1.2 Fractions and Decimals

Introduction

Imagine your local newspaper quotes a political candidate as saying, “The top half of the students are well-educated, the bottom half receive extra help, but the middle half we are leaving out ”.  You stare at the sentence for a moment and then laugh. To halve something means to split it into two. However, there are three halves here! You conclude that the speaker was not thinking carefully.

In coming to this conclusion, you are applying your knowledge of fractions. In this section, you will review fraction types, convert fractions into decimals, perform operations on fractions, and also address rounding issues in business mathematics.

Types of Fractions

To understand the characteristics, rules, and procedures for working with fractions, you must become familiar with fraction terminology. First of all, what is a fraction? A fraction is a part of a whole. It is written in one of three formats:

[latex]1/2\;\text{or}\;½\;\text{or}\;\frac12[/latex]

Each of these formats means exactly the same thing. The number on the top, side, or to the left of the line is known as the numerator. The number on the bottom, side, or to the right of the line is known as the denominator. The slash or line in the middle is the divisor line. In the above example, the numerator is [latex]1[/latex] and the denominator is [latex]2[/latex]. There are five different types of fractions, as explained in the table below.

How It Works

First, focus on the correct identification of proper, improper, compound, equivalent, and complex fractions. In the next section, you will work through how to accurately convert these fractions into their decimal equivalents.

Equivalent fractions require you to either solve for an unknown term or express the fraction in larger or smaller terms.

How to Solve For An Unknown Term

These situations involve two fractions where only one of the numerators or denominators is missing. Follow this four-step procedure to solve for the unknown:

Step 1: Set up the two fractions.

Step 2: Note that your equation contains two numerators and two denominators. Pick the pair for which you know both values.

Step 3: Determine the multiplication or division relationship between the two numbers.

Step 4: Apply the same relationship to the pair of numerators or denominators containing the unknown.

Expressing The Fraction In Larger Or Smaller Terms

When you need to make a fraction easier to understand or you need to express it in a certain format, it helps to try to express it in larger or smaller terms:

To express a fraction in larger terms, multiply both the numerator and denominator by the same number.

• Larger terms: [latex]\frac2{12}[/latex] expressed with terms twice as large would be [latex]\frac{2\times2}{12\times2}=\frac4{24}[/latex]

To express a fraction in smaller terms, divide both the numerator and denominator by the same number.

• Smaller terms: [latex]\frac2{12}[/latex] expressed with terms half as large would be [latex]\frac{2\div2}{12\div2}=\frac16[/latex]

When expressing fractions in higher or lower terms, you do not want to introduce decimals into the fraction unless there would be a specific reason for doing so. For example, if you divided [latex]4[/latex] into both the numerator and denominator of [latex]\frac2{12}[/latex], you would have [latex]\frac{0.5}3[/latex], which is not a typical format.

HOW TO 

Find numbers that divide evenly into the numerator or denominator (called factoring).

• Step 1: Pick the smallest number in the fraction.
• Step 2: Use your multiplication tables and start with [latex]1\times[/latex] before proceeding to [latex]2\times[/latex], [latex]3\times[/latex], and so on.
• Step 3: When you find a number that works, check to see if it also divides evenly into the other number.

Things To Watch Out For

With complex fractions, it is critical to obey the rules of BEDMAS.

Note in the following example that an addition sign and two sets of brackets were hidden:

You should rewrite [latex]\begin{eqnarray*}3\frac{\displaystyle¾}7\end{eqnarray*}[/latex] as [latex]\begin{eqnarray*}3+\left[\frac{\left({\displaystyle\frac34}\right)}7\right]\end{eqnarray*}[/latex] before you attempt to solve with BEDMAS.

Paths To Success

What do you do when there is a negative sign in front of a fraction, such as [latex]-\frac12[/latex]? Do you put the negative with the numerator or the denominator? The common solution is to multiply the numerator by negative [latex]1[/latex], resulting in [latex]\frac{(-1)\times1}2=\frac{-1}2[/latex].

In the special case of a compound fraction, multiply the entire fraction by [latex]−1[/latex]. Thus:

[latex]\begin{eqnarray*}&=&-1\frac12\\&=&(-1)\times(1+\frac12)\\&=&-1-\frac12\end{eqnarray*}[/latex].

Converting to Decimals

Although fractions are common, many people have trouble interpreting them. For example, in comparing [latex]\frac{27}{37}[/latex] to [latex]\frac{57}{73}[/latex], which is the larger number? The solution is not immediately apparent. As well, imagine a retail world where your local Walmart was having a [latex]\frac3{20}[/latex]th off sale! It’s not that easy to realize that this equates to [latex]15\%[/latex] off. In other words, fractions are converted into decimals by performing the division to make them easier to understand and compare.

