Clever way of showing which of these two fractions is greater?

Economister

Anyone got a nifty way to show which of the following is larger?

1) 1/(1+x)

2) (1/x) - 1

Reply 1

Pheylan

Original post by Economister

Anyone got a nifty way to show which of the following is larger?

1) 1/(1+x)

2) (1/x) - 1

For what x? :s-smilie:

Reply 2

Intriguing Alias

Suppose 1) = 2)

1/(1+x) = (1/x) - 1

1 = (x+1)/x - 1 - x
1 = 1 + 1/x - 1 - (x^2)/x
x = -x^2 + 1
x^2 + x - 1 = 0

the roots of that are

-1.61803

and

0.61803

If x > 0.61803 , 1) > 2)

If -1.61803 < x < 0 then 1) > 2)
If 0 < x < 0.61803 then 2) > 1)

If x < -1.61803 then 1) > 2)

That's the best that can be done. Neither is DEFINITELY larger than the other. But 1) is much more likely to be bigger as 2) is only bigger in a very small range.

Also they are not comparable if x = 0 or x = -1 as one will be infinite.

EDIT: I think I have corrected my inequalities now :smile:

(edited 13 years ago)

Reply 3

H.C. Chinaski

Original post by hassi94

Suppose 1) = 2)

1/(1+x) = (1/x) - 1

1 = (x+1)/x - 1 - x
1 = 1 + 1/x - 1 - (x^2)/x
x = -x^2 + 1
x^2 + x - 1 = 0

the roots of that are

-1.61803

and

0.61803

If x > 0.61803 , 1) > 2)

If -1.61803 < x < 0.61803 then 2) > 1)

If x < -1.61803 then 1) > 2)

That's the best that can be done. Neither is DEFINITELY larger than the other. But 1) is much more likely to be bigger as 2) is only bigger in a very small range.

Also they are not comparable if x = 0 or x = -1 as one will be infinite.

A rather dangerous statement me-thinks ...... I could equally argue that either statement is true for an infinite "number" of values ....

Reply 4

Intriguing Alias

Original post by H.C. Chinaski

A rather dangerous statement me-thinks ...... I could equally argue that either statement is true for an infinite "number" of values ....

Yeah you're right, but he's doing economics not maths so it would be applied more to real life and that's why I derived that statement.

But yeah I agree probably not the best statement to make :smile:

Reply 5

ttoby

Original post by hassi94

If -1.61803 < x < 0.61803 then 2) > 1)

Take x=-0.1:

1/(1+(-0.1)) = 1.1111....

1/(-0.1) - 1 = -11

2) < 1)

The mistake you made was you jumped from talking about equalities to inequalities without properly considering the signs of everything you multiply each side by, causing the inequality to flip round.

The approach I would take would be to consider 1/x - 1 - 1/(1+x) and see when it's positive, negative or undefined.

Spoiler

Reply 6

Aaron.F

1) 1/(1+x)

2) (1/x) - 1

Reply 7

Intriguing Alias

Original post by ttoby

Spoiler

Yeah sorry my bad just realised how wrong that was haha. Don't know why I didn't consider the fact I was multiplying by an unknown quantity and how that could affect the inequality.

My bad! :P

Reply 8

anshul95

Original post by hassi94

I think you have the range of values incorrect possibly because you haven't use inequalities and somewhere you may have got a sign wrong - I haven't checked fully though. For this, I would use the sign test (but before you do this you will have to collect fractions and make a table of the critical points).

Edit: LOL posted at the same time - oh well