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ABHYAS PREP: REASONING PUZZLES · VENN DIAGRAMS

Venn diagram counting (two and three sets)

Fill the Venn diagram from the inside out: the all-groups overlap first, then the two-group overlaps, then the “only” regions. Union = sum of singles − sum of pairs + triple.

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Worked examples

Example 1 · level 1

Of 67 shoppers at a mela, 33 bought bangles, 25 bought sweets and 8 bought both. How many bought only bangles?

  1. 17
  2. 25
  3. 33
  4. 42
Show solution
  1. Only bangles = 33 − 8 = 25; only sweets = 25 − 8 = 17.
  2. At least one = 33 + 25 − 8 = 50; neither = 67 − 50 = 17.
  3. Answer: 25.

Answer: 25

Example 2 · level 2

Of 67 shoppers at a mela, 39 bought toys, 32 bought bangles and 6 bought neither. How many bought only toys?

  1. 22
  2. 29
  3. 35
  4. 39
Show solution
  1. At least one = 67 − 6 = 61.
  2. Both = 39 + 32 − 61 = 10.
  3. Only toys = 39 − 10 = 29; only bangles = 32 − 10 = 22.
  4. Answer: 29.

Answer: 29

Example 3 · level 3

Of 68 employees in an office, 24 speak Telugu, 34 speak Kannada and 23 speak Tamil; 10 speak both Telugu and Kannada, 11 speak both Kannada and Tamil, 12 speak both Telugu and Tamil, and 7 speak all three. How many speak none of the three?

  1. 13
  2. 20
  3. 27
  4. 55
Show solution
  1. All three: 7. Exactly two: Telugu–Kannada 10 − 7 = 3, Kannada–Tamil 11 − 7 = 4, Telugu–Tamil 12 − 7 = 5.
  2. Only Telugu = 24 − 3 − 5 − 7 = 9; only Kannada = 20; only Tamil = 7.
  3. At least one = 81 − 33 + 7 = 55; none = 68 − 55 = 13.
  4. Answer: 13.

Answer: 13

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