2.2: Fractions, Decimals, and Rounding (Just One Slice Of Pie, Please)
- Page ID
- 22070
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.”[1] 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:
\[\text{1/2 or ½ or } \dfrac{1}{2}\nonumber \]
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 1 and the denominator is 2. There are five different types of fractions, as explained in the table below.
| Fraction | Terminology | Characteristics | Result of Division* |
|---|---|---|---|
| \(\bf{\dfrac{2}{5}}\) | Proper | The numerator is smaller than the denominator. | Answer is between 0 and 1 |
| \(\bf{\dfrac{5}{2}}\) | Improper | The numerator is larger than the denominator. | Answer is greater than 1 |
| \(\bf{3 \dfrac{2}{5}}\) | Compound | A fraction that combines an integer with either a proper or improper fraction. When the division is performed, the proper or improper fraction is added to the integer. | Answer is greater than the integer |
| \(\bf{3 \dfrac{2 / 5}{7}}\) | Complex | A fraction that has fractions within fractions, combining elements of compound, proper, or improper fractions together. It is important to follow BEDMAS in resolving these fractions. | Answer varies depending on the fractions involved |
| \(\bf{\dfrac{1}{2}}\) and \(\bf{\dfrac{2}{4}}\) | Equivalent | Two or more fractions of any type that have the same numerical value upon completion of the division. Note that both of these examples work out to 0.5. | Answers are equal |
*Assuming all numbers are positive.
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.
Solving 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.
Assume you are having a party and one of your friends says he would like to eat one-third of the pizza. You notice the pizza has been cut into nine slices. How many slices would you give to your friend?
Step 1: Your friend wants one out of three pieces. This is one-third. You want to know how many pieces out of nine to give him. Assign a meaningful variable to represent your unknown, so have \(s\) represent the number of slices to give; you need to give him \(s\) out of 9 pieces, or \(s/9\).
\[\dfrac{1}{3}=\dfrac{s}{9}\nonumber \]
Step 2: Work with the denominators 3 and 9 since you know both of them.
Step 3: Take the larger number and divide it by the smaller number. We have \(9 \div 3 = 3\). Therefore, the denominator on the right is three times larger than the denominator on the left.
Step 4: Take the 1 and multiply it by 3 to get the \(s\). Therefore, \(s = 1 \times 3 = 3\).
\[\dfrac{1 \times 3}{3 \times 3}=\dfrac{3}{9}\nonumber \]
You should give your friend three slices of pizza.
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. To express a fraction in smaller terms, divide both the numerator and denominator by the same number.
- Larger terms: \(\dfrac{2}{12}\) expressed with terms twice as large would be \(\dfrac{2 \times 2}{12 \times 2}=\dfrac{4}{24}\)
- Smaller terms: \(\dfrac{2}{12}\) expressed with terms half as large would be \(\dfrac{2 \div 2}{12 \div 2}=\dfrac{1}{6}\)
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 4 into both the numerator and denominator of, you would have, which is not a typical format. To find numbers that divide evenly into the numerator or denominator (called factoring), follow these steps:
- Pick the smallest number in the fraction.
- Use your multiplication tables and start with \(1\times\) before proceeding to \(2\times\), \(3\times\), and so on. When you find a number that works, check to see if it also divides evenly into the other number.
For example, if the fraction is \(\dfrac{12}{18}\), you would factor the numerator of 12. Note that \(1 \times 12 = 12\); however, 12 does not divide evenly into the denominator. Next you try \(2 \times 6\) and discover that 6 does divide evenly into the denominator. Therefore, you reduce the fraction to smaller terms by dividing by 6, or \(\dfrac{12 \div 6}{18 \div 6}=\dfrac{2}{3}\).
Things To Watch Out For
With complex fractions, it is critical to obey the rules of BEDMAS. As suggested in Section 2.1, always reinsert the hidden symbols before solving. Note in the following example that an addition sign and two sets of brackets were hidden: You should rewrite \(3 \dfrac{2 / 5}{7}\) as \(3+\left[\dfrac{(2 / 5)}{7}\right]\) 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 \(-\dfrac{1}{2}\)? Do you put the negative with the numerator or the denominator? The common solution is to multiply the numerator by negative 1, resulting in \(\dfrac{(-1) \times 1}{2}=\dfrac{-1}{2}\). In the special case of a compound fraction, multiply the entire fraction by \(−1\). Thus, \(-1 \dfrac{1}{2}=(-1) \times\left(1+\dfrac{1}{2}\right)=-1-\dfrac{1}{2}\).
Identify the type of fraction represented by each of the following:
- \(\dfrac{2}{3}\)
- \(6 \dfrac{7}{8}\)
- \(12 \dfrac{4 / 3}{6 \dfrac{4}{5}}\)
- \(\dfrac{15}{11}\)
- \(\dfrac{5}{6}\)
- \(\dfrac{3}{4} \& \dfrac{9}{12}\)
Solution
For each of these six fractions, identify the type of fraction.
What You Already Know
There are five types of fractions, including proper, improper, compound, complex, or equivalent
How You Will Get There
Examine each fraction for its characteristics and match these characteristics with the definition of the fraction.
Perform
- The numerator is smaller than the denominator. This matches the characteristics of a proper fraction.
- This fraction combines an integer with a proper fraction (since the numerator is smaller than the denominator). This matches the characteristics of a compound fraction.
- There are lots of fractions involving fractions nested inside other fractions. The fraction as a whole is a compound fraction, containing an integer with a proper fraction (since the numerator is smaller than the denominator). Within the proper fraction, the numerator is an improper fraction \(\left (\dfrac{4}{3} \right )\) and the denominator is a compound fraction containing an integer and a proper fraction \(\left (6 \dfrac{4}{5} \right)\). This all matches the definition of a complex fraction: nested fractions combining elements of compound, proper, and improper fractions together.
