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Ordering Fractions by Changing Them to Decimals

```Date: 12/11/2008 at 21:45:42
From: Julie
Subject: placing fractions in order

3/10, 1/2, 7/8, 4/5, 1/6, 3/4, 9/10, 1/3, 3/5, 3/8, 1/4, 1/10
Can you help me put these in order?  Please!

I just really don't understand fractions.  I know 1/4, 1/2 etc... but
when there is another number other than 1 on top, I get confused.

```

```
Date: 12/11/2008 at 23:27:37
From: Doctor Wilko
Subject: Re: placing fractions in order

Hi Julie,

Thanks for writing to Ask Dr. Math!

There are many different ways to teach or explain this.  I'll try one
way which may be different from what your daughter was taught, but it

1. Common Denominators

Before I get to your exact problem, I think it'll still be insightful
to talk a little about fractions and common denominators.  The
denominator is the number on the bottom of the fraction.  If the
denominators of a bunch of fractions are the same, the problem is
easier to handle.

For instance, in your fractions above you have 3/10, 9/10, and 1/10.
These have the same denominator, which makes them easier to compare
directly.

One way to think about fractions is in terms of a pie.  The
denominator tells you the total pieces of pie and the numerator (top
number) is, say, how many pieces of pie you are going to eat out of
the total.  So, for 3/10, this is like saying there are 10 pieces of
pie and you are going to have 3 of them.

So, it should make sense if we rank these in order of smallest to
largest, then the rank is 1/10, 3/10, and 9/10, because eating 1 out
of 10 pieces of pie is a lot less than eating 9 out of 10 pieces of pie.

You could summarize this and say: if you compare fractions with the
same denominators (bottom numbers), then the fraction with the
smallest numerator (top number) is the smallest fraction and the
fraction with the largest numerator is the largest fraction.

Again, this method is nice if you have fractions with the same
denominator.  In fact you can take fractions with different
denominators and do some math to get them to all have common
denominators and then still use this method.  For what you're doing, I
think there is a more concrete way to understand how to order the
fractions from smallest to largest.  Let's look at that now.

2. Change Fractions to Decimals

What I explained above is nice if all your fractions have the same
denominator, but unfortunately that's not the case with your problem.
You have 12 fractions all having different combinations of numerators
and denominators.

One nice thing with fractions is that any fraction can be converted to
an equivalent decimal by dividing the numerator by the denominator.

For instance, take the fraction 1/10.  In your calculator press 1,
divided by, 10.  1/10 = 0.10

So 1/10 (of a dollar) = 0.10 (this is a dime; 10 cents).

Using our example from above,

1/10 = 0.10
3/10 = 0.30
9/10 = 0.90

10 cents is less than 30 cents, which is less than 90 cents.  This
rank order also confirms our answer from above.

Now if we convert the 12 fractions from above into decimals and think
of the decimals as money, we should have a more intuitive feel for
ordering them.

1/10 = 0.10  (10 cents; smallest)
1/6 ~ 0.167
1/4 = 0.25
3/10 = 0.30
1/3 ~ 0.333
3/8 = 0.375
1/2 = 0.50
3/5 = 0.60
3/4 = 0.75
4/5 = 0.80
7/8 = 0.875
9/10 = 0.90  (90 cents; largest)

Does this help?  Please write back if you need anything else.  :-)

Also, I'd encourage you to search our archives for additional
information on fractions or for any future problems you may have--our
archives are a great resource!

- Doctor Wilko, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 12/12/2008 at 04:46:21
From: Julie
Subject: Thank you (placing fractions in order)

Thank you from the bottom of my heart for the help I so desperately
needed.  What a great explanation, I understood exactly what you
meant!  I just really appreciate that there are people out there like you!
```
Associated Topics:
Elementary Fractions

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