Converting Fractions

Convert fractions into decimals based on the fraction types and fraction rules

Proper and Improper Fractions

Resolve the division. For example, [latex]\frac34[/latex] is the same as [latex]3\div4=0.75[/latex]. As well:

[latex]\begin{eqnarray*}&\;&\frac54\\[1ex]&=&5\div4\\&=&1.25\end{eqnarray*}[/latex]

Mixed Numbers

The decimal number and the fraction are joined by a hidden addition symbol. Therefore, to convert to a decimal you need to reinsert the addition symbol and apply BEDMAS:

[latex]\begin{eqnarray*}&\;&3\frac45\\[1ex]&=&3+4\div5\\&=&3+0.8\\&=&3.8\end{eqnarray*}[/latex]

Complex Fractions

The critical skill here is to reinsert all of the hidden symbols and then apply the rules of BEDMAS:

[latex]\begin{eqnarray*}&\;&2\frac{\displaystyle\frac{11}4}{1{\displaystyle\frac14}}\\[1ex]&=&\;2+\left[\frac{(11\div4)}{(1+1\div4)}\right]\\[1ex]&=&2+\left[\frac{(11\div4)}{(1+0.25)}\right]\\[1ex]&=&2+\left[\frac{2.75}{1.25}\right]\\[1ex]&=&2+2.2\\&=&4.2\end{eqnarray*}[/latex]

Rounding Principle

Your company needs to take out a loan to cover some short-term debt. The bank has a posted rate of [latex]6.825\%[/latex]. Your bank officer tells you that, for simplicity, she will just round off your interest rate to [latex]6.9\%[/latex]. Is that all right with you? It shouldn’t be!

What this example illustrates is the importance of rounding. This is a slightly tricky concept that confuses most students to some degree. In business math, sometimes you should round your calculations off and sometimes you need to retain all of the digits to maintain accuracy.

HOW TO Apply the Rounding Principle

To round a number off, you always look at the number to the right of the digit being rounded. If that number is [latex]5[/latex] or higher, you add one to your digit; this is called rounding up. If that number is [latex]4[/latex] or less, you leave your digit alone; this is called rounding down.

• For example, if you are rounding [latex]8.345[/latex] to two decimals, you need to examine the number in the third decimal place (the one to the right). It is a [latex]5[/latex], so you add one to the second digit and the number becomes [latex]8.35[/latex].
• For a second example, let’s round [latex]3.6543[/latex] to the third decimal place. Therefore, you look at the fourth decimal position, which is a [latex]3[/latex]. As the rule says, you would leave the digit alone and the number becomes [latex]3.654[/latex]

Nonterminating Decimals

What happens when you perform a calculation and the decimal doesn’t terminate?

1. You need to assess if there is a pattern in the decimals:
   • The Nonterminating Decimal without a Pattern:

For example, [latex]\frac6{17}=0.352941176[/latex] with no apparent ending decimal and no pattern to the decimals.

• • The Nonterminating Decimal with a Pattern:

For example, [latex]\frac2{11}=0.18181818[/latex] endlessly. You can see that the numbers [latex]1[/latex] and [latex]8[/latex] repeat. A shorthand way of expressing this is to place a horizontal line above the digits that repeat. Thus, you can rewrite [latex]0.18181818[/latex] as [latex]0.\overline{18}[/latex].

2. You need to know if the number represents an interim or final solution to a problem:
   • Interim Solution

You must carry forward all of the decimals in your calculations, as the number should not be rounded until you arrive at a final answer. If you are completing the question by hand, write out as many decimals as possible; to save space and time, you can use the shorthand horizontal bar for repeating decimals. If you are completing the question by calculator, store the entire number in a memory cell.

• • Final Solution

To round this number off, an industry protocol or other clear instruction must apply. If these do not exist, then you would make an arbitrary rounding choice, subject to the condition that you must maintain enough precision to allow for reasonable interpretation of the information.

Section Exercises

Mechanics Exercises

1. For each of the following, identify the type of fraction presented.
   1. [latex]\begin{align*}\frac18\end{align*}[/latex]
   2. [latex]\begin{align*}3\frac34\end{align*}[/latex]
   3. [latex]\begin{align*}\frac{10}9\end{align*}[/latex]
   4. [latex]\begin{align*}\frac{34}{49}\end{align*}[/latex]
   5. [latex]\begin{align*}1\frac{\displaystyle\frac23}{9\;{\displaystyle\frac53}}\end{align*}[/latex]
   6. [latex]\begin{align*}\frac{56}{27}\end{align*}[/latex]
   7. [latex]\begin{align*}\frac{10\;{\displaystyle\frac15}}9\end{align*}[/latex]
   8. [latex]\begin{align*}\frac6{11}\end{align*}[/latex]
2. In each of the following equations, identify the value of the unknown term.
   1. [latex]\begin{align*}\frac{3}{4}=\frac{x}{36}\end{align*}[/latex]
   2. [latex]\begin{align*}\frac{y}{8}=\frac{16}{64}\end{align*}[/latex]
   3. [latex]\begin{align*}\frac{2}{z}=\frac{18}{45}\end{align*}[/latex]
   4. [latex]\begin{align*}\frac{5}{6}=\frac{75}{p}\end{align*}[/latex]
3. Take each of the following fractions and provide one example of the fraction expressed in both higher and lower terms.
   1. [latex]\begin{align*}\frac{5}{10}\end{align*}[/latex]
   2. [latex]\begin{align*}\frac{6}{8}\end{align*}[/latex]
4. Convert each of the following fractions into decimal format.
   1. [latex]\begin{align*}\frac{7}{8}\end{align*}[/latex]
   2. [latex]\begin{align*}15\frac{5}{4}\end{align*}[/latex]
   3. [latex]\begin{align*}\frac{13}{5}\end{align*}[/latex]
   4. [latex]\begin{align*}133\frac{\frac{17}{2}}{3\frac{2}{5}}\end{align*}[/latex]
5. Convert each of the following fractions into decimal format and round to three decimals.
   1. [latex]\begin{align*}\frac{7}{8}\end{align*}[/latex]
   2. [latex]\begin{align*}15\frac{3}{4}\end{align*}[/latex]
   3. [latex]\begin{align*}\frac{10}{9}\end{align*}[/latex]
   4. [latex]\begin{align*}\frac{15}{32}\end{align*}[/latex]
6. Convert each of the following fractions into decimal format and express in repeating decimal notation.
   1. [latex]\begin{align*}\frac{1}{12}\end{align*}[/latex]
   2. [latex]\begin{align*}5\frac{8}{33}\end{align*}[/latex]
   3. [latex]\begin{align*}\frac{4}{3}\end{align*}[/latex]
   4. [latex]\begin{align*}\frac{-34}{110}\end{align*}[/latex]

