# math problem

One hundred businesspeople gather at a meeting. 85 of them carry a cellular phone, 80 of them have a beeper, 75 of them speak at least two languages and 70 of them wear a suit. Therefore, a certain number of them have all of the above: a cell phone AND a beeper AND speak at least two languages AND wear a suit. Out of these 100 businesspeople, what is the least possible number who have all of the above?
10 - 15 - 17 - 18 - 20
this is a question on an IQ test, can someone give me the reasoning behind this answer? to view the entire test, visit

15 don’t have a cell phone, 20 don’t have a beeper, 25 don’t speak 2 languages and 30 aren’t weraring a suit. Consider these to be attributes.

Now 15+20+25+30=90, so at most 90 people are missing at least one attribute and 10 have to be doing all of them.

Glad I could help you cheat on your IQ test. And if you’re interested, there is a 35.7% chance that any one person has all the attributes and a 0.225% chance he has none.

Jim Petty
An oak tree is just a nut that stood it’s ground

I’m not sure I understand your explanation,
show me the equation.

Go wings!

I came up with 10 too. Here is my work:

85 have a cell phone. To get the smallest number of people that have a cell phone and beeper, assume that everybody who doesn’t have a cell phone have a beeper.

15 people don’t have cell phones, so subtract 15 from the 80 beepers. That leaves 65 beepers for the people who have cell phones.

So the smallest number of people who can have both a cell phone and beeper is 65.

75 people speak two languages. If everybody who doesn’t have both a cell phone and beeper speak two languages, that leaves 40 two-language-speaking-thingies for the people who have a cell phone and a beeper.
Soooo the smallest possible number of people who have a cell phone and a beeper and speak two languages is 40.

That means there are 60 people who do not have all three. If they all wear a suit, that leaves 10 of the 70 suits that must be worn by people who have cell phones, beepers, and two languages.

Trying to explain Jimpy’s original answer to red wings. Jimpy said: “15 don’t have a cell phone, 20 don’t have a beeper, 25 don’t speak 2 languages and 30 aren’t wearing a suit.” Okay.

Now, instead of answering the original question, “what is the least possible number of people who have all of these items?”, we pose the opposite question: “What is the highest possible number of people lacking at least one item?” [Finding the exact opposite is crucial to such problems!]

We find that maximum if we assume that people lacking phones are disjunct from, say, people lacking beepers (analogously for any two attributes). [This is the point where you may have to stop and think for a moment, but it should be clear then.]

Under this assumption, we get the number of people lacking any item by adding the people lacking phones to people lacking beepers etc. As Jimpy showed, this is 15+20+25+30=90.

Thus, at most 90 people lack some item, so at least 10 people have all items.

If anyone still doesn’t get it after three different explanations, I guess that IQ test is not for you…

I looked at the simple wording of the actual question ‘what is the least possible number who have all of the above?’ It seemed quite obvious to me that the answer is 1. It is therefore, I was interested in asking if anyone else saw the same answer. I guess not.

I understand your problem. You’ll see why your ‘obvious’ answer is not necessarily correct (and in the case at hand, wrong) if you change the parameters. Consider only phones and beepers, and suppose there are 99 of each for 100 people. Obviously. at least 98 (and at most 99) have both. That’s because there’s just one person without a phone and one without a beeper, and these may or may not be the same person.

The original problem is just a more complex extension of that.