math problem - can you solve this?

[Hoggirl]

The ratio of the number of distinct odd integers to the number of distinct even integers in set E is 3:5. There are 24 distinct integers in set E. Which is greater? The number of distinct odd integers in set E or 9?

 

I don't want to say what I got, but can someone do this problem for me? A friend posted this on her fb wall, and I attempted to answer it, but someone else said I was making an assumption, but didn't explain what difference it would make.

[Melinda]

I am not good at math, so there is a high likelihood I am wrong, but...

 

I would say neither is greater as E is equal to 9, because

 

If the ratio is 3:5, odd:even and the total is 24, then that would mean that there are 9 odd and 15 even, right?

 

If that is not right, somebody please explain it!

[Jenny in Florida]

I'm sure there's a more elegant way to go about this, but here's what I did:

 

If the ratio is 3:5, this means you have 8 "parts." So, I took 24 and divided by 8, meaning there are 3 in each part. Since the odd integers make up 3 parts, there are 9 of them.

 

So, neither is greater. They are equal.

[Janet in Toronto]

Have you reproduced the question completely accurately? I think the answer has to do with the word "distinct".

Yes, there would be 9 distinct integers in Set E, but if the question asks for the number of odd integers (not distinct), then it would 9 or greater.

[Hoggirl]

Have you reproduced the question completely accurately? I think the answer has to do with the word "distinct".

Yes, there would be 9 distinct integers in Set E, but if the question asks for the number of odd integers (not distinct), then it would 9 or greater.

 

So, yes, this is how it was worded. Not sure what it means by "distinct"????

 

I, too, got that it was = to 9. Then her bil posted a snarky comment about how I was assuming the set didn't include "0" or negative integers, and I didn't get what that would have to do with it at all!

 

ETA: this is what he then posted:

 

If the set includes zero, which is a valid integer, then you can't do that multiplication trick that you did to get to 9:15, you have to stick to the given 3:5, which lowers the possible number of distinct odd integers to 8, because the set is {0,1,2,3...23}. If the set includes the negative integers then it would seem like you don't have enough information to answer the question, because you must know which numbers are members of the set in order to answer the question

Edited April 22, 2009 by Hoggirl

[Jenny in Florida]

I don't get it.

 

Admittedly, I'm not a mathy sort, but I have read that comment about six times and still can't figure out what it has to do with anything. I'm more than willing to assume I'm just missing something, though.

 

So, I e-mailed the original question to my much-mathier-than-me husband. I'll let you know when he gets back to me.

[Janet in Toronto]

Okay, there is some difference of opinion over whether negative integers can be odd or even. And I take his (snarky) point about zero.

 

With 24 distinct integers, and a 3:5 ratio of odd:even, there can be 3 odd/5 even, 6 odd/10 even, or 9 odd/15 even distinct integers, if we allow negative numbers and zero to "not count". You still don't have more than 9 odd integers.

 

Example: the set could be {-7,-6,-5,-4,-3,-2,-1,0,1,3,5,7,9,11,2,4,6,8,10,12,14,16,18,20} - that is, 8 zero/negative, 6 odds, 10 evens.

 

If negative numbers can be even or odd, and you have one Zero, then that leaves 23 distinct integers, and so you can't have a ratio of evens to odds of 3:5. To have that ratio, you MUST have a number of distinct integers that is a multiple of 8. So I believe that having zero as one of the distinct integers (and permitting negative odd and even numbers) is out.

[Hoggirl]

"cannot be determined from the information given?" :D or what?

 

This is the kind of thing that drives me NUTS!

[Jenny in Florida]

. . . says we're right. The answer is 9.

 

According to him, the idea that negative numbers aren't even/odd is silly. The definition of "even" (which I found in multiple places) says that a number is even if it can be evenly divided by 2. That's it.

 

And I found more than one site arguing pretty persuasively that 0 is even, because you can divide it evenly by 2. The answer is 0.

 

Here's a snippet from Wikipedia:

 

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

 

Examples of even numbers are −4, 8, 0, and 42.

 

Examples of odd numbers are −3, 9, 1, and 5. A fractional number like 1/2 or 3.141 is neither even nor odd.

[Mommyfaithe]

I just read that problem and I swear my brain had a anuerism..no really...can you please peel me off the floor and be gentle with the fact that i NEED to use Teaching Textbooks for my kids???

 

SIGH>>>>>

Faithe:confused:

[Hoggirl]

. . . says we're right. The answer is 9.

 

According to him, the idea that negative numbers aren't even/odd is silly. The definition of "even" (which I found in multiple places) says that a number is even if it can be evenly divided by 2. That's it.

 

And I found more than one site arguing pretty persuasively that 0 is even, because you can divide it evenly by 2. The answer is 0.

