This question was on an “IQ Test” linked in this thread:
I couldn’t even begin to figure out how to compute the answer. Can someone draw it out for me, real simple-like?
I think the trick is to prevent overlap of the “nots”, meaning that you must assume that if a person is missing one item, he has all of the others. This gives the maximum number of people missing at least 1 item. So, adding together all of the people missing items, you get
15 + 20 +25 +30, which equals 90. That leaves 10 people at minimum who are missing no items, and have all of the above.
Here’s my late night WAG.
Start with the 85 who have a cell phone. Subtract the 20 who don’t have a beeper. That leaves 65. Subtract the 25 who don’t speak two languages. That leaves 40. Subject the 30 who don’t wear a suit. That leaves 10.
Same principle as Mosier, but expressed differently.
You can take them one at a time. If everyone who does not have a cellular phone has a beeper, you have 65 people with both. Then the largest number of people with both cp and b who dont speak two languages is 30, leaving you with 40 people with three items. thirty of those people might not wear a suit, so you have 10 left with all 4 items.
Exapno said that a lot more clearly than I did. I’m going to bed.
Step 1: Eliminate 17 and 18. All of your numbers are divisible by 5, so neither of those will be it. If that’s all the further you get, then when you run out of time and are forced to guess, you’ll have a 1/3 chance instead of a 1/5 chance.
Step 2: You gotta focus on the negitave here. To find the minimum that have all traits, you’ve got to find the maximum that could me missing one. Say 30 are missing trait A, a separate 25 are missing trait B, a separate 20, are missing trait C, and a separate 15 are missing trait D.
Step 3: Add 'em all up, and you get 90. That’s the maximum number of people who are missing a trait. Subtract, and you get 10 people who must have all four traits.
Step 4: Profit.
ON PREVIEW- You all suck.
ETA: (And by “suck,” I mean “are faster than I am, and likely have better internet connections.”)
Yet another way to get the answer:
I imagined all the one hundred people lined up in a row. I then imagined the 85 leftmost people had cell phones and the 80 rightmost had beepers. The overlap of these two groups, then, have a number equal to the smallest possible number of people who can have both cell phones and beepers. How many is this? Well, there are 15 phoneless people over on the right, and 10 beeperless over on the left, meaning the overlap group in the middle consists in 75 people.
Rince and repeat for the polylinguists and suit wearers, and you end up with the right answer.
I got lost on the pocket money question. Kid spends all her money in five shops each time spending one dollar more than half of what she’d had previously. How much did she spend?
Just cannot get my head around it.
Well, in each shop she starts with x dollars and spends ½x + 1, leaving her with ½x - 1. But in the final shop she spends all her money, so on that occasion x is equal to ½x + 1, i.e. x = 2. Working backwards, in the previous shop ½x - 1 = 2 so x = 6, before that x = 14, before that x = 30 and before that x = 62. So she spent $62. In general, in n shops she would spend 2[sup]n+1[/sup] - 2 dollars. Don’t take her to the mall.
maggenpye, I got tripped up on that one too. Only I took every answer and ran it through the calculator:
A-((A/2)+1) = B
B-((B/2)+1) = C
C-((C/2)+1) = D
D-((D/2)+1) = E
E-((E/2)+1) = F
Only one answer got us to F without going under 1 (actually, F=0)
Another question for everyone - what principal of mathematics am I failing to comprehend by not being able to answer the question in the OP? Not only could I not answer it, I could not even “diagram” it.
FWIW I started drawing it out like Frylock explained but I couldn’t get it right.
It seems to be related to inclusion-exclusion, but it’s not exactly the same thing. The basic idea of inclusion-exclusion is that if you’re trying to figure out how many people are in all of three groups, you start by adding up the number of people in each group, then you subtract the number of people in each pair of groups, and finally you add back in the people in all three groups. If you have more than three groups, you keep going, alternating between addition and subtraction at every step.
Wow. Y’all thought hard about that one. I saw “what is the smallest possible number” and picked the lowest number they gave as a possible answer.
I mean, it’s not like I’m trying to get into their college or anything. 
Which points out another flaw to that test: it is multiple choice. True intelligence tests contain a lot of open-ended questions. For example, I still remember when I took one in the second grade, seated one-on-one with the psychologist, and was tested to see how many folds in the paper the tester had to make and cut out a shape on before I figured out that the number of holes it made doubled with each fold.
