In a survey, one-third of the number of children like only mango and 22 do not like mango at all. Also, two-fifth of the number of children like orange but 12 like none of them. (i) Draw a Venn-diagram to show the above information. (ii) Find the total number of children in the survey. (iii) Find the number of children who like only one fruit. (iv) By how many more or less is the number of children who do not like orange at all than the number of children who do not like mango at all.
(i) Let M represent children who like mango, O represent children who like orange, and N represent children who like neither mango nor orange.
From the information given:
- One-third of the children like only mango; this is shown as M - (M ∩ O) = 1/3 Total Children
- 22 children do not like mango at all; this is shown as Total children - M = 22
- Two-fifths of the children like orange, but 12 children like none of them; this is shown as O - (M ∩ O) = 2/5 Total Children
Therefore, we can represent the relationships in the Venn diagram as follows:
A: Children who like only mango (M - (M ∩ O))
B: Children who like only orange (O - (M ∩ O))
C: Children who like both mango and orange (M ∩ O)
D: Children who like neither mango nor orange (N)
(ii) Using the given information, we can solve for the total number of children:
From M - (M ∩ O) = 1/3 Total Children,
M = 1/3 Total Children + M ∩ O
From Total children - M = 22,
Total children = 22 + M
From O - (M ∩ O) = 2/5 Total Children,
O = 2/5 Total Children + M ∩ O
Using the information from 22 children do not like mango at all, we can substitute M in terms of the total children:
Total children = 22 + 1/3 Total Children + M ∩ O
Total children = 22 + 1/3 Total Children + O - M ∩ O
Total children = 22 + 1/3 Total Children + 2/5 Total Children + M ∩ O - M ∩ O
Total children = 22 + 1/3 Total Children + 2/5 Total Children
1/3 Total Children - 2/5 Total Children = -22
5/15 Total Children - 6/15 Total Children = -22
-1/15 Total Children = -22
Total Children = 15 * 22 = 330
Therefore, the total number of children in the survey is 330.
(iii) The number of children who like only one fruit can be found by adding A and B:
A + B = 1/3 Total Children + 2/5 Total Children + M ∩ O
A + B = 1/3 * 330 + 2/5 * 330 + C
A + B = 110 + 132 + C
A + B = 242 + C
A + B = 242
Therefore, the number of children who like only one fruit is 242.
(iv) To determine the difference between the number of children who do not like orange at all and the number of children who do not like mango at all, we first need to calculate the number of children who do not like orange at all and the number of children who do not like mango at all:
Children who do not like orange at all = Total children - O = 330 - O
Children who do not like mango at all = 22 children
So, the difference is:
Children who do not like orange at all - Children who do not like mango at all
= (330 - O) - 22
= 330 - (2/5 * 330 + M ∩ O) - 22
= 330 - 132 - M ∩ O - 22
= 330 - 132 - C - 22
= 176 - C
Therefore, the number of children who do not like orange at all is more than the number of children who do not like mango at all by 176 - C.