Every Wednesday is salad bar day at Ashley's school. The most popular ingredients are cherry tomatoes, cucumber slices, and shredded carrot. Last Wednesday, out of the 27 students in Ashley's class-
14 took carrot
13 took cucumber
16 took tomato
8 took cucumber and tomato
9 took tomato and carrot
7 took carrot and cucumber
and 5 put all three on their salads.
How many students took none of these?
How many students did not take carrot?
Use a Venn diagram, put in all the data values, and your answers can just be read from the diagram
To answer these questions, we can use the principle of inclusion-exclusion. Let's break it down step by step:
1. Find the total number of students who took any of the ingredients:
- The number of students who took carrot = 14.
- The number of students who took cucumber = 13.
- The number of students who took tomato = 16.
To find the total number of students who took any of the ingredients, we need to add these individual counts together: 14 + 13 + 16 = 43.
2. Find the number of students who took two of the ingredients:
- The number of students who took cucumber and tomato = 8.
- The number of students who took tomato and carrot = 9.
- The number of students who took carrot and cucumber = 7.
To find the total number of students who took two of the ingredients, we need to add these individual counts together: 8 + 9 + 7 = 24.
3. Find the number of students who took all three ingredients:
- We are given that 5 students put all three ingredients on their salads.
4. Apply the principle of inclusion-exclusion:
To find the total number of students who took none of these ingredients, we need to subtract the number of students who took any of the ingredients from the number of students who took two of the ingredients, and then add back the number of students who took all three ingredients:
Total number of students who took none of these = Total number of students - Total number of students who took any of the ingredients + Total number of students who took two of the ingredients - Total number of students who took all three ingredients.
Total number of students = 27 (given).
Total number of students who took any of the ingredients = 43 (from step 1).
Total number of students who took two of the ingredients = 24 (from step 2).
Total number of students who took all three ingredients = 5 (given).
Now, let's substitute these values into the formula:
Total number of students who took none of these = 27 - 43 + 24 - 5 = 3.
Therefore, 3 students took none of these ingredients.
To find the number of students who did not take carrot, we can subtract the number of students who took carrot (14) from the total number of students (27):
Number of students who did not take carrot = Total number of students - Number of students who took carrot = 27 - 14 = 13.
Therefore, 13 students did not take carrot.