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Conor carries out a survey on all of the 25 students in his class (U) - Junior Cycle Mathematics - Question 3 - 2016
Question 3
Conor carries out a survey on all of the 25 students in his class (U). He asks each student if they own a pet (P), and if they own a bicycle (Q). 6 students own nei... show full transcript
Worked Solution & Example Answer:Conor carries out a survey on all of the 25 students in his class (U) - Junior Cycle Mathematics - Question 3 - 2016
Step 1
Step 1: Finds value of # (P ∩ Q)
Answer
From the given data, we know:
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Total students surveyed (U) = 25.
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Students owning neither a pet nor a bicycle = 6.
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Therefore, the number of students owning either (P or Q or both) is:
25 - 6 = 19.
28% of these students own both a pet and a bicycle. Hence, we calculate:
0.28 × 19 = 5.32, approximately 5 students own both a pet and a bicycle.
Let’s denote # (P ∩ Q) = 5.
Step 2
Step 2: Splits value in the ratio 2:1
Answer
Let # (P) be the number of students who own a pet and # (Q) be the number of students who own a bicycle. We know:
(P ∩ Q) = 5 and the ratio is given as # (P) : # (Q) = 2 : 1.
Let’s express # (Q) as x. Then, # (P) = 2x.
From the Venn diagram:
- Students owning only pets = # (P) - # (P ∩ Q) = 2x - 5.
- Students owning only bicycles = # (Q) - # (P ∩ Q) = x - 5.
Now, the total equation will be:
(2x - 5) + (x - 5) + 5 + 6 = 25. Thus:
gives us 3x - 5 + 6 = 25,
which leads us to 3x + 1 = 25. Therefore, 3x = 24, and x = 8.
Thus, # (Q) = 8 and # (P) = 16.
Step 3
Step 3: Fill in the Venn diagram
Answer
Now we can conclude:
- Students only owning a pet (P) = 16 - 5 = 11.
- Students only owning a bicycle (Q) = 8 - 5 = 3.
Finally, filling in the Venn diagram:
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(P only) = 11, # (Q only) = 3, # (P ∩ Q) = 5.
The diagram would show:
- Total in P = 16,
- Total in Q = 8,
- Students only owning neither = 6.
Each part of the Venn diagram satisfies the conditions given in the problem.