Applications Exercises

7. Calculate the solution to each of the following expressions. Express your answer in decimal format.
   1. [latex]\begin{align*}\frac{1}{5}+3\frac{1}{4}+\frac{5}{2}\end{align*}[/latex]
   2. [latex]\begin{align*}1\frac{\frac{3}{8}}{2}-\frac{11}{40}+19\frac{1}{2}\times\frac{3}{4}\end{align*}[/latex]
8. Calculate the solution to each of the following expressions. Express your answer in decimal format with two decimals.
   1. [latex]\begin{align*}\left(1+\frac{0.11}{12}\right)^4\end{align*}[/latex]
   2. [latex]\begin{align*}1-0.05\times\frac{263}{365}\end{align*}[/latex]
   3. [latex]\begin{align*}200\left[1-\frac{1}{\left(1+\frac{0.10}{4}\right)^2}\right]\end{align*}[/latex]
9. Calculate the solution to each of the following expressions. Express your answer in repeating decimal notation as needed.
   1. [latex]\begin{align*}\frac{1}{11}+3\frac{1}{9}\end{align*}[/latex]
   2. [latex]\begin{align*}\frac{5}{3}-\frac{7}{6}\end{align*}[/latex]

Questions 10–14 involve fractions. For each, evaluate the expression and round your answer to the nearest cent.

10. [latex]\begin{align*}\$134,000(1+0.14\times 23/365)\end{align*}[/latex]
11. [latex]\begin{align*}\$10,000\left(1+\frac{0.0525}{2}\right)^{13}\end{align*}[/latex]
12. [latex]\begin{align*}\frac{\$535,000}{\left(1+\frac{0.07}{12}\right)^3}\end{align*}[/latex]
13. [latex]\begin{align*}\$2,995\left(1+0.13\times\frac{90}{365}\right)-\frac{\$400}{1+0.13\times\frac{15}{365}}\end{align*}[/latex]
14. [latex]\begin{align*}\frac{\$155,600}{\left(1+\frac{0.06}{12}\right)^8}\end{align*}[/latex]

Challenge, Critical Thinking, & Other Applications Exercises

Questions 15–20 involve more complex fractions and reflect business math equations encountered later in this textbook. For each, evaluate the expression and round your answer to the nearest cent.

15. [latex]\begin{align*}\frac{\$648}{0.0575/12}\left[1-\frac{1}{\left(1+\frac{0.0575}{12}\right)^7}\right]\end{align*}[/latex]
16. [latex]\begin{align*}\frac{\$10,000}{\left(1+\frac{0.115}{4}\right)^2}+\$68\frac{\left[1-\frac{1}{\left(1+\frac{0.115}{4}\right)^2}\right]}{\frac{0.115}{4}}\end{align*}[/latex]
17. [latex]\begin{align*}\frac{\$2,000,000}{\left[\frac{\left(1+\frac{0.065}{2}\right)^{12}-1}{\frac{0.065}{2}}\right]}\end{align*}[/latex]
18. [latex]\begin{align*}\$8,500\left[\frac{1-\left(\frac{1}{1.08}\right)^4}{1.08}\right]+\$19,750\left(\frac{1}{1.08}\right)^4-\$4,350\end{align*}[/latex]
19. [latex]\begin{align*}\$15,000\left[\frac{\left(1+\frac{0.058}{4}\right)^{16}-1}{\frac{0.058}{4}}\right]\end{align*}[/latex]
20. [latex]\begin{align*}\frac{0.08}{2}\left(\$1,000\right)\left[\frac{1-\frac{1}{\left(1+\frac{0.07}{2}\right)^{10}}}{\frac{0.07}{2}}\right]+\$1,000\frac{1}{\left(1+\frac{0.07}{2}\right)^{10}}\end{align*}[/latex]

Attribution

2.2: Fractions, Decimals, & Rounding from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.