Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

The last step is very rule-minded, as it would be wrong to write as even though that technically has three decimal places. And students have to learn how to handle the special case when there are more decimal places than digits.

But with all of this, we come back to the question that we’ve hit many times in this book: Why is this the rule? Why do decimals have this strange place-counting rule that we have to learn in order to do decimal multiplication correctly? It turns out that the answer comes from fraction multiplication.

One of the main applications of decimals comes from percents. Most people know the rule that to convert a number to a percent, you move the decimal two places to the left. For example, 50% is just 0.50 (which is often written as just 0.5) and 237% would be written as 2.37. Again, we want to transition from this simply being a rule of percents and turn it into a concept relating decimals and percents.

We have talked about the literal meaning of a percent, but we haven’t talked about its practical significance. Why do we even care about percents? A percent gives us a way of thinking about values relative to other values. For example, $1000 is a lot of money when you’re buying a dinner, but it’s a small amount of money when you’re buying a home. So a percent gives us a relative framework to help us understand the size of something. In fact, it is precisely the size of the part relative to the whole.

Here are some visual examples of 50%. The total area of the figure doesn’t matter. We are just thinking about the part of it relative to the whole amount. Percents are a way of doing that in a uniform manner.

A less fortunate phrasing that students learn is "is over of." This language may help students to set up calculations when problems are worded in a specific way, but it gives very little insight into the actual meaning of percents. In real life, people don’t go around asking "What is 20\% of 45?". It usually comes in a less structured form like the following: "The bill was \$45. How much should we tip?" Although we can translate the second form into the first after we understand percents, that translation step is where the actual understanding of percents is found, and the calculation is just the execution of the idea.

PDF Version of these Worksheets

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Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

As mentioned before, most decimal multiplication is performed by calculators or computers. But computers are usually not able to perform percent calculations without a person correctly identifying the part, the whole, and the percent, and then determining what calculations are required to solve the problem. With the goal of mathematical thinking in mind, the problems in this section were set up so that calculators would not be needed.

But if you were run into a situation where you would need a calculator, as long you have the correct mathematical thinking then all you need to do is replace the mental calculation with a calculator calculation. There is no real loss (from the perspective of logic) in trading that out. Here is the light bulb problem again, but with slightly more realistic numbers:

The last batch of 1500 light bulbs had 37 defects. What is the percent of defective bulbs?

Notice that the overall process is unchanged, and it’s just a matter of using different numbers. This is very similar to the process of learning to treat variables like numbers in other calculations, such as reducing fractions. In fact, the particular area of mathematical thinking, which is sometimes called algebraic reasoning, is the whole idea that you can generalize the methods and ideas of simple examples so that they can be applied in more complex situations. You got a hint of this type of reasoning in the last few problems where you had to do a mental manipulation before putting the numbers into the parts of a whole framework. And that is the skill that you should be aiming to develop in your college level courses. You want to be more than a calculator.

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.

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.