Simplifying Mixed Fractions

Date: 10/01/1999 at 18:34:37
From: Ian 
Subject: Maths (fractions)

Simplify  10 3/4 - 8 1/2 + (2 1/2 - 1 1/8)

I could not do this. Could you send the answer back?

Thanks.

Date: 10/03/1999 at 22:15:54
From: Doctor Sandi
Subject: Re: Maths (fractions)

Hi Ian, 

I'll help you with this one and although I will use different whole 
numbers I will use the fractions in your question so as not to 
confuse you.

Suppose you have a similar question:

     6 3/4 - 4 1/2 + (6 1/2 - 3 1/8).

There is actually quite a lot involved in this question but I'll go 
through it with you step by step and then you can apply it to your own 
question.

The first thing I would do (as always with questions of this type) is 
to get rid of the brackets. The way to get rid of the brackets is to 
perform the calculations within them. So first off we'll look at

     6 1/2 - 3 1/8

These are what are called mixed fractions because each consists of a 
whole number and a proper fraction. What we need to do first is to get 
them into fraction form. It would be really helpful if you could write 
down the math as we go along.

With the 6 1/2, first multiply the denominator (bottom number in the 
fraction) by the whole number (6). The answer is 12. Then to the 12, 
add the numerator (the top half of the fraction). So you have 13. This 
then becomes the numerator of the fraction and the 2 remains as the 
denominator. So you have 13/2. You can tell that this is correct 
because if you divided 13 by 2, you would have 6 1/2, wouldn't you? 
So, so far we have 13/2.

I'll do the next one as well. This one is 3 1/8. So what do we do? 
Multiply the denominator by the whole number, in this case 3. The 
answer is 24. Add the numerator, 25. The denominator stays as 8, so we 
have 25/8.

So the numbers inside the bracket now look like this (13/2) - (25/8). 
Now whenever you subtract or add fractions they ALWAYS have to have 
the same denominator. That means that we have to find a number for the 
denominator that they both have in common. In this case notice that 2 
goes into 8 four times, so we must make the denominator of the first 
fraction 8. Then we can subtract the first fraction from the second as 
required. In order to make the denominator of the first fraction 8, we 
make use of the fact that 2 goes into 8 four times. So, if we multiply 
the NUMERATOR (the top half of the fraction remember) by 4 (making 
52), and the denominator by 4, we will end up with 52/8. This now 
makes it much easier to subtract (or add if that is what you want to 
do).

In this case we have (52/8) - (25/8). To subtract the second fraction 
from the first, all we have to do is subtract the NUMERATORS. So we'll 
have (52-25)/8 = 27/8. 27/8 can actually be made into a mixed number 
but because we have other things to do we won't do that yet; we'll 
leave it in that form for now.

After having done all of this your equation is:

     6 3/4 - 4 1/2 + 27/8 

Now please put the 6 3/4 and the 4 1/2 into fraction form by yourself, 
using exactly the same method that I did earlier. Don't take any 
notice of the negative sign when you do this. Your answers should be 
(27/4) and (9/2). Now that you've got this far, you're doing great.

Finally have the equation all in fraction form and it looks like this:

     (27/4) - (9/2) + (27/8)

Now remember what I said earlier about how when you subtract or add 
fractions, they all have to have the same denominator, so we have to 
look at the denominators and find a number that they have in common. 
This is called the Lowest Common Denominator or (LCD).

In this case 4 and 2 and 8 are related, in that 8 can be divided by 
both 4 and 2. You will find when you come to do your own question that 
the denominators are exactly the same, so what we're about to do now 
can be applied straight to your question as well - that's why I chose 
a question where the fractions were the same.

Now remembering we want the denominator to be 8, for (27/4): 4 goes 
into 8 twice, so 2 x 27 = 54, and 2 x 4 = 8, so this fraction becomes 
(54/8). See if you can give the fraction (9/2) a denominator of 8 
yourself. You should get (36/8). For (27/8) the denominator of this 
fraction is already 8, so leave this one alone - nothing needs to be 
done to it. Now we have all of the fractions over 8:

     (54/8) - (36/8) + (27/8)

Incidentally, this is the same as writing

     (54 - 36 + 27)/8

Either way it is just a calculation problem now: 54-36+27 = 45. This 
is the numerator that goes over the denominator 8. The answer is 
therefore 45/8. Because the numerator is greater than the denominator, 
we can make it into a mixed fraction. Your question was given in terms 
of mixed fractions so your answer should be in terms of mixed 
fractions too.

How many times does 8 go into 45? 5 with 5/8 left over, right? 5 5/8 
is the final answer to the question that I've posed above. I hope that 
you are able to follow it and then to apply it directly to the 
question you have asked about. The procedure for converting a mixed 
fraction into a fraction (actually its called an "improper" fraction 
when the numerator is greater than the denominator) is always the 
same, and so is finding a lowest common denominator so that you can 
add or subtract, so I don't anticipate that you will have any 
problems.

However if you do, please write back to Dr Math right away and we will 
help you. Also, if you have any other questions that you would like to 
ask, please don't hesitate to send them too.

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