50 children have a choice of beans, plantain and rice. 21 of them took beans, 24 took plantains and 18 took rice, 3 took beans only, 9 took plantains only, 2 took rice only and 5 took all three items of food. Illustrate it in a Venn diagram, find the number of children took plantain only, rice and beans only, non of the three items
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14 took none
14 took none of the food
?? You say
9 took plantains only
find the number of children took plantain only
well, duh -- 9
So, did you fill in your Venn diagram?
If we let
x = # who took beans and plantains only
y = # who took plantains and rice only
z = # who took beans and rice only
then we have
x+z+5+3 = 21
x+y+5+9 = 24
y+z+5+2 = 18
now solve for x,y,z
18 took none of these
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I need the answer.
Let's start by drawing a Venn diagram representing the given information:
```
B
/ \
/ \
/ \
P-------R
\ /
\ /
\ /
A
```
where B, P and R represent beans, plantain and rice respectively and A represents the region outside the three sets (or those who didn't take any of the three items).
From the given information, we can fill in parts of the diagram:
- 3 took beans only, so we write 3 in the region of B that doesn't overlap with P or R.
- 9 took plantains only, so we write 9 in the region of P that doesn't overlap with B or R.
- 2 took rice only, so we write 2 in the region of R that doesn't overlap with B or P.
- 5 took all three items of food, so we write 5 in the region where all three sets overlap.
```
B
/ \
/ \
/ \
P--5--R
\ /
\ /
\ /
A
```
Now we can use the remaining information to fill in the rest of the diagram. Let x be the number of children who took beans and plantains only, y be the number who took plantains and rice only, and z be the number who took beans and rice only.
- 21 took beans, which includes the 3 who took beans only, and x who took both beans and plantains, and z who took both beans and rice. So we have:
```
B
/|\
/ | \
/ | \
P--x--R
\ | /
\| /
\ /
A
```
- 24 took plantains, which includes the 9 who took plantains only, and x who took both beans and plantains, and y who took both plantains and rice. So we have:
```
B
/|\
/ | \
/ | \
P--x--R
\ | / \
\|/ \
\y----A
```
- 18 took rice, which includes the 2 who took rice only, and z who took both beans and rice, and y who took both plantains and rice. So we have:
```
B
/|\
/ | \
/ | \
P--x--R
\ | / \
\|/ \
\y----A
/ \
/ \
/ \
z-------A
```
Now we can find the values of x, y and z:
- The total number of children who took beans is 21 (including x and z). Subtracting the 3 who took beans only and the 5 who took all three items gives us:
```
21 - 3 - 5 = 13
```
So we have x + z + 13 = 21, or x + z = 8.
- The total number of children who took plantains is 24 (including x and y). Subtracting the 9 who took plantains only and the 5 who took all three items gives us:
```
24 - 9 - 5 = 10
```
So we have x + y + 10 = 24, or x + y = 14.
- The total number of children who took rice is 18 (including y and z). Subtracting the 2 who took rice only and the 5 who took all three items gives us:
```
18 - 2 - 5 = 11
```
So we have y + z + 11 = 18, or y + z = 7.
Now we can fill in the values of x, y and z in the Venn diagram:
```
B
/|\
/ | \
/ | \
P--3--R
\ | / \
\|/ \
\y----A
/ \
/ \
/ \
z--2----A
```
Finally, we can answer the questions:
- The number of children who took plantain only is 9.
- The number of children who took rice and beans only is 0 (since we don't have any value for x).
- The number of children who took none of the three items is A, which is the region outside the three sets. This includes 14 children (since the total number of children is 50 and we've accounted for 3+9+2+5+13+3+10+2+5+3+9+2+7=61).