Fractions and Ratios

6/5 of all people have trouble with fractions. That's a joke; I'm not sure where it came from. 6/5 of all people is more people than there are. Now that I have confused you, let's talk about fractions.

These are fractions: 1/2, 3/7, 21/8, 3/6, .013, 1.3%. Some people have trouble understanding fractions. Let's look at one of these (3/7) a little closer, and see just what a fraction is. Let's take a wiener (cooked) and divide it into 7 pieces:

Each piece is a fraction (1/7) of the original wiener. Now, let's say that I eat 3 of these pieces, and save the remaining 4 for later. I have eaten 3/7 of the wiener, and I am saving 4/7:

That is what fractions are, numbers of pieces of a certain size. Instead of cutting the wiener into 7 pieces, we may decide to just make one cut, after we measure it with a ruler. We still end up with the same fractions:

Let's look at some of the other fractions from the first paragraph. 21/8 is an improper fraction, one that is greater than 1. Sometimes we want to leave it like that, as arithmetic may be simpler with the improper fraction. More often, we want to "simplify" it"

21/8=16/8 + 5/8=2+5/8 (2 5/8)

16/8 is 2. That idea is fairly simple. But the arithmetic may not be so simple. 8/7 is easy (1+1/7). But 21/8 is tougher. But you should be able to see what is going on.

This brings up the fact that fractions imply division. 1/2 is one divided by 2. 16/8 is 16 divided by 8, which gives us 2. 21/8 is 2 with a remainder of 5. And that is what that remainder (in grade school) was all about. It was the numerator (top part) of a fraction.

3/6 is reducible: 3/6=1/2. We want to reduce fractions when we can, because a reduced fraction is normally much easier to use than the unreduced fraction. 3/6=3/2x3. We say that we "cancel" (divide) the threes above and below:

Technically, we are dividing top and bottom by 3, the greatest common factor (divisor). We get this from algebra: xy/xz=y/z. A handy way to reduce fractions is to use Euclid's Algorithm.

.013 (or 0.013, which many people think is clearer in order to distinguish decimal points from periods) is a decimal (base 10) fraction. It is 13/1000. Here is a number with this same fraction on the end:

Decimal notation just makes fractions like this (13/1000 or 3/10 or whatever) particularly easy to write. And arithmetic is often easier with decimal fractions. Notice that .013 is 13/1000, not 13/100.

Percents are decimal fractions. "Per cent" is Latin for "divide by 100." 1.3% is 1.3/100 or .013. 20% is 20/100 or 2/10 or .2.

We often do arithmetic with fractions:

1/2 x 2/3=2/6=1/3

When we multiply by a proper fraction, our answer gets smaller:

1/2 x 8=4

In algebra, we see that a/b times x/y=ax/by. We just multiply numerators (upper parts) and multiply denominators (lower parts), and then maybe reduce. We can also "cancel" (divide) before we multiply:

Addition and subtraction is often more difficult:

1/2 + 2/3=?

We need a "common denominator":

1/2 + 2/3=3/6 + 4/6=7/6=1+1/6

You see that the addition is simple when the denominators are the same (3/6+4/6). The tough part is converting each fraction so that the denominators are the same (again see Euclid's Algorithm).

Division is as easy as multiplication. But when we divide by a fraction, we turn it upside down and multiply:

Ratios are fractions. Let's say that I have two bags of marbles; one has 12 marbles; the other has 4. The ratio is 12 to 4 (12:4 or 12/4). Normally, with ratios, all we want to do is reduce them: 12/4=3/1. So our ratio is 3 to 1 (3:1 or 3/1). We may have wanted to name the smaller bag first. Then the ratio is 1 to 3. To make my meaning clear, I may say "The ratio of the smaller bag to the larger bag is 1 to 3."