TEK 6.4 C/D, TEK 6.5 A - Ratios
TEK 6.4 C Learning Goal: I can give examples of ratios as multiplicative comparisons of two quantities describing the same attribute.
TEK 6.4 D Learning Goal: I can give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
TEK 6.5 A Learning Goal: I can represent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions.
Vocabulary
Ratio: Comparison of two quantities, or numbers.
Rate: Comparison of two quantities, or numbers, that have different units.
Unit Rate: A rate that, in its fraction form, has a denominator of 1.
Unit Price: The cost per item of a product, similar to unit rate.
Proportion: A statement that two ratios are equal.
Scaling: Multiplying or dividing two related quantities by the same number.
Scale factor: The number by which you multiply while scaling.
Ratio table: A table whose columns all have pairs of numbers that have the same ratio.
Equivalent ratios: Ratios that express the same relationship between quantities. You can write a proportion between them.
Ratios
A ratio is a multiplicative comparison of two quantities. Right now, we will just focus on quantities that have the same units, because that is typically what we are talking about when we mention ratios. Often, the two quantities being compared in a ratio are just objects that don't even have units. Let's say a construction crew was building a house, and they decided that they need 3 hammers for every 12 nails.
This is an example of a ratio because two quantities (3 and 12) are being compared. When we said that ratios are a multiplicative comparison, that means that the two quantities will have a multiplicative relationship (see the lesson before this for help on multiplicative relationships.) This means if we were to get 3 more hammers, we would need 12 more nails, which creates a total of 6 hammers and 24 nails.
If the relationship is multiplicative, then it will run through zero on a graph. With (0, 0) and (3, 12) as coordinates, you can actually determine the equation for this relationship. Again, go to the lesson before this if you need help with multiplicative relationships.
You can keep adding as many hammers and nails as you want, as long as you follow the same multiplicative relationship. There are four main ways to write a ratio:
Words
We used words to describe a ratio in the example above. When we said there are 3 hammers for every 12 nails, that was describing the ratio in words. What this means is that every 3 hammers has a set of 12 nails attached to it. If we had 12 hammers, there would be 4 sets of 3 hammers. So, there would be 4 sets of 12 nails (48 nails) that would have to go with the 12 hammers.
Another way ratios are written with words is using the word "to". You might write "3 hammers to 12 nails", or, more commonly, just "3 to 12". This means the same thing as when we wrote it out in a sentence.
Colon
Using a colon to represent ratios is almost the same as using the word "to" (such as in "3 to 12"). You just replace the "to" with a colon. The ratio would now be 3:12.
Fraction
To write a ratio as a fraction, you take the first number (before the colon) and put it over the second number (after the colon.) For 3:12, you would have 3 over 12 because 3 is before the colon and 12 is after it.
If possible, simplify the fraction. The greatest common factor of 3 and 12 is 3, so divide both the top and the bottom by 3:
Even if you simplify a ratio, it is still the same ratio. A ratio and all of its unsimplified forms are all equivalent, or equal.
Decimal
Remember that fractions basically just represent division problems. So, if you want to write a ratio in decimal form, you actually do the division problem. The fraction 1/4 is just 1 divided by 4, so do that operation. You probably already know that 1/4 is equal to 0.25. For different problems, you may have to do long division, use a calculator, or use prior knowledge you have from experience. No matter how you get there, 0.25 is our ratio.
It doesn't matter if you simplify the ratio before dividing. You could have just used 3/12 and you would end up with the same answer.
Order Is Important!
In all these forms of ratios, order is very important. It may not seem important in the written form, but once you start dividing numbers, you want to make sure you have the right numbers on top and on bottom. If you have this ratio, where A and B both represent quantities, or numbers:
A will always be first or on top, and B will always be second or on bottom.
If you are writing the ratio using a colon, A will be first and B will be second:
If you are using a fraction, A will be on top:
When you are finding the decimal form, remember to put A first in the division problem and B last, or put A in the dividend spot and B in the divisor spot.
Rates
Rates are a special type of ratio that compare quantities with different units. Most often, one of the quantities in a rate will have a unit of time (seconds, minutes, hours, etc.), but not always. The other quantity could be anything - distance (miles, feet, meters, etc.), heartbeats, amount of items sold, etc.
Unit Rate
Most often, rates are written as unit rates because they are easiest to understand. Unit rates are rates in which the second quantity being compared, or the quantity on the bottom in the fraction form, is equal to 1 unit. Unit rates are typically written as the first quantity per 1 unit of the second quantity.
For example, if your family is on a road trip, and you notice that you guys have driven 180 miles in the last 3 hours, your rate can be expressed as 180 miles to 3 hours, or 180/3 in fraction form. That sounds kind of weird because most people don't write it like that. Most people simplify the fraction to 60/1 (divide both the top and bottom of 180/3 by 3, the greatest common factor). Then they write it as 60 miles per hour, instead of 60 miles to one hour. The word "per" basically represents a division symbol (/). 60 miles per hour is the unit rate, and the phrase "miles per hour" probably sounds familiar to you because unit rates are used a lot in everyday life.
Unit rates are always expressed using "per" or a division symbol in place of "per." To find a unit rate, just simplify the rate until the bottom number is one.
Unit Price
A specific type of unit rate is the unit price. The unit price is the cost per one unit of a product (cost/unit). Unit price is one of the cases where a rate can have no unit of time. You find a unit price exactly the same way as a unit rate: simplify until the denominator of the rate in fraction form (which should be where the product quantity is) is one.
