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### Shortcut for Comparing Fractions

```Date: 01/14/2003 at 07:50:56
From: Joe
Subject: Comparing fractions

We are going over comparing fractions in class and my teacher showed a
little shortcut. He used the example of 2/3 and 3/5. He said you can
cross multiply and get 9/10. The numerator (the product of the
interior terms, 3 and 3) always corresponds to the second of the two
fractions, and the denominator (the product of the exterior terms, 2
and 5) corresponds to the first fraction.

He showed us many examples and I have tried many myself and this
always works. 9 is less than 10, so 3/5 is less than 2/3. My question
is WHY?
```

```
Date: 01/14/2003 at 11:20:56
From: Doctor Ian
Subject: Re: Comparing fractions

Hi Joe,

Suppose I have two quantities, and I want to know which is larger.
One way to do that is to divide one by the other, and see if I end up
with something less than, greater to, or equal to 1.

This is trivial when you do it with integers:

7 / 5 > 1  so 7 > 5

3 / 4 < 1  so 3 < 4

But it also works with fractions.  Suppose I use '?' to represent '=',
'>', or '<'.  That is, I'm going to use it as a 'variable' for these
relational operators.  Then if I want to compare two fractions, a/b
and c/d, I can do this:

a/b
----- ? 1
c/d

Of course, to divide by a fraction, I invert and multiply, so this is
the same as

a   d
- * - ? 1
b   c

If I get something greater than 1, a/b must be larger. If I get
something less than 1, a/b must be smaller. If I get 1, the two
fractions must be equal.

Let's try it with your example:

2/3
----- ? 1
3/5

2   5
- * - ? 1
3   3

10
-- > 1
9

So 'cross-multiplying' is just what you have to do to divide the first
fraction by the second. Comparing the result to 1 tells you whether
the numerator (first fraction) or denominator (second fraction) is
larger.

Does that make sense?

You can get a better feel for what is going on if you use letters
instead of numbers. For example, suppose my two fractions are a/b and
(a+1)/b.  I know that the second one has to be larger, right?  Let's
see what happens when we divide:

a/b
--------- ? 1
(a+1)/b

a     b
- * ----- ? 1
b   (a+1)

a
----- < 1
(a+1)

Let's try comparing two equivalent fractions. If we have a fraction
a/b, we can make an equivalent fraction by multiplying by k/k, where k
is anything other than zero:

a/b
--------- ? 1
(ka)/(kb)

a    kb
- * ---- ? 1
b    ka

kab
--- = 1
kab

For fun, you might consider trying to see which is larger: a/b, or
(a+k)/(b+k). That is, when you increment both the numerator and
denominator of a fraction by the same amount, is the resulting
fraction larger, or smaller?

or anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra
Middle School Fractions

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