GEAR UP FOR LIFE
MATH MENTOR:
Intro to Ratios
We use ratios to make comparisons between two things.
When we express ratios in words, we use the word "to." We say "the ratio of something to something else." Are you hungry? Look at the items below.
Ratios can be written in several different ways: in words, as a fraction, and with a colon.
6 pieces onion to 8 pieces of meat = 6/8 =6:8
| 16 graduates to 85 undergraduates | ||
| 4 male workers to 8 female workers | ||
| 7 dogs to 27 cats | ||
| 8 technicians to 58 computers |
Express the following relationship in another form.
A foot has 12 inches.
Two yards have six feet.
There are 6 girls for every 2 boys in the class.
I'll pay you six cents for every dollar I make on the sale.
Now you are ready to apply what you know about ratios.
If a foot has twelve inches? How many inches do 2 feet have? Yes, 24. It makes sense that if you want to keep the same ratio, you will multiply or divide both sides by the same amount. That's what you did to get the answer.
For example:
If you have 12 eggs to a
carton (12:1), you will have 24 eggs in 2 (24:2) cartons, and 36 eggs in
3 cartons (36:3). When you multiply or divide each side of the ratio by
the same amount, you have different numbers but the ratio, the
relationship, stays the same. Ratios that change numbers by equal
amounts are called equal ratios. They are also called
proportions.
A proportion is a set of numbers
that means the same thing, the same amount.
A proportion is
an equation that states that two ratios are
equal, such as
12/1 =
24/2 = 36/3 These are equal ratios or
proportions.
Which rows below have the same ratios of gas to oil?
| GAS |
OIL |
|
| 1 |
 |
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| 2 |
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| 3 |
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| 4 |
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| 5 |
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What would be the gear ratios in the following gears when they are connected?
1. 70 teeth and 25 teeth
2. 80 teeth and 20 teeth
3. 90 teeth and 30 teeth
If you know what one ratio is, you find the value in an equal ratio by multiplying or dividing.
You know that you have to add one cup of milk to every two cups of flour in a recipe to feed 4 people. You plan on 8 people. What do you need to do? Sure, double the recipe: 2 cups of milk and 4 cups of flour! You just created a proportion!
Now let's approach this from another point of view.
You know that your engine requires 2 ounces of oil for every gallon of gas. Your engine holds 3 gallons of gas. How much oil do you need to fill our tank? Apply what you know. Multiply both sides by equal amounts.
2/1 so ?/3
That's one way to find a missing number. To always be able to find the missing number in a proportion, you need a process.
Let's turn that into a process that you can use.
So here's the process: Apply
Cross Products
Find the missing number in a proportion
Yes, cross products cross. And yes, they multiply (product=multiplication).
LOOK AT THE FOLLOWING PROPORTION
How do you know the statement is a proportion? If you reduce (divide to the smallest number) both sides, they will equal 1/3.
15 ÷ 5 = 3 5 ÷ 5 = 1 = 1/3
6 ÷ 2 = 3 2÷2 = 1 = 1/3
Proportions reduce to the same fraction for each ratio. 15:5 = 6:2. Fifteen to five is the same ratio as 6 to two.
When you understand that rule, you can begin to solve problems. You apply cross products to find a missing number.
To find the missing number, which we will call X...
Step 1. State the proportion in fractions
5/15 = x/3
Step 2. Multiply the opposites: top of one X bottom of the other; left of one x right of the other. The numbers should always equal the same if they are equal proportions.
So x must be1 since 15 X 1 = 15
Try it: A welding class has 8 men and 2 women. The auto mechanics class the same ratio among its students. It has 10 men. How many women?
8/2 = 24/x
8 X 6 = 48 so X = 6
If you are not sure about how to figure out the value of X when you have an equation like 8x=48, you can divide.
48 ÷ 6 so x = 6 Two different ways for finding the same answer. There are many ways to find answers. Your way is always right if the answer is the same!
And that's how easy it is. Now practice working with lots of ratios.
