A workshop manager reviews every booking sheet from the past week. The Venn diagram shows the number of cars in the workshop this week that need brake work,...
Question 1 Report
A workshop manager reviews every booking sheet from the past week. The Venn diagram shows the number of cars in the workshop this week that need brake work, set \(B\), and tyre work, set \(T\).
- Write down \(n(B \cap T)\) (1)
- Find \(n(B \cup T)\) (1)
- Find \(n(\mathcal{E})\), the total number of cars in the workshop this week (1)
Answer Details
The Venn diagram shows two overlapping circles, \(B\) and \(T\), inside a rectangle for the universal set, with a number in each of the four regions: only \(B\), the overlap, only \(T\), and outside both circles.
(a) \(B \cap T\) is the overlap region, which the diagram shows directly: \(n(B \cap T) = 5\). [1 mark]
(b) \(B \cup T\) is every car needing at least one type of work, so it is the sum of the only-\(B\), overlap and only-\(T\) regions, leaving out the region outside both circles: \(n(B \cup T) = 9 + 5 + 7 = 21\). [1 mark]
(c) The total number of cars, \(n(\mathcal{E})\), includes the region outside both circles as well, so all four regions are added: \(n(\mathcal{E}) = 9 + 5 + 7 + 4 = 25\). [1 mark]
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