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### Borrowing and Subtracting with Mixed Numbers

```Date: 09/03/2008 at 16:40:09
From: Emerald
Subject: 3 1/3 - 2 2/5

How do I make the first numerator larger than the second numerator
when I'm subtracting mixed fractions?

On one website it says: Make the first numerator larger than the
second, and it gives this example: 5 1/3 = 4 4/3. I don't understand
how 5 1/3 became 4 4/3. I looked for a step by step method on how
this was done but I couldn't find one.

I would normally start solving this problem: 3 1/3 - 2 2/5 like this:
Starting with 3 1/3, I would multiply the denominator (3) with
the integer (3) then add the numerator (1) and end up with 10/3
(3x3=9+1=10) Then I would do the same thing with 2 2/5 (5x2=10+2=12)
which leaves me with 12/5. Now I'm at 10/3 - 12/5. This is where I
get totally lost. I think I'm supposed to multiply the two
denominators together (3x5=15). And then subtract the two
numerators which leaves me with something like an improper 8 (10-
12=8) leaving me with 8/15. Now I strongly feel like this is
incorrect, because I believe my answer should either be a mixed
fraction or just a smaller proper fraction. My brain is quite frozen

```

```
Date: 09/03/2008 at 18:47:01
From: Doctor Ian
Subject: Re: 3 1/3 - 2 2/5

Hi Emerald,

You wrote:

>How do I make the first numerator larger than the second numerator
>when I'm subtracting mixed fractions?

The same way you would do it if making change.  Suppose you have 4
dollar bills, and 3 dimes, and you want to give someone 1 dollar bill
and 7 dimes.  You could trade 1 dollar bill for 10 dimes, giving you 3
dollar bills and 13 dimes.  Then it's easy, right?

Same thing here.  Suppose I want to subtract

4 3/10 - 1 7/10

I can change the initial number this way:

4 3/10 = 4 + 3/10

= 3 + 1 + 3/10

= 3 + 10/10 + 3/10

= 3 + 13/10

= 3 13/10

So now I have

3 13/10 - 1 7/10 = ...

which I can do easily.

It's really the same thing we do when subtracting:

43
- 17
----

I trade 1 group of 10, for 10 "groups" of 1,

3 [13]
- 1   7
--------

and now I can proceed.

So you see, it's all the same idea, over and over again--just
breaking up groups, so we can use the pieces.

>On one website it says: Make the first numerator larger than the
>second, and it gives this example: 5 1/3 = 4 4/3. I don't
>understand how 5 1/3 became 4 4/3. I looked for a step by step
>method on how this was done but I couldn't find one.

Does it make more sense now?  One key to this is remembering that a
mixed number is really an implied addition:

5 1/3 = 5 + 1/3

We just leave out the "+" as a convenience.

>I would normally start solving this problem: 3 1/3 - 2 2/5 like
>this: Starting with 3 1/3, I would multiply the denominator (3) with
>the integer (3) then add the numerator (1) and end up with 10/3
>(3x3=9+1=10) Then I would do the same thing with 2 2/5 (5x2=10+2=12)
>which leaves me with 12/5. Now I'm at 10/3 - 12/5. This is where I
>get totally lost.

This is a perfectly valid way to do it.   You just need to find a
common denominator--one that is divisible by both the denominators
you have.

The smallest number divisible by 3 and 5 is 15, so you'll multiply the
first fraction by 5/5 (which is the same as multiplying by 1, so the
appearance changes, but the value doesn't),

10   5   50
-- * - = --
3   5   15

And you'll multiply the other fraction by 3/3,

12   3   36
-- * - = --
5   3   15

And now you just subtract the numerators:

10   12   50   36   50 - 36   14
-- - -- = -- - -- = ------- = --
3    5   15   15      15     15

As a check, we can think about whether that's in the right ballpark.
We're subtracting 2 and something from 3 and something, so we should
get an answer around 1.  Which we did.

>I think I'm supposed to multiply the two
>denominators together (3x5=15). And then subtract the two
>numerators which leaves me with something like an improper 8 (10-
>12=8) leaving me with 8/15. Now I strongly feel like this is
>incorrect, because I believe my answer should either be a mixed
>fraction or just a smaller proper fraction. My brain is quite frozen

Note that you could have also just changed the fractions to have
common denominators,

3 1/3 - 2 2/5

= 3 5/15 - 2 6/15

= 2 20/15 - 2 6/15

= (2 - 2) + (20/15 - 6/15)

= 0 + 14/15

= 14/15

Does this help?

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

```

```
Date: 09/18/2008 at 16:49:28
From: Emerald
Subject: Thank you (3 1/3 - 2 2/5)

Thanks man. That was awesome! I used this knowledge when I took my
placement exam and got a high enough score so that I didn't have to
pay for a remedial math class. I'll definitely be coming back asking

Emerald
```
Associated Topics:
Elementary Fractions

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