2.1 Rounding Whole Numbers and Decimals
Introduction
Throughout this text we will be deriving multiple formulas related to the mathematics of business and finance. Before we can understand how these formulas work and how to properly apply them it is essential that we gain confidence in performing arithmetic operations in the right order, using whole numbers, decimals, and fractions.
In this chapter you will review the basic arithmetic skills necessary for these business and finance applications.
Rounding Rules
One of the most common sources of difficulties in math is that different people sometimes use different standards for rounding. This seriously interferes with the consistency of final solutions and makes it hard to assess accuracy. So that everyone arrives at the same solution to the exercises/examples in this textbook, these rounding rules apply throughout the book:
- Never round an interim solution unless there is a logical reason or business process that forces the number to be rounded. Here are some examples of logical reasons or business processes indicating you should round:
- You withdraw money or transfer it between different bank accounts. In doing so, you can only record two decimals and therefore any money moving between the financial tools must be rounded to two decimals.
- You need to write the numbers in a financial statement or charge a price for a product. As our currency is in dollars and cents, only two decimals can appear.
- When you write nonterminating decimals, show only the first six (or up to six) decimals. Use the horizontal line format for repeating decimals. If the number is not a final solution, then assume that all decimals or as many as possible are being carried forward.
- Round all final numbers to six decimals in decimal format and four decimals in percentage format unless instructions indicate otherwise.
- Round final solutions according to common business practices, practical limitations, or specific instructions.
- For example, round any final answer involving dollar currency to two decimals. These types of common business practices and any exceptions are discussed as they arise at various points in this textbook.
- Generally avoid writing zeros, which are not required at the end of decimals, unless they are required to meet a rounding standard or to visually line up a sequence of numbers.
- For example, write [latex]6.340[/latex] as [latex]6.34[/latex] since there is no difference in interpretation through dropping the zero.
Paths To Success
Does your final solution vary from the actual solution by a small amount? Did the question involve multiple steps or calculations to get the final answer? Were lots of decimals or fractions involved? If you answer yes to these questions, the most common source of error lies in rounding. Here are some quick error checks for answers that are “close”:
-
- Did you remember to obey the rounding rules laid out above? Most importantly, are you carrying decimals for interim solutions and rounding only at final solutions?
- Did you resolve each fraction or step accurately? Check for incorrect calculations or easy-to-make errors, like transposed numbers.
- Did you break any rules of BEDMAS?
Rounding Whole Numbers and Decimal Numbers
Rounding numbers makes them easier to work with and easier to remember. Rounding changes some of the digits in a number but keeps its value close to the original. It is used in reporting large quantities or values that change often, such as in population, income, expenses, etc.
The process of approximating numbers is called rounding. Numbers are rounded to a specific place value depending on how much accuracy is needed. Saying that the population of Canada is approximately 37 million means we rounded to the millions place. The place value we round to depends on how we need to use the number.
HOW TO
Round Whole Numbers
Step 1: Identify the digit to be rounded (this is the place value for which the rounding is required)
Step 2: If the digit to the immediate right of the required rounding is less than [latex]5[/latex] do not change the value of the rounding digit.
If the digit to the immediate right of the required rounding digit is [latex]5[/latex] or greater than [latex]5[/latex], increase the value of the rounding digit by one (Round the number up).
Step 3: Change the value of all the digits that are to the right of the rounding digit to [latex]0[/latex].
Place Value of Whole Numbers
When reading whole numbers, the position of each digit in a whole number determines the place value for the digit. Table 2.1.1 below illustrates the place value of the ten digits in the whole number [latex]3,957,261, 840[/latex].
| [latex]3[/latex], | [latex]9[/latex] | [latex]5[/latex] | [latex]7[/latex], | [latex]2[/latex] | [latex]6[/latex] | [latex]1[/latex], | [latex]8[/latex] | [latex]4[/latex] | [latex]0[/latex] |
|---|---|---|---|---|---|---|---|---|---|
| Billions | Hundred million | Ten Million | Millions | Hundred Thousand | Ten Thousand | Thousands | Hundreds | Tens | Ones |
In this example, [latex]2[/latex] is in the Hundred of Thousands place value and represents, [latex]200,000[/latex], whereas [latex]9[/latex] is in the Hundred of Millions place value and represents [latex]900,000,000[/latex].
