A survey has been conducted on methods of commuter travel. Each respondent was…
2018
A survey has been conducted on methods of commuter travel. Each respondent was asked to check Bus, Train and Automobile as a major method of travelling to work. More than one answer was permitted. The results reported were as follows :
Bus 30 people; Train 35 people; Automobile 100 people; Bus and Train 15 people; Bus and Automobile 15 people; Train and Automobile 20 people; and all the three methods 5 people. How many people completed the survey form?
- A.
120
- B.
165
- C.
160
- D.
115
Attempted by 6 students.
Show answer & explanation
Correct answer: A
Inclusion–exclusion formula for three sets:
Total = Bus + Train + Automobile − (Bus and Train) − (Bus and Automobile) − (Train and Automobile) + (Bus and Train and Automobile)
Sum of single counts: 30 + 35 + 100 = 165.
Sum of pairwise overlaps to subtract: 15 + 15 + 20 = 50. Subtracting gives 165 − 50 = 115.
Add back the triple overlap (was subtracted three times): +5 → 115 + 5 = 120.
Total respondents = 120
Explanation: When subtracting pairwise overlaps we remove people who use all three methods three times, so we must add the triple-overlap back once.