Question

A survey of 255 adults found that during the last year, 40 traveled by plane but not by train, 75 traveled by train but not by plane, 50 traveled by bus but not by plane or by train, 55 traveled by bus and plane, 35 traveled by all three, and 170 traveled by plane or train. How many did not travel by any of these modes of transportation?

Answer 1

To find the number of people who did not travel by any of these modes of transportation, we can use the principle of inclusion-exclusion. The number of adults who did not travel by any of these modes is 70.

Step-by-step explanation:

To find the number of people who did not travel by any of these modes of transportation, we can use the principle of inclusion-exclusion. First, let's calculate the total number of people who traveled by at least one mode of transportation.

Total = Number of people who traveled by plane or train + Number of people who traveled by bus but not by plane or train

Total = 170 + 50 = 220

Next, let's subtract the number of people who traveled by all three modes of transportation.

Total = Total - Number of people who traveled by all three

Total = 220 - 35 = 185

Therefore, 185 adults traveled by at least one mode of transportation. To find the number of adults who did not travel by any of these modes, we subtract this number from the total number of surveyed adults.

Number of adults who did not travel by any of these modes = Total number of surveyed adults - Number of adults who traveled by at least one mode

Number of adults who did not travel by any of these modes = 255 - 185 = 70

Answer 2

35 adults did not travel by any of these modes of transportation

Step-by-step explanation:

I add a drawing of the sets

Let's write the equations system reading the exercise :

`A+B+C+D+E+F+G+H=255\\B+E=40\\C+F=75\\D=50\\E+G=55\\G=35\\B+C+E+F+G+H=170`

We must find A.

We solve the system equations :

`E+G=55\\G=35\\Then\\E+35=55\\E=20`

`B+E=40\\E=20\\Then\\B+20=40\\B=20`

`B+C+E+F+G+H=170\\B=20\\E=20\\G=35\\C+F=75`

`B+(C+F)+E+G+H=170\\20+75+20+35+H=170\\H=20`

`A+B+C+D+E+F+G+H=255\\A+B+(C+F)+D+E+G+H=255\\A+20+75+50+20+35+20=255\\A=35`