6.1+Ratios

Section 6.1 - Ratios

 * A Ratio Compares Similar Things**
 * A ratio describes the relationship between two amounts where the units are generally similar. **

The photo of the soccer game gives us a chance to count people, and to compare those counts to each other. For example, there are nine total players shown in this photo. As we begin to compare different sub-groups, however, it becomes more meaningful. Sometimes we use the word "to" as a comparison term. Here are some comparisons that can be made:

Five South China (red) players **to** nine total soccer players. Four Ranger (blue) team members **to** nine total soccer players. Five South China (red) players **to** four Ranger (blue) players.

We compared "red players" to "blue team members" and to "total players", but in reality, each of these categories could be described as "soccer players". We were comparing one kind of soccer player to another.

There are several different ways to write ratios:



An important skill will be translating between these three forms. You will often find the word form, or a colon, in the problem, but will need to translate into the fraction form to do any calculations. If you look at the examples above, you can see how these three different forms relate to each other.

When translating a ratio into a fraction format, the quantity that comes first is always the numerator and the second quantity is always the denominator.
 * A Ratio as a Fraction**

//Question// - Look at the image of nuts and bolts. If you were going to write a ratio that compares them, which number goes first (numerator) - the nuts or bolts?

//Answer// - It sort of matters and it kind of doesn't. What matters is the units. If you say six bolts to 23 nuts, or 23 nuts : 6 bolts, they are both accurate. Just make sure the fractions match up (don't mix up the numbers and units.)


 * Try a few! **



Last year the Old Dominion Logging Company harvested 93,000 Douglas Fir trees and 38,000 Spruce trees from a site in Western Oregon. In an effort to replace the harvested trees, they planted six seedlings for each tree they took - six firs for each fir tree and six spruce for each spruce.

1. What is the ratio of Spruce trees to Douglas Fir trees that were harvested?

2. What is the ratio of Douglas Fir trees taken to the total number of trees harvested?

3. What is the ratio of Spruce trees planted to the number of Spruce trees harvested?

The last question gives you a fairly large fraction to deal with. 228,000 to 38,000 is a challenging fraction to write, to calculate, perhaps even to comprehend. However, it can be simplified just by reducing the fraction to lowest terms. Using repeated division or any other method for reducing fractions, you could reduce that to a ratio of 6 to 1. That actually makes a lot of sense, because the problem told us that they planted six trees for each one that was harvested.

Although reducing the fraction always helps, and should always be done, this problem brings out a way in which ratios are different than fractions. In order to describe a ratio as a comparison between two quantities, you need to have both a numerator and a denominator. So, even though you could reduce a fraction like 6 over 1 to just 6, you need to leave it in the form of 6 over 1.


 * Always reduce ratios to their simplest form, but make sure that they stay in fraction format, with both a numerator and denominator that are whole numbers. **

Which brings up another interesting difference between fractions and ratios - ratios break the "no division by zero" rule.

Four young gymnasts visited Coach Karolyi's gym in Texas. What is the ratio of female gymnasts to male gymnasts that visited that day? Four to Zero could be written as 4 over 0, which would certainly not be acceptable as a fraction, but is perfectly fine as a ratio.

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 * Proportions**

If two thirds of the people in a room are wearing black shoes and you look at three people's shoes, you would expect to see two pairs of black shoes and one pair that was not black. If you looked at thirty people, you would expect that twenty would be wearing black shoes. The ratio of black shoes to total shoes, 2 to 3, would be consistent regardless of the number of people. There is a special tool that allows you to calculate the same ratio for a different sample size -- it's called a proportion.

A proportion is usually written as two equivalent fractions that are equal to each other, but where one of the quantities is unknown.



If you look at these as equivalent fractions, you can see how the missing value is found. In the first example above, if you multiply the denominator by 10 you get the second fraction, so multiply the numerator by 10 as well and you find that the missing value is 20. In the second example, the numerator is multiplied by 25, so when the denominator is multiplied by 25, you find that 75 is the answer. However, the answers don't always come out evenly, and sometimes the multiplication is more challenging, so there is a better way to solve these problems.

To solve a proportion, use what is called "cross products". The example below shows equivalent fractions and how cross products work.

So, if one of the values is unknown, it works the same way and becomes a "solve for x" problem.

It doesn't matter where the X is -- cross products still works.

media type="youtube" key="k1k4UycrmSU?fs=1" height="390" width="480" See if you can find the proportion in the following problems and solve using cross products: One out of every three people who walk pass the candy shop stops in, though they don't always buy something. If 34 people stop in, how many people walked by? If 420 people walk by, how many would you expect to stop in?


 * Similar Triangles**

Triangles are considered to be "similar triangles" if the sides (and the angles actually) are mathematically proportional. If triangles are similar, you will be able to find the length of one of the sides by knowing the measurement of the corresponding side in the other triangle, as well as the relationship between the triangles. This is found by knowing the ratio of one to the other sides to the corresponding side in the other triangle.




 * It is critical that corresponding sides match up in the proportion! ** However, there are two ways to do this.



Both will result in the same answer. The corresponding sides must either be the two numerators, or the numerator and denominator of the same fraction. As long as they aren't opposite each other, it will work.

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 * Chapter Six Practice Problems**

**Where to from here?**
6.1 Ratios 6.2 Rates 6.3 Percents Chapter Six Summary Chapter Six Homework