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### Simplifying Mixed Fractions

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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.
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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
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

- Doctor Sandi, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Elementary Fractions

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