College

A survey of 1,000 high school seniors showed the following:

- 100 students took the SAT, ACT, and AP classes.
- 200 students took the SAT and AP classes.
- 250 students took the ACT and AP classes.
- 150 students took the ACT and SAT.

Additionally:

- A total of 600 students took the ACT.
- A total of 300 students took the SAT.
- A total of 550 students took AP classes.

How many students took none of these (neither the SAT, nor the ACT, nor any AP classes)?

Answer :

According to the given information, we can calculate the number of students who took none of the SAT, ACT, or AP classes.

Let's break down the information provided:

- The total number of students who took the SAT is 300.

- The total number of students who took the ACT is 600.

- The total number of students who took AP classes is 550.

We can use the principle of inclusion-exclusion to find the number of students who took none of these exams or classes.

Total number of students who took at least one of the exams or classes = Number of students who took the SAT + Number of students who took the ACT + Number of students who took AP classes

= 300 + 600 + 550

= 1450

Now, let's calculate the number of students who took both the SAT and ACT:

Number of students who took both SAT and ACT = Number of students who took the SAT + Number of students who took the ACT - Number of students who took both SAT and ACT and AP classes

= 300 + 600 - 150

= 750

Next, let's calculate the number of students who took both the SAT and AP:

Number of students who took both SAT and AP = Number of students who took the SAT + Number of students who took AP classes - Number of students who took both SAT and ACT and AP classes

= 300 + 550 - 200

= 650

Similarly, let's calculate the number of students who took both the ACT and AP:

Number of students who took both ACT and AP = Number of students who took the ACT + Number of students who took AP classes - Number of students who took both SAT and ACT and AP classes

= 600 + 550 - 100

= 1050

Now, let's calculate the number of students who took all three exams or classes:

Number of students who took all three exams or classes = Number of students who took both SAT and ACT + Number of students who took both SAT and AP + Number of students who took both ACT and AP - Number of students who took both SAT and ACT and AP classes

= 750 + 650 + 1050 - 100

= 2350

Finally, we can calculate the number of students who took none of these exams or classes:

Number of students who took none of these = Total number of students - Number of students who took at least one of the exams or classes

= 1000 - 1450

= 450

Therefore, 450 students took none of the SAT, ACT, or AP classes.

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