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 
on the subject. Please Help...

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 
>on the subject. Please Help...

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
for more advice. Thanks again. 

Your new fan,
 Emerald