A farm keeps 90 animals. Set \(C\) is the animals kept in the north field and set \(D\) is the animals that are milked. Each region of the Venn diagram is w...
Question 1 Report
A farm keeps 90 animals. Set \(C\) is the animals kept in the north field and set \(D\) is the animals that are milked. Each region of the Venn diagram is written in terms of \(y\).
- Write an expression, in terms of \(y\), for the total number of animals. (1)
- Find the value of \(y\). (2)
- Find \(n(C \cap D')\). (1)
- Find \(n((C \cup D)')\). (1)
Answer Details
The four regions of the diagram do not overlap and between them account for every animal, so their expressions add to the total.
(a)
\[2y + (y - 4) + (y + 6) + 12\]Collecting the \(y\) terms gives \(2y + y + y = 4y\), and the constants give \(-4 + 6 + 12 = 14\):
\[= 4y + 14\][1]
The \(-4\) must be subtracted, not added; treating it as \(+4\) is the usual slip.
(b) The farm keeps \(90\) animals, so
\[4y + 14 = 90 \implies 4y = 76\][1]
\[y = 19\][1]
(c) \(C \cap D'\) means in \(C\) but not in \(D\), which is the part of the \(C\) circle outside the overlap:
\[n(C \cap D') = 2y = 38\][1]
(d) \((C \cup D)'\) is everything outside the union, so it is the region beyond both circles:
\[n((C \cup D)') = 12\][1]
Substituting \(y = 19\) checks the whole diagram: the regions hold \(38\), \(15\), \(25\) and \(12\) animals, totalling \(90\).
Note that \((C \cup D)'\) and \(C' \cap D'\) describe the same region, a result known as one of De Morgan's laws: being outside both circles is the same as being outside their union.
Download The App On Google Playstore
Everything you need to excel in your exams