Let's say 4 movie tickets cost $36.60. Movie tickets is the product being sold here, so the rate is $36.60/4 tickets because the rate should follow this form: cost/unit. Now all you need to do is simplify until the number of products (4) is one:
Proportions
A proportion is a statement that two ratios are equal, or equivalent. Usually, both ratios will be written in fraction form and set equal to each other, like this proportion about how many Oreos are eaten in a given time:
This proportion says that 5 Oreos eaten in 1 minute is an equal ratio to 10 Oreos eaten in 2 minutes. If you simplify 10/2 by dividing the top and bottom by the greatest common factor, 2, you do get 5/1, so these ratios really are equal.
You can easily check that a proportion is true by cross multiplying. Cross multiplying is multiplying the numerator of the first fraction by the denominator of the second fraction, and multiplying the denominator of the first fraction by the numerator of the second fraction. The number on each side of the equal sign after you have multiplied should be the same.
In our Oreo problem, we can also cross multiply.
We end up with 10 on both sides, so we know this proportion is correct.
Sometimes, you will have to find a missing part of the proportion. Let's say you went to the store and you were able to buy 15 apples for $12 total. How much would 5 apples cost? First, set up the ratios:
Now, cross multiply:
You need both sides to be equal, so you need to figure out what number is multiplied by 15 to equal 60. That is the same as asking what 60 divided by 15 is (4). Or, you can just divide both sides by 15:
15 divided by 15 is just one and 60 divided by 15 is four, so the ? becomes equal to 4. $4 is how much 5 apples would cost.
Ratio Tables
A ratio table is a table filled with columns that each have pairs of numbers with equivalent ratios, meaning they all simplify to the same basic ratio. The first number in the ratio goes on the top row, and the second number goes on the bottom row, so that for each column you are dividing the number on top by the number on bottom to find the ratio. Take this ratio table of how much pizza is needed for a pizza party based on the number of kids that attend, for example:
Each column has the same ratio when you divide the number in the top row by the number in the bottom row. The first row has a ratio of 3 kids/5 slices. The second row has a ratio of 6 kids/10 slices. 6/10 reduces to 3/5 if you divide the top and bottom by 2, the greatest common factor. So, 3/5 and 6/10 are equivalent ratios. 9 kids/15 slices reduces to 3/5 as well when you divide the top and bottom by 3, the greatest common factor. So, all three columns represent equivalent ratios.
Since each column represents an equivalent ratio, then any two columns can be set equal to each other to make a proportion. If we wanted to use the first two columns, for example, it would look like this:
You can cross multiply to make sure the ratios are equal:
Scaling
Scaling is multiplying or dividing both parts of a ratio by the same amount. The scale factor is the number you multiply the ratio by when you are doing this. Across a ratio table, each column is a scaled version of the first column. In the pizza example:
The first column is the most simplified version of the ratio, 3/5. As you go right across the table, each column is scaled differently. The second column has a ratio of 6/10. If you multiply both the top and bottom of 3/5 by 2, you get 6/10. Since you multiplied both parts of the ratio 3/5 by the same number (2) to get 6/10, 6/10 is 3/5 scaled by a factor of 2. The third column, 9/15, is scaled by a factor of 3, because you multiply the top and bottom of 3/5 by 3 to get to 9/15.
If the table was reversed:
You would divide instead of multiply as you move from left to right across the table. 9/15 divided by 1.5 on both the top and bottom would give you 6/10, and 9/15 divided by 3 on both the top and bottom would give you 3/5.
You may be wondering what the scale factor is if you are dividing instead of multiplying. Dividing by a fraction is the same thing as multiplying by its reciprocal, so the scale factor is the reciprocal of the number you divide by. For example, if you are moving from 9/15 to 3/5, you divide the top and bottom by 3, so the reciprocal of 3, which is 1/3, is the scale factor. Look at the lesson on multiplying fractions if you need help with this.
To find a scale factor, you can always divide the final or new number by the original number.
Sometimes, you will have a ratio table that is only partly complete, like this one describing a girl's heartbeats over time:
If each column is a scaled version of the first, then if the bottom of the second column was multiplied by 5, so will the top:
You could also set up a proportion between the two columns, if you wanted to.
Sometimes, it won't be this easy. Take this table, for example:
It isn't clear what the scale factor should be here because there is no whole number you can multiply 8 by to get 10. What you have to do in these cases is scale down and then up. Scaling down using the greatest common factor is always a good start.
For the milk problem, divide both the top and bottom of the first column, 8/80, by 8, the greatest common factor between 8 and 80. Now the ratio is 1/10.
Next, we will scale back up because we need the top of 1/10 to equal 10. So, we will multiply both the top and bottom by 10 because 1 times 10 is 10. We now have the ratio 10/100 for the third column:
So, 10 ounces of milk is 100 calories.
Graphs
In the lesson before this, we talked about multiplicative and additive relationships. Since ratios are multiplicative relationships, they can also be expressed in a graph form. To make a graph, put the first number in the ratio (on top of the fraction or on the top row of the table) on the x-axis, and the second number (on the bottom of the fraction or on the bottom row of the table) on the y-axis. The relationship between the x and y values of any coordinate on the graph should be the same ratio. Here is the extended ratio table for the milk problem:
And here is the graph that goes along with it:
Graphs of ratios:
- Are always straight lines
- Always go through the point (0, 0)
- Always have the first/top quantity on the x-axis, and the second/bottom quantity on the y-axis
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