Social Question

LostInParadise's avatar

Test of mathematical intuition?

Asked by LostInParadise (31905points) 2 months ago

How many pairs of numbers x and y would you guess that there are there such that 1/x + 1/y = 1/10?

I will show a way of finding them all.

Observing members: 0 Composing members: 0

7 Answers

JLeslie's avatar

Deleted by me. I have to rethink it.

LifeQuestioner's avatar

1+x/y=x/10
y+x=xy/10
10y+10x=xy
Looking at this, and I could be wrong, I would say that the only solution is if x and y both equal 0.

Zaku's avatar

My intuition says not to care.

My mathematical intuition plus grade school line equation knowledge, says that’s a curve equation, and that there would be an infinite number of point solutions x & y, which also passes the test of imagining plugging in various numbers for x & y.

LostInParadise's avatar

The first thing to realize is that x and y both have to be greater than 10.
Next, notice that half of 1/10 is 1/(20). One of 1/x and 1/y is greaer than or equal to 1/(20) and the other is less than or equal to 1/(20). We can arbitrarily choose 1/x >= 1/(20). That means that x<=20. Therefore possible x values go from 11 to 20, meaning there are at most 10 possible pairs. Using brute force we can check which values of x give us integer values for y. We get the following pairs:
1/11, 1/110
1/12, 1/60
1/14, 1(/35)
1/15, 1/(30)
1/(20), 1/(20)

Zaku's avatar

I’m confused by your answer.

First, what are the parentheses doing?

Second, where did you constrain answers to integers? Seems to me you wrote numbers, and numbers include non-integers.

So how about x = 25, y = 16 + ⅔?

If you don’t constrain to integers, it seems to me there are an infinite number of pairs of values for x and y (the reciprocals of all pairs of numbers that add to 1/10). I don’t think x or y are bounded, either, since for any value x, y is the number such that their reciprocals add to 1/10.

Zaku's avatar

Here is a graph, FWIW.

LostInParadise's avatar

I should have specified that x and y must be integers. Otherwise, there are infinitely many possibilities.

As for the parentheses, they were needed in order to have the numbers displayed prpoperly.

Answer this question

Login

or

Join

to answer.
Your answer will be saved while you login or join.

Have a question? Ask Fluther!

What do you know more about?
or
Knowledge Networking @ Fluther