Read the following data carefully and answer the following question. There are 210 persons in a party, and all of them eat three different types of flavoured icecreams. 40 people eat only butterscotch, 30 people eat all three flavoured icecream,there are total 130 people who eat butterscotch and 100 people who eat vanila. 40 people eat butterscotch and vanila only, 10 people eat chocolate and vanila only.

What is the number of people who eat only Chocolate?

Question

Read the following data carefully and answer the following question. There are 210 persons in a party, and all of them eat three different types of flavoured icecreams. 40 people eat only butterscotch, 30 people eat all three flavoured icecream,there are total 130 people who eat butterscotch and 100 people who eat vanila. 40 people eat butterscotch and vanila only, 10 people eat chocolate and vanila only.

What is the number of people who eat only Chocolate?

Accepted Answer

**Solution:**

1. Define the sets:
   - Let *B* = number of people who eat Butterscotch
   - Let *V* = number of people who eat Vanilla
   - Let *C* = number of people who eat Chocolate

2. Given data:
   - Total people, *T* = 210
   - Only Butterscotch = 40
   - All three flavours = 30
   - Total Butterscotch eaters, |B| = 130
   - Total Vanilla eaters, |V| = 100
   - Only Butterscotch and Vanilla = 40
   - Only Chocolate and Vanilla = 10

3. Let the number of people who eat only Chocolate be *x*.

4. Let the number of people who eat Butterscotch and Chocolate only be *y*.

5. Using the total Butterscotch eaters:
   $$|B| = 	ext{Only B} + 	ext{B and V only} + 	ext{B and C only} + 	ext{All three}$$
   $$130 = 40 + 40 + y + 30$$
   $$130 = 110 + y$$
   $$y = 20$$

6. Using the total Vanilla eaters:
   $$|V| = 	ext{Only V} + 	ext{B and V only} + 	ext{C and V only} + 	ext{All three}$$
   We don't know only Vanilla yet, but let's find it later if needed.

7. Total people count:
   Sum of all disjoint groups = Total people

Groups are:
   - Only B = 40
   - Only V = ? (let's call it *v*)
   - Only C = *x* (to find)
   - B and V only = 40
   - B and C only = 20
   - C and V only = 10
   - All three = 30

So,
   $$40 + v + x + 40 + 20 + 10 + 30 = 210$$
   $$140 + v + x = 210$$
   $$v + x = 70$$

8. Using total Vanilla eaters:
   $$100 = v + 40 + 10 + 30 = v + 80$$
   $$v = 20$$

9. Substitute *v = 20* in step 7:
   $$20 + x = 70$$
   $$x = 50$$

**Answer:** Number of people who eat only Chocolate is **50**.

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