Word Problem Question From AoPS Volume 1

[dtmowns]

Could someone try to explain why the answer to the following Q is 10,000? Doesn't exceeds mean it should be 10,000.01?

*The formula N = (8)*(10^8)*(x^-3/2) gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals?

 

[JeffM]

Could someone try to explain why the answer to the following Q is 10,000? Doesn't exceeds mean it should be 10,000.01?

*The formula N = (8)*(10^8)*(x^-3/2) gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals?

You are correct. Either the question or the answer is badly worded, or you have not quoted one of them exactly.

 

[dtmowns]

Hmm, interesting. Thanks Jeff

 

[stapel]

Could someone try to explain why the answer to the following Q is 10,000? Doesn't exceeds mean it should be 10,000.01?

*The formula N = (8)*(10^8)*(x^-3/2) gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals?

"N" is the number of people; "x" is the income being exceeded. That is, "x" means "the income is more than this number of dollars". The answer, being "x = 10,000", means "the income for these people is more than ten thousand dollars".

That is, the "exceeds" part is built into the very definition of x.

 

[dtmowns]

"N" is the number of people; "x" is the income being exceeded. That is, "x" means "the income is more than this number of dollars". The answer, being "x = 10,000", means "the income for these people is more than ten thousand dollars".

That is, the "exceeds" part is built into the very definition of x.

I see. So you're saying just by writing x = 10,000 your also saying that the answer to the question is $10,000.01? True of course, though seems a bit strange, because the problem specifically asks for the lowest possible income. Ah well. Splitting hairs.

 

[JeffM]

I see. So you're saying just by writing x = 10,000 your also saying that the answer to the question is $10,000.01? True of course, though seems a bit strange, because the problem specifically asks for the lowest possible income. Ah well. Splitting hairs.

I am afraid that I disagree with stapel's answer.

If the question is "what is the smallest possible value of the lowest income," the word "exceeds" is not in the question. The smallest possible value (in terms of US currency) that exceeds $10,000 is $10,000.01. The question IS badly worded.

 

[stapel]

If the question is "what is the smallest possible value of the lowest income," the word "exceeds" is not in the question.

But the word "exceeds" is in the question, so the variable does not stand for "the smallest possible value of the lowest income". Instead, it means what it says: the number N of individuals whose income exceeds x dollars. "Exceeds" means "is greater than".

I see. So you're saying just by writing x = 10,000 your also saying that the answer to the question is $10,000.01? True of course, though seems a bit strange, because the problem specifically asks for the lowest possible income. Ah well. Splitting hairs.

No, I'm saying that "x = 10,000" means that there are N = 800 people who income exceeds (or "is more than") ten thousand dollars. The "more than" is built into the definition. The solution (N, x) = (800, 10000) means:

List out all the incomes, in order from least to greatest. Now look at the values of the last eight hundred items in the list. These values represent the incomes of the top eight hundred earners (N = 800) in this group. All of these values will be above ten thousand dollars (x = 10,000).
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[JeffM]

No, I'm saying that "x = 10,000" means that there are N = 800 people who income exceeds (or "is more than") ten thousand dollars. The "more than" is built into the definition. The solution (N, x) = (800, 10000) means:

List out all the incomes, in order from least to greatest. Now look at the values of the last eight hundred items in the list. These values represent the incomes of the top eight hundred earners (N = 800) in this group. All of these values will be above ten thousand dollars (x = 10,000).
​

I agree. The question posed, however, is

*The formula N = (8)*(10^8)*(x^-3/2) gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals?

The word "is" does not mean "exceeds." If the writer meant "exceeds," then it was sloppy of the the writer to use "is." The question could have been worded "What is the highest income necessarily exceeded by the lowest income of the 800 individuals with the highest income." (Note that "wealth" was also misused.) In that case, the answer would indeed be $10,000. Part of what math teaches, however, is the need to be precise in how we formulate and express ideas. A student should be applauded for parsing a question exactly, not told that the question is to be interpreted according to the answer given. It is true that there is a meaningful question to which the answer is $10,000, but that seems not to have been the question posed.