The 90 men that visited restaurant take one of either pounded yam, rice or Amala. 10 take pounded yam and rice 19 take rice and Amala and 29 take pounded yam and Amala, x men take all the three menu. 2x men each take one food item. How many men take all three food items?
To find the number of men who take all three food items, we can use a Venn diagram to visually represent the given information. Let's break down the information step-by-step:
Step 1: Let's define the following:
P = number of men who take pounded yam
R = number of men who take rice
A = number of men who take Amala
x = number of men who take all three menu items
Step 2: Based on the information given, we can create the following equations:
P + R + A - (10 + 19 + 29) = 90 (Total number of men who visited the restaurant)
P + R + A - 58 = 90
Step 3: We know that 2x men each take one food item. This means that the total number of men who take only one food item is 2x * 3 = 6x.
Step 4: Using the Venn diagram, we can represent the given information and solve for x.
Let's represent the Venn diagram:
P R A
/ \ / \ /|
/___\ /____\ / |
10 19 |
\ / 29
________
Step 5: From the Venn diagram, we can see that:
P + R + 29 = 10 + 19 + x
P + A + 10 = 19 + 29 + x
R + A + 19 = 10 + 29 + x
Step 6: Simplify the equations:
P + R + x = 58 - 29 = 29
P + A + x = 58 - 10 = 48
R + A + x = 58 - 19 = 39
Step 7: Substitute the values of P + R + x, P + A + x, and R + A + x into the equation from Step 4:
29 + 48 + 39 - 6x = 90
116 - 6x = 90
-6x = -26
x = 26 / 6
x = 4.33
Since we are dealing with the number of men, we can't have a fraction of a man. Therefore, we can conclude that x = 4.
So, 4 men take all three food items.
To find the number of men who took all three food items, we can use a set theory approach. Let's break down the information given:
Let P represent the set of men who took pounded yam.
Let R represent the set of men who took rice.
Let A represent the set of men who took Amala.
We are given the following information:
- The total number of men who visited the restaurant is 90.
- 10 men took both pounded yam and rice. This means there are 10 men in the intersection of sets P and R: |P ∩ R| = 10.
- 19 men took both rice and Amala. This means there are 19 men in the intersection of sets R and A: |R ∩ A| = 19.
- 29 men took both pounded yam and Amala. This means there are 29 men in the intersection of sets P and A: |P ∩ A| = 29.
- The number of men who took only one food item is 2x. This means there are 2x men in the union of sets P, R, and A, excluding those in the intersections.
To find x, the number of men who took all three food items, we need to find the number of men in the intersection of sets P, R, and A: |P ∩ R ∩ A|. Since we don't have this information directly, we need to derive it based on the given information.
Using the principle of inclusion-exclusion, we can calculate |P ∪ R ∪ A|, the number of men who took at least one of the food items.
|P ∪ R ∪ A| = |P| + |R| + |A| - |P ∩ R| - |P ∩ A| - |R ∩ A| + |P ∩ R ∩ A|
Given:
|P ∪ R ∪ A| = 90
|P ∩ R| = 10
|P ∩ A| = 29
|R ∩ A| = 19
Now we can substitute these values into the equation above and solve for |P ∩ R ∩ A|:
90 = |P| + |R| + |A| - 10 - 29 - 19 + |P ∩ R ∩ A|
90 = |P| + |R| + |A| - 58 + |P ∩ R ∩ A|
|P| + |R| + |A| + |P ∩ R ∩ A| = 148
We are given that the number of men who took only one food item is 2x. Since there are three food items (pounded yam, rice, and Amala), the number of men who took only one food item is:
2x = |P - (R ∪ A)| + |R - (P ∪ A)| + |A - (P ∪ R)|
= |P| - |P ∩ R| - |P ∩ A| + |P ∩ R ∩ A| + |R| - |P ∩ R| - |R ∩ A| + |P ∩ R ∩ A| + |A| - |P ∩ A| - |R ∩ A| + |P ∩ R ∩ A|
= 2|P ∩ R ∩ A|
We know that |P ∩ R ∩ A| = x, so we can rewrite the equation as:
2x = 2x
This implies that x can have any value, as long as x is a non-negative integer.
Therefore, the number of men who took all three food items (pounded yam, rice, and Amala) can be any non-negative integer.
(10 + 19 + 29) take two items ... x take three ... 2 x take one
(10 + 19 + 29) + x + 2x = 90
something wrong with the numbers