There are 450 people in a society who like three different types of foods: X, Y, and Z. The ratio of the number of people who like only food X, only food Y, and only food Z is 5 : 7 : 3 respectively. The number of people who like all three foods is 50\% of the number of people who like only food Y. The total number of people who like only food Y & food Z combined with those who like only food X & food Y is 50. The number of people who like only food X & food Z is 30.
What is the difference between the number of people who like only food Y and the number of people who like only food X & food Y?

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Accepted Answer

**Solution:**

1. Let the number of people who like only food X, Y, and Z be $5k$, $7k$, and $3k$ respectively.

2. Total people who like only X, Y, Z = $5k + 7k + 3k = 15k$.

3. People who like all three foods = 50\% of people who like only food Y = $0.5 	imes 7k = 3.5k$.

4. Let the number of people who like only food Y & Z be $a$, and those who like only food X & Y be $b$.

5. Given, $a + b = 50$.

6. People who like only food X & Z = 30.

7. Total people = sum of all categories:
   $$5k + 7k + 3k + a + b + 30 + 3.5k = 450$$
   $$15k + (a + b) + 30 + 3.5k = 450$$
   $$18.5k + 50 + 30 = 450$$
   $$18.5k + 80 = 450$$
   $$18.5k = 370$$
   $$k = \frac{370}{18.5} = 20$$

8. Now, number of people who like only food Y = $7k = 7 	imes 20 = 140$.

9. We need to find the difference between number of people who like only food Y and number of people who like only food X & Y, i.e., $140 - b$.

10. Since $a + b = 50$, and $a$ and $b$ are unknown individually, we cannot find the exact value of $b$.

**Therefore, the difference cannot be determined with the given data.**

**Answer:** Can’t be determined

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