In a society of 450 people, there are three types of foods: X, Y, and Z. The number of people who like only food X, only food Y, and only food Z are in the ratio 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 and food Z combined and those who like only food X and food Y combined is 50. The number of people who like only food X and food Z is 30.
What is the sum of the number of people who like only food Z and only food X?

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**Solution:**

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

2. Total people = 450, so:
   $$5k + 7k + 3k + 	ext{(people who like exactly two foods)} + 	ext{(people who like all three foods)} = 450$$

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

4. Sum of people who like only food Y & food Z and only food X & food Y is 50. Let:
   - People who like only food Y & food Z = $a$
   - People who like only food X & food Y = $b$
   Then:
   $$a + b = 50$$

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

6. Now, total people:
   $$5k + 7k + 3k + a + b + 30 + 3.5k = 450$$
   $$15k + (a + b) + 30 + 3.5k = 450$$
   $$15k + 50 + 30 + 3.5k = 450$$
   $$18.5k + 80 = 450$$
   $$18.5k = 370$$
   $$k = \frac{370}{18.5} = 20$$

7. Number of people who like only food X = $5k = 5 	imes 20 = 100$

8. Number of people who like only food Z = $3k = 3 	imes 20 = 60$

9. Sum of people who like only food Z and only food X:
   $$100 + 60 = 160$$

**Answer:** 160

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