 

Here's a snippet from Wikipedia:

 

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

 

Examples of even numbers are −4, 8, 0, and 42.

 

Examples of odd numbers are −3, 9, 1, and 5. A fractional number like 1/2 or 3.141 is neither even nor odd.

 

If 0 is even then we must be right, right? :D

[Hoggirl]

9 is the correct answer. She is studying for the GRE. I asked her to please post it on her thread or whatever it is called on fb. I have this PRIDE issue!:D I want Mr. Snarky to know I was right.

[kiana]

This goes to show why you shouldn't be snarky ... because being both snarky and wrong amuses everyone else ;)

[Hoggirl]

9 is the correct answer. She is studying for the GRE. I asked her to please post it on her thread or whatever it is called on fb. I have this PRIDE issue!:D I want Mr. Snarky to know I was right.

 

Right and non-snarky is preferable, isn't it?

 

However, he won't let it go and has now moved to her wall with "you (meaning my friend/his sister-in-law who obviously messaged him that my answer was right - he removed HIS comment in the thread about "tutoring" me!) are making an unsubstantiated assumption that they are counting integers."

 

What's the difference between a "counting integer" and a plain ol' integer????

 

At any rate, I have determined that she and I would easily complete the math section of the GRE, while he would be writing a dissertation-like footnote about the invalid assumptions necessary to complete the test; and he would never finish the stinkin' GRE. :D

[Jenny in Florida]

What's the difference between a "counting integer" and a plain ol' integer????:D

 

As far as I can tell, nothing. Again, from Wikipedia:

 

The integers (from the Latin integer, literally "untouched", hence "whole": the word entire comes from the same origin, but via French[1]) are natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, −2, −3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers. In other terms, integers are the numbers one can count with items such as apples or fingers, and their negatives, as well as 0.

[Spock]

So, yes, this is how it was worded. Not sure what it means by "distinct"????

 

I, too, got that it was = to 9. Then her bil posted a snarky comment about how I was assuming the set didn't include "0" or negative integers, and I didn't get what that would have to do with it at all!

 

ETA: this is what he then posted:

 

If the set includes zero, which is a valid integer, then you can't do that multiplication trick that you did to get to 9:15, you have to stick to the given 3:5, which lowers the possible number of distinct odd integers to 8, because the set is {0,1,2,3...23}. If the set includes the negative integers then it would seem like you don't have enough information to answer the question, because you must know which numbers are members of the set in order to answer the question

 

 

Depending on which math program you learned with, zero can either be considered "even" or "neither even nor odd". If zero is considered even, then your answer is correct. If zero is considered neither even nor odd, your answer could be slightly off.

 

I get the impression that the person criticizing your answer also considers negative numbers to be neither even nor odd, though I am not familiar with the rationale for that.

 

I also don't see the need for the word "distinct" in the problem, unless it means that some integers are repeated in the set (maybe a set like {2, 5, 2, 7, ...}?). In that case, even though "2" appears twice, it would only count as one "distinct" even integer. However, if that is the case, there is insufficient information to solve the problem.

[kiana]

I am not sure which math program is teaching that 0 is neither even nor odd, but this is incorrect although a common misconception.

 

Mathematically speaking, a non-zero integer a divides an integer b if and only if there exists an integer k such that ak = b. Clearly, if b = 0, k = 0 works for any integer a. Thus 0 is divisible by every integer.

 

Another way to look at it is that 0 is a multiple of 2 since 0 x 2 = 0, and integral multiples of 2 are even.

 

Another way to look at it is using your standard rules of addition of evens and odds, that is, E + E = E, O + O = E, O + E = O. 1 + (-1) = 0, 2 + (-2) = 0. There is no way to add an even and an odd and get 0.

 

Mr. Snarky is probably calling the set of positive integers the counting integers, but he is still incorrect about the parity of non-positive integers.

[Hoggirl]

I am not sure which math program is teaching that 0 is neither even nor odd, but this is incorrect although a common misconception.

 

Mathematically speaking, a non-zero integer a divides an integer b if and only if there exists an integer k such that ak = b. Clearly, if b = 0, k = 0 works for any integer a. Thus 0 is divisible by every integer.

 

Another way to look at it is that 0 is a multiple of 2 since 0 x 2 = 0, and integral multiples of 2 are even.

 

Another way to look at it is using your standard rules of addition of evens and odds, that is, E + E = E, O + O = E, O + E = O. 1 + (-1) = 0, 2 + (-2) = 0. There is no way to add an even and an odd and get 0.

 

Mr. Snarky is probably calling the set of positive integers the counting integers, but he is still incorrect about the parity of non-positive integers.

 

 

I am too much of a techno-idiot to only quote part of a post, but your last bit about parity, etc. rocks. Although I don't get what it means :D I just think it sounds SMART! :lol: