(a) In a market survey, 100 traders sell fruits, 40 sell apples, 46 oranges, 50 mangoes, 14 apples and oranges, 15 apples and mangoes and 10 sell the three ...
Question 1 Report
(a) In a market survey, 100 traders sell fruits, 40 sell apples, 46 oranges, 50 mangoes, 14 apples and oranges, 15 apples and mangoes and 10 sell the three types of fruits. Each of the 100 traders sells at least one of the three fruits.
(i) Represent the information in a Venn diagram ; (ii) Find the number that sell oranges and mangoes only.
(b) Find the value of x for which \(312_{four} + 52_{x} = 96_{ten}\)
Answer Details
(a) Apples \(A=40\), Oranges \(O=46\), Mangoes \(M=50\). \(A\cap O=14\), \(A\cap M=15\), all three \(=10\); every trader sells at least one, total \(=100\).
(i) Venn diagram. Three intersecting circles; centre (all three) \(=10\).
(ii) Let \(O\cap M\) (total) \(=y\). By the inclusion-exclusion principle:
\(|A|+|O|+|M|-|A\cap O|-|A\cap M|-|O\cap M|+|A\cap O\cap M|=100\).
\(40+46+50-14-15-y+10=100 \Rightarrow 117-y=100 \Rightarrow y=17\).
Oranges and mangoes only \(=y-10=17-10=\mathbf{7}\).
(b) \(312_{four}+52_x=96_{ten}\).
\(312_{four}=3(16)+1(4)+2=54\). So \(52_x=96-54=42\).
\(52_x=5x+2=42 \Rightarrow 5x=40 \Rightarrow x=\mathbf{8}\).
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