Suppose the following data represents a survey of 500 students who were enrolled in a freshman college mathematics course

450 passed the course

10 of those who failed the course liked it,

25 of those who failed the course signed up for another mathematics course,

55 of those who liked the course signed up for another mathematics course

60 of those who passed the course signed up for another mathematics course

350 of those who passed the course liked it,

300 of those who passed the course liked it but did not sign up for another mathematics course

(a) Give a Venn diagram summarizing this data

(b) How many of the students who failed disliked the course and did not sign up for another mathematics course

(c) How many of the students liked the course?

(d) How many of the students who did not like the course passed it?

$\begin{align} & n(\xi )=500 \\ & \text{Let those that passed be denoted by }P \\ & \text{Let those that liked the Maths course be denoted by }L \\ & \text{Let those that signed up for another Maths course be denoted by }S \\ & n(P)=500 \\ & n(P'\cap L)=10 \\ & n(P'\cap S)=25 \\ & n(L\cap S)=55 \\ & n(P\cap S)=50 \\ & n(P\cap L)=350 \\ & n(P\cap L\cap S')=300 \\ & n(P\cap L)=n(P\cap L\cap S')+n(P\cap L\cap S) \\ & 350=300+x \\ & x=n(P\cap L\cap S)=50 \\ & n(L\cap S)=55 \\ & n(L\cap S\cap P')=55-50=5 \\ & n(P\cap S)=60 \\ & n(P\cap S\cap L')=n(P\cap S)-n(L\cap S\cap P)=60-50=10 \\ & n(P\cap S'\cap L')=450-(10+50+300)=90 \\ & \text{Consider }P\cup L \\ & n(L\cap S'\cap P')=n(P'\cap L)-n(S\cap L\cap P')=10-5=5 \\ & \text{Consider }P\cup S \\ & n(P'\cap L'\cap S)=n(P'\cap S)-n(S\cap L\cap P')=25-5=20 \\ & Note:\text{ }n(P\cup L\cup S)=480\text{ from the venn diagram} \\ & (b)\text{ The number of those who failed disliked the course and did not sign up } \\ & \text{for another mathematics course}=n(P'\cap L'\cap S') \\ & \\ & n(P'\cap L'\cap S')=n(\xi )-n(P\cup L\cup S)=500-480=20 \\ & \\ & (c)\text{Those that liked the course }n(L) \\ & n(L)=n(L\cap P'\cap S')+n(P\cap L\cap S)+n(S\cap L\cap P')+n(P\cap L\cap S') \\ & n(L)=5+50+5+300=360 \\ & \\ & (d)\text{ Those who did like the course but passed it}=n(L'\cap P) \\ & n(L'\cap P)=n(P\cap S'\cap L')+n(P\cap S\cap L')=90+10=100 \\\end{align}$