Multiple guess questions should be structured so that a guess based upon false reasoning from the question’s directions should not be possible. That question should have had something like “8” as the least possible answer, so that your reasoning would have resulted in an incorrect scoring. 
For an example of someone’s Stanford Binet test kit, look here.
I don’t know if there is a particular mathematical principle that has a name (though I agree with ultrafilter that it’s related to the “inclusion-exclusion principle”), but, I think, the key insight has to do with complementarity: if you know how many people do fall into a certain category (or what percentage do, or the probability that someone does), you can easily calculate how many do not fall into that category. In this particular question, then, you’d look for the least possible overlap among those complementary groups.
A Venn diagram is sometimes helpful in situations like this, but it gets trickly to use when there are four overlapping sets.
Yeah, a Venn diagram was what I tried to draw out before I got stuck. I had the feeling that some sort of over-lapping was going to be the key to finding the answer, but I quickly realized that my two little circles with “85” and “80” written in them and an overlap of “15” was completely wrong.
Lately, I’ve been trying to understand where my “blocks” are when it comes to math. I’m a fairly intelligent person and I did excel in math in school - but only up to Algebra II. Not only did I fail that class, I never took another (calc, trig, etc.) I am definitely good with everyday math (say, money-related math) but often I run in to concepts such as the one in this question where my mind just STOPS. Even though everyone’s examples made sense (in that they worked out mathematically), I still couldn’t really understand until Frylock stepped in with a nice “picture” example.
Another example of a “block” I come to with math is this - and this is really simple:
My company pays 2% city tax. Quickbooks tells me I owe $180 in city tax this month. I need to fill in the City Tax Form with the amount of tax I owe AND the amount in payroll paid this month. Quickbooks doesn’t tell me the amount of payroll right there on the screen, so I need to do some sort of multiplication to know what the payroll was.
X*.02 = $180
Easy, right? But every month I end up opening up the calculator, and multiplying 180 by 5, 20 and 50 until one of my results “works” when plugged in to the above calculation.
Is the above concept anywhere near the concept of the OP? Is the concept above just “algebra”?
I’m trying to identify what concepts cause me the most trouble so I can either work on those concepts - or try to avoid them completely.
What is my major malfunction? 
It’s just algebra. And given that you excelled through Alg I in High School*, I’m honestly suprised you’re plugging in guesses. You solve this exactly the same way you would solve something like 2x = 4.
.02x = 180
divide both sides by .02, so
x = 180/.02
If you want to use a calculator, that’s what you should plug in.** No guesswork needed!
-FrL-
- I have a similar background actually! I was a “math genius” all the way through a class called “Algebra II w. Pre-Calculus.” I say I was a “math genius” because starting in Sixth Grade, I was pulled out of class and allowed to teach myself eighth grade algebra instead of what everyone else was doing, and from then on I was always one to two years ahead of my class in math.
Then came calculus. And I was just not getting something. I was failing from practically day one. I was really, really upset about this. “Math genius” was part of my self-identity back then. (Hey, I was a kid.)
Eventually I just dropped it and comforted myself in the knowledge that I was a genius in every other way as well. 
**Here’s how not to use a calculator and do it in your head. You need to divide 180 by .02. Recall that dividing by fractions is the same as multiplying by their reciprocals. Well, .02 is obviously the same as 2/100, the reciprocal of which is 100/2. Multiplying by 100 is easy–just add two zeros at the end. Then dividing by two is almost as easy.
So multiply 180 by 100–18,000–then divide that by two–9000.
-FrL-
Or you could just multiply by 50… :dubious:
I wouldn’t think it is as easy to multiply 18 by 5 than it is to divide it by two.
18/2–memorized by many children in elementary school. 18*5–not so much.
-FrL-
Nevermind…
I think the biggest ‘trick’ or ‘block’ in the OP’s question was that you had to realize your goal. it was not to minimize the “alls”, it was to maximize the “nots”. I did what Mosier did. I mentally saw a bunch of name tags that said “not a suit” or something. Those were my tickets out of the finals. I found that I had 25, 20, 15, and 30 of those. That’s 90 “get out of my count” tickets, leaving 10 people stuck in my count.
There’s a classic “X people in a room, they all shake hands once w/ everyone else. How many handshakes?” Some people say “X shake w/ X-1, but that’s counting each twice, so divide by 2” I say “Person 1 shakes everyone then goes and eats the buffet, never to be bothered again.” and count them up that way, like “7+6+5+4…”
Ah, but the flaw in that approach is, what if there isn’t a buffet?