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Fractions
Fractions are made up of two parts: a numerator and a denominator. The numerator is written as the
top part and the denominator is written as the bottom part... like so: N / D
You can think of fractions in a couple of different ways. One is like slices of a pie (or pizza, if you like
pizza!) The denominator (bottom) is the number of times you slice the pie so that every slice is the same
size. The numerator (top) is the number of slices of that particular size that you have.
Another way to think of fractions is as a division problem. Basically, you divide the top by the bottom.
One half, then, is 1 divided by 2 (two goes into one). This way of thinking about a fraction is really
helpful when you start thinking about changing a fraction to a decimal! ½ = 0.5 Why? Because 2 goes
into 1.0 (which looks a lot like 10, don’t y’ think?!!!) 0.5 times.
There are basically three classes of fractions: proper fractions, improper fractions, and mixed numbers.
A proper fraction is a smaller number over (divided by) a larger number (e.g.: 4/7). An improper fraction
is a larger number over (divided by) a smaller number (e.g.: 7/4). A mixed number is what happens
when you take an improper fraction and split it up into its whole number and proper fractional
components (parts) (e.g.: 7/4 is the same as 1 and ¾).
You can write every whole number as a fraction.
1 = any number divided by itself: 1/1, 2/2, 3/3, 4/4, and so on
2= twice any number divided by itself: 2/1, 4/2, 6/3 and so on
The rest follow a similar pattern (do you see how fractions are basically division problems?).
Equivalent Fractions
Equivalent fractions are different ways to use numbers to write the same sized piece of pie. To convert
a smaller fraction to a larger equivalent fraction, simply multiply that fraction times 1 . . . and by 1, I
mean the fractional equivalent of one that will give you the denominator you’re looking for.
Here’s how: say you have the fraction 2/5 and you want to make the denominator be 100. What
number do you multiply 5 by to get 100? That’s right, 20! So far, what you have is 2/5 * ?/20 = ?/100
Remember, we’re multiplying 2/5 by some fractional equivalent of 1, so really, we’re multiplying 2/5 by
20/20. The problem now looks like: 2/5 * 20/20 = 40/100 Whaddayaknow, 2/5 of $1 is $0.40 which is
what we got by converting 2/5 to an equivalent fraction, 40/100!!!
1/2 = 2/4 = 3/6 = 4/8 = . . .
1/3 = 2/6 = 3/9 = 4/12 = . . .
2/3 = 4/6 = 6/9 = 8/12 = . . .
Adding Like Fractions (Fractions with the same denominator)
When the denominators of a fraction are the same, all you have to do is add the numerators! The
denominator stays the same!
1/3 + 2/3 = 3/3 = 1
Adding Unlike Fractions (Fractions with different denominators)
If you try to add fractions with different denominators, you have to make the denominators the same
before they may be added together.
1/2 + 1/3 = ?
First, find an equivalent fraction for each basic fraction so that both denominators are the same.
1/2 = 2/4 = 3/6
1/3 = 2/6
In this case, 6 is the least common (means the same; they both “have something in common”)
denominator (least common denominator will henceforth be referred to as LCD)
Then change the problem so that the fractions with different denominators now are equivalent fractions
with the same denominator!
1/2 + 1/3 = 3/6 +2/6
Now you may add the two fractions together! 5/6 is the answer.
Changing a Mixed Number to an Improper Fraction and Back Again
If you remember, mixed numbers are of the form 1 ¾ (with a whole number and a fraction) and
improper fractions are of the form 7/4 (with the number of slices being more than the number of cuts).
I will now show you how a mixed number, like 1 ¾ is equal to 7/4 in two ways. Then, you’ll learn a
shortcut.
First, think about pizza again (or pie, yes, I know you’re hungry now. . .). If you have a whole pizza, cut
into four slices (quarters), and three more quarters of another pizza, how many total slices of pizza do
you now have? That’s right, seven slices of pizza cut into quarters. . . 7/4!!!
OK, looking at the same thing from a different angle, I will prove that 1 ¾ = 7/4 using fractions.
Remember that 1 may be rewritten as a fraction (e.g.: 1/1). I’m going to set up the problem as follows:
1/1 + 3/4 = 7/4 If you recall, to add fractions, the denominators must be the same, and I want the
bottom of 1/1 to be 4. I’m going to multiply 1*1 to get 1 like this. . . 1/1 * 4/4 = 4/4 Now the problem
reads: 4/4 + 3/4 = 7/4 Now, the denominators are the same, so I can add the numerators! 4 +3 = 7;
therefore 7/4 = 7/4 which is true! Tada!!!
(The Shortcut)
Here’s the shortcut I promised. You will convert a mixed number to an improper fraction and back
again. Here’s the mixed number: 2 ½
Take the denominator of the fraction and multiply it by the whole number, then add the numerator.
That number will be the numerator of the improper fraction, while the denominator stays the same.
2 * 2 +1 = 5 therefore, 5/2 is the improper fraction of 2 ½!!!
Now, you’ll take an improper fraction and convert it to a mixed number. Here’s the improper fraction:
10/3.
Remember that a fraction is the same thing as a division problem, so simply divide 10 by 3 and you get 3
with a remainder of 1. The remainder is written as a fraction where the denominator stays the same as
it was with the improper fraction: 1/3 Therefore, the improper fraction 10/3 is the same as the mixed
number 3 and 1/3.