Math Is Fun Forum

InspectorCthulhu

Member

Registered: 2016-06-02

Posts: 7

Can't see how to prove this statement about fractions

Someone showed me a trick for determining which of 2 fractions is larger (where this isn't obvious). It works on examples I've tried, but I've not been able to think of a general proof to show it always works. Can anyone help?

The assertion is:
If a/b <c/d, then da<bc

For example - is 6/8 < 5/6?
da = 6*6 = 36
bc = 8*5 = 40
da<bc, therefore a/b <c/d (and 6/8 < 5/6).

Thanks for your help & I hope this is a fun thing to think about!

zetafunc

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Registered: 2014-05-21

Posts: 2,436

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Re: Can't see how to prove this statement about fractions

Generally what you want to do is to suppose the converse holds, then try to get a contradiction.

In your case, you could suppose that

, which would imply that

, a contradiction. This tells you that your original assertion was false, and therefore it must be that

InspectorCthulhu

Member

Registered: 2016-06-02

Posts: 7

Re: Can't see how to prove this statement about fractions

Thank you zetafunc. Ah, I think I've just seen how it works.

If I wanted to get both fractions over a common denominator in order to compare them, I can do this:

a/b = da/bd
c/d  = bc/bd

so my original a/b ? c/d------------ (where '?' means 'what is the relationship between?' - I don't know if there's a formal maths symbol for that?)
becomes da/bd ? bc/bd
= da ? bc

So that's how and why it works (I think)!

Last edited by InspectorCthulhu (2016-06-02 03:27:02)

zetafunc

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Registered: 2014-05-21

Posts: 2,436

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Re: Can't see how to prove this statement about fractions

Yes, that will make it easier to compare the fractions. Although bear in mind that multiplying across an inequality by something negative, will reverse the direction of the inequality.

(where '?' means 'what is the relationship between?' - I don't know if there's a formal maths symbol for that?)

I don't know of such a symbol, but sometimes we use a ~ b or aRb to denote an equivalence relation -- but > or < are not equivalence relations.

Welcome to the forum!

InspectorCthulhu

Member

Registered: 2016-06-02

Posts: 7

Re: Can't see how to prove this statement about fractions

Thank you - the forum already seems like a friendly place!