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Fractions
Addition
In order
to add fractions, you must have a common denominator. Another way to say this
is that you must have the same unit (or label).
I usually ask my students the following question: “What’s 1 + 1?”. Of
course, you might think the answer is 2, but consider the related question,
“What’s 1 minute + 1 second?”. The answer is not two. Why not? They are not
the same units. In order to add them, you must first convert them to the same
unit. Thus, we change 1 minute to 60 seconds, then add to get 61 seconds.
For instance, take 1/4 + 1/8 . The top number (the numerator) tells you
how many of those items you have (you have one “fourth”), while the bottom
number (the denominator) is the type of piece you are working with (a “fourth”
of a pie or pizza).
Now, fourths are not the same as eighths. If you draw a picture, a fourth
of a pie is twice as big as an eighth (as long as we are talking about the same
pie, that is).
You can change fourths to eighths real easy, though. Take the fourth and
cut it down the middle, and you will get two “eighths”. That’s right, 1/4 =
2/8. Now add the other eighth to get 3/8.
How about a tougher one? 3/5 + 1/7. Now, fifths are not the same as
sevenths. Can fifths be easily changed into sevenths? The answer: no. But
... they both can be changed into 35ths, otherwise known as a “common
denominator”. Now 3/5 has to be changed into 35ths. How do you do that? Cut
each piece into 7 slices, but that means you must cut the 3 pieces into 7 slices
as well. Thus, 3/5 = 21/35 ( a former student of mine gave this a most
amusing name of “upducing” since it is the opposite of “reducing”). Change 1/7
into 35ths by multiplying the numerator and denominator by 5 to get 5/35.
Now we can them: 21/35 + 5/35 gives a total of 26/35.
Next question: Why don’t you add the denominators as well? How come
21/35 + 5/35 doesn’t give 26/70? After all, you add the numerators, but why not
the denominators? Answer: the denominator is like the label: 1 second + 3
seconds = 4 seconds (same label or unit).
How do tougher fractions work? Answer: same way.
Let’s take 1/x + 1/y. First of all, notice that they are two fractions
that are added, thus we still must have a common denominator.
But, what do both ‘x’ and ‘y’ go into? After all, we don’t even know what
number the variables stand for, do we?
The same principles apply: what did 5 and 7 both go into? Answer: 35
(or 5 times 7). Thus,we take ‘x’ times ‘y’ to get a denominator of ‘xy’.
What about the numerator? Well, to the first fraction, we need to multiply
the top and bottom by ‘y’, thus we get y/xy. The second one has to be
multiplied by x/x , thus we get x/xy. Now we can add (yippee!)
y/xy + x/xy yields (y + x )/ xy
Now I know that this doesn’t give you a number as an answer, but keep
in mind that I started with variables, so it’s not too surprising that my
“answer” has variables in it too.
Basic rule: get a common denominator by multiplying numerator and
denominator by the same number, then add the numerators.
Subtraction
Same as addition, just take away instead of plussing.
Multiplication
This is quite different. You do not need a common denominator. You
can multiply straight across.
1/3 * 1/4 is 1/12. How is that possible? Well, a third of a fourth
would yield a twelfth. In other words, if you take part of a part, you get an
even smaller part of the original.
What about 2/3 * 6/7? Well, you could take 2 * 6 to get 12 in the
numerator, and 3 * 7 to get 21 in the denominator. But I usually make the
students always reduce (12/21) to lowest terms. That means to take out a
common factor, in this case 3. Well, you know what? You can take out that
common factor BEFORE you multiply instead of AFTER multiplying. It’s commonly
called “cross cancelling” (not cross multiplying). You are cancelling a common
factor of 3. We cancel out the 3 and put a 1 (since three goes into three once)
and we cancel out the 6 and put a 2 (since three goes into six twice).
Then you just have 2 * 2 in the top (4) and 1 * 7 in the denominator
(7). You still get 4/7, but I think it’s a lot quicker and easier.
What about variables?
How about 2/x * x/4 ?
Since it is multiplying, you don’t need a common denominator, and you could
just multiply across.
But are there common factors we can cancel?
Look at the 2 and the 4. They both have a common factor of 2, so divide
both of them by 2. That leaves a 1 on the left and a 2 on the right.
What about the ‘x’ and the ‘x’? Since they are the same number, they
cancel out right away, leaving just 1’s.
Thus you have 1/1 * 1/2, or just 1/2.
Doesn’t it matter what number ‘x’ is? Answer: no, since they are both the
same number, they will cancel out no matter what they are. The only time you
get in trouble is when ‘x’ might be 0, since you can’t divide by 0.
Division
Guess what?
Division is the opposite of multiplication, so they should be related, and they
are!
You don’t need a common denominator, and you can go straight across. The
problem is that division is so different that hardly ever does it work out
nicely going straight across.
Take 1/4 divided by 1/2. In other words, how many halves does it take to
make a fourth? Let me remind you that 6 divided by 3 is 2 because it takes 3
two’s to make 6.
The simplest way is to invert and multiply by the reciprocal.
What?
Keep the first fraction the same (1/4) but multiply it by the reciprocal
of 1/2 (which is just a fancy way of saying “switch the numerator and
denominator” .
Thus you have 1/4 * 2/1. Guess what? Here we go again with
multiplying! Cross cancel the 4 and the 2, and you’ll have
1/2 * 1/1 to get an answer of 1/2.+
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