In this lesson, you'll learn what ratios are and discover their relationship with fractions. You'll also learn what makes ratios equivalent and look at some examples.
A ratio is a relationship between two numbers (usually involving some kind of measurement). For example, when people drive, they travel at a certain speed. We usually refer to that speed as miles per hour. That's a ratio because it's a relationship between distance and time.
So if you're driving 60 mph, that means that for each hour you drive, you're traveling 60 miles.
There are all kinds of ratios. On a map, there's usually a map scale that might indicate, for instance, that 1 inch = 100 miles. Each inch on the map represents 100 miles in the real world. This is another relationship between two numbers, which makes it another ratio.
Ratios are often written in one of two ways. Let's use our map scale. One way of writing out that ratio is:
We could also write the ratio as a fraction:
What are Equivalent Ratios?
So if you're driving 60 mph (60:1 or 60/1), then in 2 hours, you'll have driven 120 miles. If we wrote out that ratio, it would be 120:2 or 120/2.
60/1 and 120/2 look different. They definitely include different numbers. But they're really the same ratio written different ways. In fact, they're called equivalent ratios, which are ratios that express the same relationship between two numbers.
The ratios 60/1 and 120/2 are equivalent because the relationship between the two parts of the ratios didn't change. According to the ratio 60/1, you travel 60 miles for every hour you drive. That relationship between the two numbers stays the same when we write 120/2. Your still driving 60 miles for every hour you drive. But since you've driven 2 hours now, you've traveled 120 miles. So even though the numbers in 120/2 are different, they still describe the same relationship between distance and time that 60/1 does.
In the same way, 180/3 is also equivalent to 60/1, because you're still traveling 60 miles every hour. The only difference is that you've driven for 3 hours now, so you've traveled a total of 180 miles.
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There's also another connection between 60/1 and 180/3. Look at the first number in both ratios, which is also the numerator (or top number) in the fractions we've written. To get from 1 to 3, we multiply by 3. We do the same thing with the denominator (or bottom number) in both fractions: we multiply by 3 again. So when you multiply both parts of a ratio by the same number, you make an equivalent ratio.
All we're really doing is making equivalent fractions, which are two different fractions that are equal.
We could, in fact, multiply the numerator and denominator by any number and get an equivalent fraction. If we wanted to, we could even multiply our ratio by fifty over fifty:
So when we're dealing with a ratio, if we're multiplying (or dividing) both parts of it by the same number, we're creating equivalent ratios.
Johnny and Michelle are running a race. In the time that Johnny ran 2 meters, Michelle ran 6 meters. We can set up a ratio of Johnny's distance to Michelle's distance as 2:6 or 2/6.
What if we want to know how far Michelle ran when Johnny had run only one meter? Well, in our original ratio, Johnny ran 2 meters. To go from 2 meters to 1 meter, we divide by 2. So we'd have to divide Michelle's distance by 2, as well.
The distance Michelle ran in this new ratio is 3 meters. So when Johnny had run 1 meter, Michelle had run 3 meters. Look out, Johnny!
A ratio is a relationship between two numbers, such as miles per hour or inches per mile. It can be written two ways; for example, 1:10 or 1/10.
Equivalent ratios (which are, in effect, equivalent fractions) are two ratios that express the same relationship between numbers. We can create equivalent ratios by multiplying or dividing both the numerator and denominator of a given ratio by the same number.
Equivalent ratios are fractions used to convert between units by representing equal measures. The exercises below will be used to assist you in further understanding the use of equivalent ratios and their role in conversions. Solutions are provided below
1. Are the fractions 3/20 and 9/60 equivalent?
2. What is the numerator of 3/65? What is the denominator?
3. If John takes 4 hours to do his homework, how many minutes did he take? What is the equivalent ratio to convert between minutes and hours?
4. Setup and solve the equivalent ratio for the following scenario. If Jamie was going on a road trip across the country and drove at a constant pace of 75 miles per hour, how many hours did it take him to travel 975 miles?
5. Setup and solve the equivalent ratio for the following scenario. If Betty makes $10.50 an hour, how much money will she make after working 80 hours?
6. Is 2/9 equivalent to 6/20?
7. Setup and solve the equivalent ratio for the following scenario. Justin can eat ten mini cheeseburgers as one meal. If they cost $1.25, how much does his meal cost?
1. Yes, the 3/20 is a reduced form of 9/60
2. The numerator is 3, the denominator is 65
3. It will take him 240 minutes, 1 hour = 60 minutes so the ratio is 1/60
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