In a group of 140 students, 85 know how to use a Macintosh, 100 know how to use an IBM-compatible computer, and 10 do not know how to use either kind. How many students know how to use both kinds of computers?
A) 75
B) 40
C) 55
D) 90
E) 130

Question

Quizplus · Accepted Answer

We can use a Venn diagram to visualize the information given in the problem. Let's use two circles to represent Macintosh and IBM-compatible computer users, and let the overlap between them represent the students who know how to use both kinds of computers.

From the problem, we know that:
- 85 students know how to use a Macintosh
- 100 students know how to use an IBM-compatible computer
- 10 students do not know how to use either kind

Let's fill in the Venn diagram accordingly:

```
                Macintosh
          _____/¯¯¯¯\_____
         |                |
    IBM- |     BOTH       |    Neither
 Compatible|                |   
         |_____\__/____|
                    IBM-Compatible
```
We know that the total number of students is 140, so if we add up the numbers in the diagram, we get:
- Macintosh only: x
- IBM-compatible only: y
- Both: z
- Neither: 10

x + y + z + 10 = 140
x + y + z = 130

We don't know x or y, but we do know that:
- 85 students know how to use a Macintosh. This includes both Macintosh only students (x) and students who also know how to use IBM-compatible computers (z). So:
    x + z = 85
- 100 students know how to use an IBM-compatible computer. This includes both IBM-compatible only students (y) and students who also know how to use Macintosh computers (z). So:
    y + z = 100

We can solve this system of equations by subtracting the second equation from the first:

x + z - (y + z) = 85 - 100
x - y = -15
y - x = 15

We can use either equation to solve for one of the variables in terms of the other:
y - x = 15
y = x + 15

Now we can substitute this expression for y into one of the earlier equations to solve for z:
y + z = 100
(x + 15) + z = 100
z = 85 - x

Substituting this expression for z into the other equation, we get:
x + z = 85
x + (85 - x) = 85
85 - x + x = 85
z = 55

So 55 students know how to use both kinds of computers. Answer: C.

In a Group of 140 Students, 85 Know How to Use