We read and write numbers from left to right. A comma or a space separates every three digits into groups, starting from the “ones” which makes the whole number easier to read.
| Billions | Hundred Million | Ten Million | Millions | Hundred Thousand | Ten Thousand | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|---|---|---|
| 1,000,000,000 | 100,000,000 | 10,000,000 | 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
| [latex]10^9[/latex] | [latex]10^8[/latex] | [latex]10^7[/latex] | [latex]10^6[/latex] | [latex]10^5[/latex] | [latex]10^4[/latex] | [latex]10^3[/latex] | [latex]10^2[/latex] | [latex]10^1[/latex] | [latex]10^0[/latex] |
The place value of “ones” is [latex]1(=10^0)[/latex] and each place has a value [latex]10[/latex] times the place value to its right, as shown in Table 2.1.2 above.
Place Value of Decimal Numbers
The position of each digit in a decimal number determines the place value of the digit. Table 2.1.3 below illustrates the place value of the five digit decimal number of [latex]0.75396[/latex].
| Ones | Tenths | Hundreths | Thousanths | Ten Thousandths | Hundred Thousandths | |
|---|---|---|---|---|---|---|
| [latex]0[/latex] | . | [latex]7[/latex] | [latex]5[/latex] | [latex]3[/latex] | [latex]9[/latex] | [latex]6[/latex] |
Example 2.1.1
What is the place value of the digit [latex]{\color{red}{3}}[/latex] in each of the following numbers and what amount does it represent?
-
- [latex]67{\color{red}{3}},542[/latex]
- [latex]5{\color{red}{3}},721,890[/latex]
- [latex]251.{\color{red}{3}}47[/latex]
- [latex]64.0{\color{red}{3}}7[/latex]
Solution
a.
Step 1: Find where the digit [latex]3[/latex] is in the number.
[latex]\begin{eqnarray*}&=&67{\color{red}{\overbrace3}},542\\&=&{\color{red}{3}}\text{,}\overset{\color{blue}{1}}5\overset{\color{blue}{2}}4\overset{\color{blue}{3}}2\\&=&{\color{red}{3\text{,}000}}\end{eqnarray*}[/latex]
Step 2: Make a statement
The [latex]3[/latex] is in the thousandths place.
b.
Step 1: Find where the digit [latex]3[/latex] is in the number.
[latex]\begin{eqnarray*}&=&5{\color{red}{\overbrace3}},721\text{,}890\\&=&{\color{red}{3}}\text{,}\overset{\color{blue}{1}}7\overset{\color{blue}{2}}2\overset{\color{blue}{3}}1\text{,}\overset{\color{blue}{4}}8\overset{\color{blue}{5}}9\overset{\color{blue}{6}}0\\&=&{\color{red}{3\text{,}000\text{,}000}}\end{eqnarray*}[/latex]
Step 2: Make a statement.
The [latex]3[/latex] is in the millionths place.
c.
Step 1: Find where the digit [latex]3[/latex] is in the number.
[latex]\begin{eqnarray*}&=&251.{\color{red}{\overbrace3}}47\\&=&.{\color{red}{\overset{\color{blue}{1}}3}}47\\&=&{\color{red}{0.3}}\end{eqnarray*}[/latex]
Step 2: Make a statement.
The [latex]3[/latex] is in the tenths place.
d.
Step 1: Find where the digit [latex]3[/latex] is in the number.
[latex]\begin{eqnarray*}&=&64.0{\color{red}{3}}7\\&=&.\overset{\color{blue}{1}}0{\color{red}{\overset{\color{blue}{2}}3}}7\\&=&{\color{red}{0.03}}\end{eqnarray*}[/latex]
Step 2: Make a statement.
The [latex]3[/latex] is in the hundredths place.
Try It
1) Identify the digit that occupies the following place values in the number [latex]320,948.751[/latex]:
- Hundred thousandths
- Hundredths
- Tenths
Solution
- [latex]3\;({\color{red}{3}}20,948.751)[/latex]
- [latex]9\;(320,{\color{red}{9}}48.751)[/latex]
- [latex]7\;(320,948.{\color{red}{7}}51)[/latex]
Example 2.1.2
Round the following whole numbers to the indicated place value: [latex]19,456[/latex] to the nearest tenth.
Solution
Step 1: Identify the rounding digit in the tens place.
[latex]19,4{\color{red}{5}}6[/latex]
Step 2: Check the digit to the right. Do we increase?
The digit to the immediate right is [latex]6[/latex], which is greater than [latex]5[/latex].
Therefore, we do increase the value of the rounding digit by one, from [latex]5[/latex] to [latex]6[/latex].
Step 3: Do we change the value?
Yes, we change the value of the digits to the right of the [latex]5[/latex] to [latex]0[/latex].
Step 4: Write as a statement.
Therefore the rounded number becomes [latex]19,460[/latex].
Attribution
“Rounding Whole Numbers” from Mathematics for the Liberal Arts Corequisite by Deborah Devlin and Lumen Learning is licensed under a Creative Commons Attribution 4.0 license unless otherwise noted.
