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 neither a pet nor a bicycle. 28% of the students own both a pet and a bicycle. The ratio #P(Q) : #Q(P) = 2 : 1. Use this information to fill in the Venn diagram.

Junior Cycle Mathematics Probability: Conor carries out a survey on al

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

Finds #P ∩ Q

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Answer

To find the number of students who own both a pet and a bicycle (#P ∩ Q), we first determine the total number of students who own neither. There are 25 students total, and 6 own neither, leading to 25 - 6 = 19 students who own either a pet or a bicycle (or both).

If 28% of the students own both a pet and a bicycle, we can calculate:

$#P \cap Q = 0.28 imes 25 = 7$

Step 2

Splits value in the ratio 2 : 1

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Answer

Given the ratio #P(Q) : #Q(P) = 2 : 1, this means:

Let #P(Q) = 2x and #Q(P) = x. The total can be expressed as:

$#P \cap Q + #P(Q) + #Q(P) = 19$ Substituting the values, we get:

$7 + 2x + x = 19$ This simplifies to:

$3x = 12$ So,

$x = 4$ Thus:

#P(Q) = 2x = 8
#Q(P) = x = 4

Step 3

Fill in the Venn diagram

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Answer

Using the values found:

#P (owning pets) = #P(Q) + #P ∩ Q = 8 + 7 = 15
#Q (owning bicycles) = #Q(P) + #P ∩ Q = 4 + 7 = 11
The number of students who own neither remains 6.

We can now fill in the Venn diagram as follows:

U = 25
#P = 15
#Q = 11
#P ∩ Q = 7
Students neither = 6.

Conor carries out a survey on all of the 25 students in his class (U) - Junior Cycle Mathematics - Question 3 - 2016

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

Finds #P ∩ Q

Splits value in the ratio 2 : 1

Fill in the Venn diagram

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