Learning Objectives
- Understand the relationship between decimals and fractions.
- Add and subtract decimals.
Chapter 21 Decimal Addition and Subtraction
Section 21.1 Making Sense with Dollars and Cents
Although fractions are important and useful for algebraic manipulations, many of the day-to-day numbers that people encounter are decimals. But it turns out that decimals are just a special way of writing parts of a whole, which means that they’re also just fractions. Most students don’t have a good sense of the interplay between the two notations, and quite often think of them as mostly unrelated to each other.
Where do decimals even come from? It turns out that this is an extension of the way we write numbers. We use what a place value system which means that our numbers depend on both the symbol we use (the digits 0-9) and the position of that digit within the number itself.
Activity 21.1. Representations of Numbers.
Let’s think about the number 237. We have been trained to understand that when we put digits side-by-side like this to make larger numbers, that each digit refers to a different group size.
Try it!
Write the number 8367 in expanded form and label each of the parts with the corresponding unit (as shown above).
Solution.
The choice to use ones, tens, and hundreds (and also thousands, ten thousands, and so forth) is because each of those numbers turns out to be a power of 10.
Notice that as we go down the list, the exponents increase and so do the numbers. If we go up the list, the exponents get decrease and so do the numbers. And in the same way we used this pattern to develop negative exponents, we can develop decimals. The naming of these units are a little awkward to say out loud, but the mathematics behind them is simple. Tenths are the size you get when you take one object and break it into ten pieces. Hundredths are the size you get when you take one object and break it into one hundred pieces.
Activity 21.2. Expanded Form of a Number.
Try it!
Write the number 35.79 in expanded form and label each of the parts with the corresponding unit.
Solution.
It turns out that decimals have multiple representations. This comes out of thinking about multiple representations of the same fraction.
\begin{equation*}
\begin{aligned}
0.3 = \frac{3}{10} \amp = \frac{30}{100} = 0.30 \\
\amp = \frac{300}{1000} = 0.300 \\
\amp = \frac{3000}{10000} = 0.3000
\end{aligned}
\end{equation*}
Activity 21.3. The Connection Between Fractions and Decimals.
There is a simple connection between the number of decimals and the powers of ten. The number of zeros in the power of in the denominator corresponds to the number of decimal places for that representation of the number. For example, the number 0.0365 (four decimal places) corresponds to \(\frac{365}{10000}\) (four zeros in the denominator). In fact, this number also corresponds to the exponent of the in the denominator: \(\frac{365}{10000} = \frac{365}{10^4}\text{.}\) Thinking about numbers this way will help to avoid certain types of errors.
Try it!
Write the number as a fraction.
Solution.
\begin{equation*}
0.086 = \frac{86}{1000} = \frac{86}{10^3}
\end{equation*}
Most of the challenges with addition and subtraction of decimals are resolved by simply ensuring that all your numbers have the same number of decimal places. The best analogy for this is money. American currency is always written with two decimal places (if decimal places are being used). And what this does is that it helps us think about the coins relative to same base unit (1 cent) all the time, and there’s no confusion about whether a dime ($0.10) is the same as a penny ($0.01).
Activity 21.4. Adding Decimals.
It can be helpful to ensure that all of the numbers are written to the same number of decimals if the calculation is intended to be performed mentally or by hand. This will help to reinforce the underlying concept as well as avoid computational errors.
\begin{equation*}
3.04 + 1.1 = 3.04 + 1.10 = 4.14
\end{equation*}
Try it!
Calculate \(2.5 + 1.22\text{.}\)
Solution.
\begin{equation*}
2.5 + 1.22 = 2.50 + 1.22 = 3.72
\end{equation*}
Activity 21.5. Subtracting Decimals.
The exact same trick applies to subtraction.
\begin{equation*}
2.8 - 1.06 = 2.80 - 1.06 = 1.74
\end{equation*}
Try it!
Calculate \(4.77 - 2.3\text{.}\)
Solution.
\begin{equation*}
4.77 - 2.3 = 4.77 - 2.30 = 2.47
\end{equation*}
In practice, most decimal calculations are done by calculator or computer. In fact, certain disciplines (such as physics and chemistry) use the number of decimals as an indication of the quality of a measurement, so that 4.21 is not the same measurement as 4.21000. So we will not be spending a lot of time performing large decimal calculations by hand. Instead, this section should be interpreted as developing a conceptual basis, not a computational basis, for decimals.
Section 21.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/21-Worksheets.pdfWorksheet Worksheet 1
1.
Write the number 34.72 in expanded form and label each of the parts with the corresponding unit.
2.
Write the number 20.709 in expanded form and label each of the parts with the corresponding unit.
3.
Write the number 0.73 as a fraction.
4.
Write the number 0.029 as a fraction.
5.
Write the number 1.05 as an improper fraction. Explain why you think your answer is correct.
