Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.
What are Ratio Tables?
It was a great presentation. You got to see a real heart beating and watch the simulated blood flow. Then you got an idea. What about your own blood? If your heart really works so hard to push blood through your arteries, you should be able to feel it. Asking your teacher how to find your pulse, you decided to measure and record how fast your heart was beating. You did the test several times, counting your heartbeats for different lengths of time.
|Number of Beats||How Long?|
|150 beats||2 minutes|
|225 beats||3 minutes|
|300 beats||4 minutes|
Every one of those pairs of numbers represents a ratio, which is two numbers or quantities being compared (usually in fraction form, with one on the numerator and the other on the denominator). A ratio table is a structured list of equivalent (equal value) ratios that helps us understand the relationship between the ratios and the numbers. Rates, like your heartbeat, are a special kind of ratio, where the two compared numbers have different units. Let's look at some examples of ratio table problems.
Bike Ride Example
Maybe you like to ride your bike around town, and you're curious whether your speed is consistent during your various adventures. So, as you take each trip, you use a stopwatch to measure how long it takes to make the trip and your bike's odometer to measure the distance. Then, you set up a table, filling in the information you collected: where you went, distance to those various destinations (in feet), and the time it took to make the trip on your bike (in seconds).
|Destination||Time it takes||How far it is|
|Grocery Store||250 seconds||5000 feet|
|Quick Stop||300 seconds||6000 feet|
|Library||1200 seconds||24,000 feet|
|Basketball Court||750 seconds||15,000 feet|
To find out what your average speed is for each trip and to determine if it's the same for every ratio in the table, you'll divide distance by time for each trip.
5,000 feet / 250 seconds = 20 feet per second
6,000 feet / 300 seconds = 20 feet per second
24,000 / 1,200 seconds = 20 feet per second
15,000 / 750 seconds = 20 feet per second
Yup, you're consistent! You tend to travel at 20 feet per second, no matter where you go on your bike. So, assuming that's true, could you figure out how long it would take you to get to a new place? Absolutely!
If you know that your friend's house is three miles away, you could grab your calculator and do some arithmetic. There are 5,280 feet in a mile, so 3 x 5,280 means your friend's house is 15,840 feet away. Dividing by your 20 feet per second, you find that it will take you 792 seconds to get to your friend's house.
In the last example, we had a complete table of values to work with. Now let's complete a table where some of the values are missing.
Since we know that the ratios are equivalent, we can find the missing values. Take a look at Figure 1.
In Figure 1, there are letters in place to represent each value that we're missing. We'll use the ratio that's complete (5/3) to find the other values. Here's how it works:
If you have the denominator (number on the bottom of the fraction), you'll multiply your denominator by the number on top of the complete ratio, then divide by the number on the bottom. This means that we will multiply by 5 and divide by 3. Notice in Figure 1 how you took your 6, multiplied it by the 5, then divided by the 3 to get your missing numerator and complete the fraction (10/6).
On the second ratio, you have the numerator (number on the top of the fraction) and are looking for the denominator. This means you'll have to reverse the process you did last time. Now you'll multiply by the bottom number and divide by the top one. You can see in the figure how you multiply your 15 by the 3 on the bottom, then divide by the 5 on the top. This gives you a 9 for your missing denominator, completing the fraction (15/9).
For the third ratio, you're missing the numerator again, so you do it the same way as the first one. Take your 24, multiply by the 5, then divide by the 3.
Remember, on each of these, the idea is to cross-multiply the known numbers, then divide by the remaining number. Picture the little 'hook' or 'check mark' that appears in the diagram, then remember ''cross-multiply, then divide''.
Filling in the table, we now have:
Let's check our results: 5/3 = 10/6 = 15/9 = 40/24 = about 1.67. We're correct!
A ratio table is a structured list of ratios that are all equivalent (remember that ratios are two numbers or measurements that are being compared). You can check to see if they're equivalent by dividing out each pair. The result should be the same for each one. A rate is a special ratio where the units are different, like beats per minute or miles per hour.
In math problems involving ratio tables, you can find the missing numerator values by multiplying your denominator by the number on top of a complete ratio, then dividing by the number on the bottom. You can find missing denominators by multiplying your numerator by the number on the bottom of a complete ratio, then dividing by the number on top. Remember, always cross-multiply (go top to bottom or bottom to top), then divide by what's left. Once you get it down, ratio tables aren't too bad!
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