea? How many drink only Coffee?

Answer: Only Tea = 30, Only Coffee = 20

3. In a group of 150 people, 70 like Cricket, 60 like Football, 50 like Tennis. 20 like both Cricket and Football, 15 like both Football and Tennis, 18 like both Cricket and Tennis, and 8 like all three sports. How many like at least one of the three sports?

Answer: 135

Where this comes up

Venn Diagram is a regular topic in the Reasoning/General Intelligence section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk.

Practice more Venn Diagram questions

This page covers the concept and a handful of worked examples. Pariksha Saathi has full topic-wise practice sets for Venn Diagram (and every other Reasoning/Quant/English topic) with instant scoring and explanations — free, no account required.

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