Question

50 children had a choice of beans, plantain and rice. 21 of them took beans, 24 took plantain and 18 took rice. 3 took beans only, 9 took plantain only, 2 took rice only and 5 took all three items of food.

a. illustrate this information in a Venn diagram.
b. Use your diagram to find the number of children who took;
Plantain and beans only, rice and beans only and none of the three items
​

Answer 1

Plantain and beans only: 6 kids. Rice and beans only: 7 kids. None: 14 kids

Explanation:

As seen in the image:

First, the "one item only" and "all 3" choices can be separated. Then, to determine the total amount left for the combined choices (beans + plantain, beans + rice, plantain + rice), we subtract the "one item only" and "all 3" choice from the respective choices.

21 total beans choosers. 21 - 3 (bean only choosers) - 5 (all 3 choosers) = 18.

24 total plantain choosers. 24 - 9 (plantain only choosers) - 5 (all 3 choosers) = 10.

18 rice choosers. 18 - 2 (rice only choosers) - 5 (all 3 choosers) = 11.

Now we need to solve a system of equations. This is because we know how many kids chose each food, but we do not know the combinations (ex. 13 leftover bean choosers, how many chose beans and rice and how many chose beans and plantain?).

BEANS + PLANTAIN area = x. PLAINTAIN + RICE area = z. RICE + BEANS area = y.

Looking at the diagram and equations for the combination choosers, we can see that

x + y = 13

y + z = 11

x + z = 10

Isolate x and y so that x = 13 - y and y = 11 - z. Substitute x in the last equation, then y.

13 - y + z = 10. 13 - (11 - z) + z = 10. Solve this new equation.

13 - 11 + z + z = 10. 2 + z + z = 10. 2z + 2 = 10. 2z = 8. z = 4.

Now we can substitute z in the original first two equations. y + 4 = 11. y = 7 x + 4 = 10. x = 6.

Now we can insert these numbers into the Venn Diagram. We can check the amount of kids who chose each food. Beans: 3 + 6 + 5 + 7 = 21. Plantain: 9 + 6 + 5 + 4 = 24. Rice: 2 + 7 + 5 + 4 = 18.

Plantain and Beans only: 6 kids. Rice and Beans only: 7 kids.

Now we can figure out the kids that chose none of the options. Add the amount that chose (the numbers in each section) subtracted from the total kids (50). 3 + 6 + 5 + 7 + 9 + 4 + 2 = 36. 50 - 36 = 14.

None: 14 kids