Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 22.1 What Percent Do You Understand?

There is a method that most students learn for multiplying decimals.

• Step 1: \(11 \cdot 36 = 376\text{.}\)

• Step 2: The number has one decimal and the number has two decimals, resulting in three total decimals.

• Step 3: Write with three decimal places:

For example, to calculate \((1.1) \cdot (0.36)\text{:}\)

• Step 1: \(11 \cdot 36 = 376\text{.}\)

• Step 2: The number has one decimal and the number has two decimals, resulting in three total decimals.

• Step 3: Write with three decimal places:

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.

Activity 22.1.2. Converting Between Decimals, Percents, and Fractions (Part 1).

The word "percent" can be interpreted literally as "per hundred." So when we write we're saying parts per hundred. Notice that this reflects the language of parts of a whole, which leads us to think about this as a fraction. And once we write this as a fraction, we can see why it's the same as just moving the decimal two places to the left.

\begin{equation*} 75\% = \frac{75}{100} = 0.75 \end{equation*}

Also notice that \(\frac{75}{100}\) can be reduced to \(\frac{3}{4}\text{.}\) You may want to revisit Worksheet 21.2 for help with some of these fraction to decimal conversions.

In general, you will want to be able to move fluidly between all three of the notations (decimals, fractions, and percents).

Try it!

Complete the chart. Reduce the fractions where possible.

Solution.

Activity 22.1.3. Converting Between Decimals, Percents, and Fractions (Part 2).

When all the numbers are written with just two decimal places, it makes the conversion to percents and fractions simple. And while there is nothing different when working with other numbers of decimals, they are prone to more errors. Just be careful and use your multiple ways of thinking about these numbers be a guide to help you determine if you have made an error.

Try it!

Complete the chart. Reduce the fractions where possible.

Solution.

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.

When we talk about percentages, we often talk about a percent "of" something else, and the something else represents the whole. But sometimes we have to use context and reading comprehension to determine what the part is.

• 10% of students don't understand percents. (The whole is "all students" and the part is "the students that don't understand percents.")

• 70% of the budget went to salaries. (The whole is "the budget" and the part is "salaries.")

• Save 20% off of the regular price! (The whole is "the regular price" and the part is "the discount.")

Algebraically, we might write the relationship as

which could be written equivalently as

This second version can be read as "the part is the percent of the whole." And this phrasing is a helpful reminder of the meaning of percents.

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.