Of the employees in Company X, 95% have desktop computers, 75% have both desktop and laptop, and 60% have both desktop amd PDA. At lease what percent of the employees in the company have all of the three kinds of devices.

a) 33%

b) 35%

c) 37%

d) 40%

e) 60%

OA after discussion.

## Overlapping sets

##### This topic has expert replies

- GMATGuruNY
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Let's plug in that there are 100 people.ov25 wrote:Of the employees in Company X, 95% have desktop computers, 75% have both desktop and laptop, and 60% have both desktop amd PDA. At lease what percent of the employees in the company have all of the three kinds of devices.

a) 33%

b) 35%

c) 37%

d) 40%

e) 60%

OA after discussion.

The big idea with overlapping groups:

*subtract the overlap*.

**Number who have a desktop and a laptop but not a PDA = total who have a desktop and a laptop - number who have all 3.**

Number who have a desktop and a PDA but not a laptop = total who have a desktop and a PDA - number who have all 3.

Number who have a desktop and a PDA but not a laptop = total who have a desktop and a PDA - number who have all 3.

We can plug in the answer choices, which represent the number who have all 3 devices. In order to minimize the number who have all 3, we should assume that of the 95 people with a desktop, no one has just a desktop: each person owns at least 2 of the 3 devices.

Since all the values in the problem are multiples of 5, answer choices A and C are unlikely to be correct. Let's start with B.

**Answer choice B: 35 own all 3.**

Desktop and laptop but not PDA = 75-35 = 40.

Desktop and PDA but not laptop = 60-35 = 25.

Desktop and laptop but not PDA + Desktop and PDA but not laptop + all 3 = 40+25+35 = 100.

Too big, because total with desktop = 95. Eliminate B.

**Answer choice D: 40 own all 3.**

Desktop and laptop but not PDA = 75-40 = 35.

Desktop and PDA but not laptop = 60-40 = 20.

Desktop and laptop but not PDA + Desktop and PDA but not laptop + all 3 = 35+20+40 = 95.

Success!

The correct answer is D.

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- Scott@TargetTestPrep
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ov25 wrote: ↑Mon Dec 13, 2010 7:26 amOf the employees in Company X, 95% have desktop computers, 75% have both desktop and laptop, and 60% have both desktop amd PDA. At lease what percent of the employees in the company have all of the three kinds of devices.

a) 33%

b) 35%

c) 37%

d) 40%

e) 60%

OA after discussion.

**Solution:**

Let x be the percent of employees who have both Desktop and Laptop, y be the percentage of employees who have both Desktop and PDA, z be the percent of employees who have all three and w be the percent of employees who have only Desktop.

Since 95% of the employees have a Desktop, x + y + z + w = 95. Since 75% have a Desktop and a Laptop, x + z = 75. Similarly, since 60% have a Desktop and a PDA, y + z = 60. Adding the last two equations together, we obtain:

x + y + 2z = 135

Subtracting x + y + z + w = 95 from this equation, we obtain:

z - w = 40

To minimize the value of z, we should choose w as small as possible. Letting w = 0, we see that the smallest value of z (which is the percent of employees who own all three) is 40.

**Answer: D**

**Scott Woodbury-Stewart**

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[email protected]

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- gmatter2012
- Master | Next Rank: 500 Posts
**Posts:**136**Joined:**Sat May 05, 2012 7:03 pm**Thanked**: 1 times

Just a small clarification, to minimize Z shouldn't we need to maximize W and to Maximize Z shouldn't we need to minimize W, so when we make W=0 aren't we maximizing Z ?Scott@TargetTestPrep wrote: ↑Tue Feb 23, 2021 3:27 pmov25 wrote: ↑Mon Dec 13, 2010 7:26 amOf the employees in Company X, 95% have desktop computers, 75% have both desktop and laptop, and 60% have both desktop amd PDA. At lease what percent of the employees in the company have all of the three kinds of devices.

a) 33%

b) 35%

c) 37%

d) 40%

e) 60%

OA after discussion.Solution:

Let x be the percent of employees who have both Desktop and Laptop, y be the percentage of employees who have both Desktop and PDA, z be the percent of employees who have all three and w be the percent of employees who have only Desktop.

Since 95% of the employees have a Desktop, x + y + z + w = 95. Since 75% have a Desktop and a Laptop, x + z = 75. Similarly, since 60% have a Desktop and a PDA, y + z = 60. Adding the last two equations together, we obtain:

x + y + 2z = 135

Subtracting x + y + z + w = 95 from this equation, we obtain:

z - w = 40

To minimize the value of z, we should choose w as small as possible. Letting w = 0, we see that the smallest value of z (which is the percent of employees who own all three) is 40.

Answer: D