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. 4 0 people eat butterscotch and vanila only, 10 people eat chocolate and vanila only.

People eating chocolate and butterscotch only are what percent of people eating only butterscotch?

The correct answer is 50% (Option 1).

Let us break down the given information using a Venn diagram approach for the three ice cream flavours: Butterscotch (B), Vanilla (V), and Chocolate (C).

From the question, we are given the following values for the different regions:
1. Total number of persons at the party = 210
2. People who eat only Butterscotch, n(Only B) = 40
3. People who eat all three flavours, n(B ∩ V ∩ C) = 30
4. People who eat only Butterscotch and Vanilla, n(B ∩ V only) = 40
5. People who eat only Chocolate and Vanilla, n(C ∩ V only) = 10
6. Total number of people who eat Butterscotch, n(B) = 130
7. Total number of people who eat Vanilla, n(V) = 100

The total group of people who eat Butterscotch is made up of four distinct regions:
- People who eat only Butterscotch
- People who eat only Butterscotch and Vanilla
- People who eat only Butterscotch and Chocolate
- People who eat all three flavours

We can express the total number of people who eat Butterscotch as:

$n\left(B\right)=n\left(\text{Only B}\right)+n\left(\text{B and V only}\right)+n\left(\text{B and C only}\right)+n\left(\text{All 3}\right)$

Substitute the known values into the equation:

$130=40+40+n\left(\text{B and C only}\right)+30$

Simplify the right side:

$130=110+n\left(\text{B and C only}\right)$

Subtract 110 from both sides to find the number of people who eat only Chocolate and Butterscotch:

$n\left(\text{B and C only}\right)=130-110=20$

So, 20 people eat only Chocolate and Butterscotch.

Now, we need to calculate what percent the number of people eating only Chocolate and Butterscotch (20) is of the number of people eating only Butterscotch (40).

$\text{Percentage}=\left(\frac{n\left(\text{B and C only}\right)}{n\left(\text{Only B}\right)}\right)\times 100$

Substitute the values into the formula:

$\text{Percentage}=\left(\frac{20}{40}\right)\times 100=0.5\times 100=50\%$

Thus, the people eating Chocolate and Butterscotch only are 50% of the people eating only Butterscotch.