Section 22.5 Going Deeper: Proportional Reasoning
In this section, we focused on the standard type of percent problem and some minor variants of it. With some of the more complicated problems, you had to do a bit more thinking to correctly identify the components of the basic percent relationship. This way of thinking about percents is a specific example of a broader framework known as ratios. A ratio is a general type of mathematical relationship where two (or more) quantities are held in a constant proportion with each other.
For example, if you are buying packages of hot dog buns where each package contains 8 buns, then there is a constant ratio between the number of packages you purchase and the total number of buns you have. We can set this up using the following word equation:
When you compare this to the percent equation, you'll see that it has a similar structure, though the name of the components are different.
We can set this up a little more generally:
We usually prefer to use symbols instead of words because it takes up a lot less space, so if we let \(y\) represent the number of item \(Y\text{,}\) \(x\) represent the number of item \(X\text{,}\) and let \(k\) be the ratio of item \(Y\) to item \(X\) then this becomes
This type of relationship is one of the standard models that we use for talking about how two quantities are related to each other. We call this a linear (or direct) relationship between the variables. In many cases, it's more useful to think of the equation in the form
which more directly shows us that is the ratio of the number of item \(Y\) to the number of item \(X\text{.}\)
We can set up this relationship between any two collections of objects. In example above, it was the ratio of hot dog buns to the number of packages. When we're talking about percents, it's the ratio of "the part" (the number of a specific type of object) to "the whole" (the total number of objects in the collection). We also saw this ratio when looking at slopes of lines (the amount of "rise" to the amount of "run"). When we think about speed, we think about the ratio of how far something travels for a given amount of time, which is why speed has units such as "miles per hour." This shows that idea of a ratio is fundamental to algebraic reasoning and is useful in many applications.
It is often useful to think about the distinction between a ratio and a proportion. A ratio is the relationship between two quantities. A proportion is when we say that two ratios are the same. For example, if you went to the store to buy two packages of hot dog buns and your friend went to the store to buy two packages of hot dots, you may end up with different numbers of buns and dogs even though you bought the same number of packages. The reason for this is that the buns and dogs may have different numbers of objects per package. They are not proportional to each other.
Proportional reasoning can be difficult for many people because the the importance of one quantity is measured relative to the size of another. For example, losing $1000 can be very detrimental to a household's income, but a multi-billion dollar company would hardly be bothered by the loss. And on the other side of things, the difference between the price of milk being $2.99 per gallon or $3.09 per gallon is negligible for a household's budget, but this can be a large additional expense for a company that needs to buy millions of gallons of milk.
This idea can be looped back around to percents by thinking about percent change. The idea of a percent change is that we're looking at the ratio of the amount of change to the quantity relative to the quantity itself. This helps us to think about how much of an impact something has on the overall situation. For example, a pay raise of $1 per hour means a lot to someone who is earning $10 per hour, but it means very little to someone earning $100 per hour, even though the raise is the same size. The difference is that it's a larger percent increase in wages to the person earning less money (a 10% pay raise compared to a 1% pay raise).
An important note about percent change is that it can lead to error if you're not careful. An example of this can be seen if we think about something doubling. For example, let's say that the person making $10 per hour finds a different job and doubles their wages to $20 per hour. What is the percent change of their wages? Some people immediately latch on to the number 2 (because the wages doubled) and convert 2 into a percent to get a percent change of 200%. But this is wrong. We have to go back and think about what the definition of a percent increase is. We have to look at the total amount of change in the wages, which is $10 more per hour, and then use the base pay as the denominator of the ratio, which is also $10. This means that there was a 100% increase in their wages.
This is not intuitive for many people. It's a specific type of thinking that requires time and practice in order to become proficient. As you continue to take quantitative classes, especially science and social science courses, you will see proportional reasoning start to seep into the basic language you use to describe the world around you, and the more prepared you are, the more you will get out of those other classes.