Worksheet Worksheet 2
1.
Calculate \(2.3 + 3.08\text{.}\)
2.
Calculate \(4.2 + 1.03\text{.}\)
3.
Calculate \(4.6 - 2.22\text{.}\)
4.
Calculate \(3.6 - 1.38\text{.}\)
5.
A student calculates \(2.5 + 1.04\) and gets 3.09 as the result. How would you explain the error to them?
Worksheet Worksheet 3
1.
Calculate \(2.03 + 1.98\text{.}\)
2.
Calculate \(1.013 + 3.92\text{.}\)
3.
Calculate \(11.11 + 1.111\text{.}\)
4.
Calculate \(6.1 - 3.28\text{.}\)
5.
Calculate \(4.28 - 1.005\text{.}\)
6.
Calculate \(5.006 - 3.09\text{.}\)
Worksheet Worksheet 4
1.
The word decimal comes from the Latin root for ten. This is because numbers can be written in terms of powers of ten. This gives us a different way of writing the expanded form of a number:
\begin{equation*}
\begin{aligned}
238 \amp = 200 + 30 + 8 \\
\amp = 2 \cdot 100 + 3 \cdot 10 + 8 \cdot 1 \\
\amp = 2 \cdot 10^2 + 3 \cdot 10^1 + 8 \cdot 10^0
\end{aligned}
\end{equation*}
With this framework in mind, write the number 21.84 using expanded form and showing the powers of ten.
2.
Write the number 107.509 using expanded form and showing the powers of ten.
3.
Write the number 3100000000 using expanded form and showing the power of ten.
4.
Write the number 0.000000079 using expanded form and showing the power of ten.
5.
Describe some of the challenges that you faced in the last two calculations.
Worksheet Worksheet 5
1.
There are certain decimals that come up often enough that it is useful to be able to convert between decimals and fractions. Convert each of the decimals below into fractions, and then completely reduce them. Put the reduced fraction into the charts.
2.
There are fractions that lead to decimals that repeat a pattern forever. We can indicate repeating decimals in two different ways. One notation writes out enough of the number so that the pattern is "obvious" and then uses an ellipsis to indicate that the pattern continues. The other way uses a bar over the part of the number that repeats. Using a calculator or long division, complete the chart of values.
3.
Some decimals have interesting patterns that can be explored. One of the more surprising examples of this happens with fractions with 7 in the denominator.
What pattern do you observe in these decimals?
Section 21.3 Deliberate Practice: Adding and Subtracting Decimals
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.
Focus on these skills:
- Write the original expression.
- Focus your attention on the position of the digits relative to the decimal point.
- Present your work legibly.
Worksheet Worksheet
Instructions: Perform the indicated calculation.
1.
Calculate \(3.14 + 1.08\text{.}\)
2.
Calculate \(2.4 + 3.81\text{.}\)
3.
Calculate \(4.07 - 2.78\text{.}\)
4.
Calculate \(3.3 - 1.07\text{.}\)
5.
Calculate \(5.32 + 2.079\text{.}\)
6.
Calculate \(1.6 + 2.504\text{.}\)
7.
Calculate \(2.054 - 1.23\text{.}\)
8.
Calculate \(3.13 - 1.307\text{.}\)
9.
Calculate \(12.08 + 3.127\text{.}\)
10.
Calculate \(11.37 - 2.037\text{.}\)
Section 21.4 Closing Ideas
For the overwhelming majority of real life situations (especially work situations) that you can imagine, if there are any parts of a whole involved, it will probably be done with decimals. And in most of those real life situations, if you need to do a calculation, you would do it with a calculator. So why do we need to learn how to do these basic calculations, and what is the value of relating things back to fractions?
Remember that the ability to calculate a number and the ability to understand that number are two separate skills. If you type in some calculations and get 5.07 and 5.1 as the results, you may still need to decide which of those is greater. You might be surprised (or maybe not) that a fair number of adults will get that comparison wrong. Part of this is simply on the level of numerical and computational literacy, which is to help you correctly understand and use information in the real world.
The importance of being able to work with fractions is that there’s a lot of information that is better communicated and more accurately communicated using fractions than decimals. For example, there are situations where it’s easier to think about getting 8 items for $3 than it is to think about getting each individual item for approximately $0.38 each. And that ratio is much more useful as the unreduced fraction \(\frac{\$ 3}{\text{8 items}}\) if you need to buy in increments of 8 (like hot dog buns).
The value is not only in having two different ways of thinking about it, but also being able to relate the two together. Sometimes fractions are the better tool and sometimes decimals are the better tool. And so it is beneficial to be able to take the information you have and apply the right tool to solve the problem instead of forcing yourself to use the wrong tool because you don’t know how to use the other one. There’s a well-known saying to this effect: "If all you have is a hammer, everything looks like a nail."
