In this chapter, we're going to start our work with ratio and proportion. I've written over here on the board the definition for the ratio of two numbers:
if a and b are any two numbers, then the ratio of a to b is a/b, where b ≠ 0. So, as you can see, ratios are fractions.
For example, the ratio of 6 to 8 is the fraction 6/8, which of course, we can reduce to lowest terms as 3/4.
So, the ratio of 6 to 8 is equivalent to the fraction 6/8, which reduces down to 3/4. Likewise, the ratio of 15 to 3 is 15/3, or 5/1. Now,
a lot of times when we work with ratios, they'll say 15/3, that looks like this is a fraction, then, that would reduce normally just to 5.
But many times working with ratios, we leave that 1 in the denominator.
Let's go to the board now and try some problems where we take ratios and convert them to fractions.
Problem number 1: we have 64 to 12; and I want to write this ratio as a fraction in lowest terms. So, 64 is in the numerator; 12 is in the denominator.
And then, I simply have to reduce this to lowest terms. So, let's see, I'll divide this numerator by 4 and I get 16;
and I'll divide the denominator by 4, and I end up with 3. So, the ratio of 64 to 12 is the same as the fraction 16/3, in lowest terms.
Now, what we want to do is look at a second problem that involves the ratio of two fractions: so, I have 3/4 to 1/4.
I want to write that ratio as a fraction in lowest terms. Well, this is the same as 3/4,� that will be in the numerator;
and then, that number will be in the denominator. Now, I have a complex fraction. So, I'm going to write this as 3/4 times,
and I'll multiply by the reciprocal just like I did when I simplified complex fractions. Those 4's divide out, and I end up with simply 3/1.
And since I'm working with ratios, I think I'll keep that as 3/1. If I was working with just regular numbers, I might simplify it down to just 3.
But since we're working with ratios, let's leave it as 3 to 1. Here's our next problem: how about the ratio of 2 2/3 to 5/3?
Well, here I have a mixed number; here, I have a fraction. But the idea behind these doesn't change.
It's still 2 2/3 in the numerator, and 5/3 in the denominator. Now, I'm going to change this numerator,
that mixed number into a improper fraction: 3(2) = 6, + 2 = 8. So, I have 8/3 in the numerator divided by 5/3 in the denominator.
Now, instead of dividing by 5/3, I'm going to multiply by its reciprocal. So, I have 8/3 times the reciprocal of this, which is 3/5.
The 3's divide out, and I end up with just 8/5 for that. So, that improper fraction, 8/5, is in lowest terms.
So, I think I'll just leave the problem that way, as 8/5. If I was just working with numbers, I might change this into a mixed number.
But since we're working with ratios, that is a ratio, or a fraction, in lowest terms; we'll just leave it at 8/5. Here's our next one:
how about 2.5 to 3/2, that ratio? Well, here I have a decimal and a fraction.
But still, the first number goes in the numerator; the second number goes in the denominator. So, it's 2.5 over 3/2. Let's see if we can simplify that.
How about 2.5 over, 3/2 = 1 1/2 = 1.5? Let's see, I can multiply numerator and denominator by 10. I would have 25/15.
I'll divide the numerator and denominator by 5, and I'll end up with 5/3 as that fraction in lowest terms.
So, this time, it turned out to be the best thing was to change this fraction into a decimal.
And then get, multiply numerator, denominator by 10 to get a little bit easier numbers to work with; then, reduce to lowest terms,
and I end up with that fraction, again, an improper fraction, but still, it's the lowest-terms equivalent of this ratio, right here.