Venn diagram problem

Question

62 students were surveyed on the fruits they liked;

34 liked apples

30 liked bananas

33 liked strawberries

11 liked apples &bananas

15 liked bananas &strawberries

17 liked apples &strawberries

19 liked exactly two of the following fruits; apples, bananas, and strawberries

A. How many students liked all of the three fruits?

B. How many liked apples only?

Answer 1

For part (a), we are told that 19 students liked exactly two fruits so, we

 

Let x equal to the number of students who liked apples and bananas ONLY

Let y equal to the number of students who liked bananas and strawberries ONLY

Let z equal to the number of students who liked apples and strawberries ONLY

 

that way, the equation would be:

   x + y + z = 19

 

 

Now we let c equal to the number of students who liked all three fruits

 

Given that
- 11 liked apples and bananas means...
   x + c = 11
- 15 liked bananas and strawberries means...
   y + c = 15
- 17 liked apples and strawberries means...
   z + c = 17

 

Now solve for x, y, z

x = 11 - c

y = 15 - c

z = 17 - c

 

Then, substitute it in the first equation and solve for c

x + y + z = 19

(11 - c) + (15 - c) + (17 - c) = 19 

-3c = 19 - 11 - 15 - 17

c = -24/-3

c = 8

 

So the number of students who liked all three fruits is 8

 

With that answer, we can now solve for part (b)

 

Let a equal to the number of students who liked apples ONLY

 

Given that 

- 34 liked apples means...

   a + x + z + c = 34

 

Now we know that 

c = 8

x = 11 - c = 3
z = 17 - c = 9

 

So we substitute it into our equation and solve for a

 

a + x + z + c = 34

a + 3 + 9 + 8 = 34

a = 14

 

So the number of students who liked apples only is 14