- The numerator is larger than the denominator. This matches the characteristics of an improper fraction.
- The numerator is smaller than the denominator. This matches the characteristics of a proper fraction.
- There are two proper fractions here that are equal to each other. If you were to complete the division, both fractions calculate to 0.75. These are equivalent fractions.
Of the six fractions examined, there are two proper fractions (a and e), one improper fraction (d), one compound fraction (b), one complex fraction (c), and one equivalent fraction (f).
- Solve for the unknown term \(x\): \(\dfrac{7}{12}=\dfrac{49}{x}\)
- Express this fraction in lower terms: \(\dfrac{5}{50}\)
Solution
- Find the value of the unknown term, \(x\).
- Take the proper fraction and express it in a lower term.
What You Already Know
The needed fractions in a ready-to-solve format are provided.
How You Will Get There
- Apply the four-step technique to solving equivalent fractions. The first step has already been done for you, in that the equation is already set up.
- Find a common divisor that divides evenly into both the numerator and denominator. As only 1 and 5 go into the number 5, it makes sense that you should choose 5 to divide into both the numerator and denominator. Note that 5 factors evenly into the denominator, 50, meaning that no remainder or decimals are left over.
Perform
Step 2: You have both of the numerators, so work with that pair.
Step 3: Take the larger number and divide by the smaller number, or \(49 \div 7 = 7\). Therefore, multiply the fraction on the left by 7 to get the fraction on the right.
Step 4: Applying the same relationship, \(12 \times 7 = 84\).
- \(\dfrac{5 \div 5}{50 \div 5}=\dfrac{1}{10}\)
Result
- The unknown denominator on the right is 84, and therefore \(\dfrac{7}{12}=\dfrac{49}{84}\).
- In lower terms, \(\dfrac{5}{50}\) is expressed as \(\dfrac{1}{10}\).
Converting to Decimals
Although fractions are common, many people have trouble interpreting them. For example, in comparing \(\dfrac{27}{37}\) to \(\dfrac{57}{73}\), which is the larger number? The solution is not immediately apparent. As well, imagine a retail world where your local Walmart was having a th off sale! It’s not that easy to realize that this equates to 15% off. In other words, fractions are converted into decimals by performing the division to make them easier to understand and compare.
How It Works
The rules for converting fractions into decimals are based on the fraction types.
Proper and Improper Fractions
Resolve the division. For example, \(\dfrac{3}{4}\) is the same as \(3 \div 4 = 0.75\). As well, \(5/4=5 \div 4=1.25\).
Compound Fractions
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:
\[3 \dfrac{4}{5}=3+4 \div 5=3+0.8=3.8\nonumber \]
Complex Fractions
The critical skill here is to reinsert all of the hidden symbols and then apply the rules of BEDMAS:
\[2 \dfrac{11 / 4}{11 / 4}=2+\left[\dfrac{(11 \div 4)}{(1+1 \div 4)}\right]=2+\left[\dfrac{(11 \div 4)}{(1+0.25)}\right]=2+\left[\dfrac{2.75}{1.25}\right]=2+2.2=4.2\nonumber \]
Convert the following fractions into decimals:
- \(\dfrac{2}{5}\)
- \(6 \dfrac{7}{8}\)
- \(12 \dfrac{9 / 2}{1 \dfrac{2}{10}}\)
Solution
Take these fractions and convert them into decimal numbers.
What You Already Know
The three fractions are provided and ready to convert.
How You Will Get There
- This is a proper fraction requiring you to complete the division.
- This is a compound fraction requiring you to reinsert the hidden addition symbol and then apply BEDMAS.
- This is a complex fraction requiring you to reinsert all hidden symbols and apply BEDMAS.
Perform
- \(\dfrac{2}{5}=2 \div 5=0.4\)
- \(6 \dfrac{7}{8}=6+7 \div 8=6+0.875=6.875\)
- \(12 \dfrac{9 / 2}{10}=12+\left[\dfrac{9 \div 2}{1+2 \div 10}\right]=12+\left[\dfrac{9 \div 2}{1+0.2}\right]=12+\left[\dfrac{4.5}{1.2}\right]=12+3.75=15.75\)
In decimal format, the fractions have converted to 0.4, 6.875, and 15.75, respectively.
Rounding Principle
Your company needs to take out a loan to cover some short-term debt. The bank has a posted rate of 6.875%. Your bank officer tells you that, for simplicity, she will just round off your interest rate to 6.9%. 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 It Works
To round a number off, you always look at the number to the right of the digit being rounded. If that number is 5 or higher, you add one to your digit; this is called rounding up. If that number is 4 or less, you leave your digit alone; this is called rounding down.
For example, if you are rounding 8.345 to two decimals, you need to examine the number in the third decimal place (the one to the right). It is a 5, so you add one to the second digit and the number becomes 8.35.
For a second example, let’s round 3.6543 to the third decimal place. Therefore, you look at the fourth decimal position, which is a 3. As the rule says, you would leave the digit alone and the number becomes 3.654.
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, \(\dfrac{6}{17}=0.352941176\)... with no apparent ending decimal and no pattern to the decimals.
- The Nonterminating Decimal with a Pattern: For example, \(\dfrac{2}{11}=0.18181818\)... endlessly. You can see that the numbers 1 and 8 repeat. A shorthand way of expressing this is to place a horizontal line above the digits that repeat. Thus, you can rewrite 0.18181818 ... as \(0 . \overline{18}\).
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.
Important Notes
To assist in your calculations, particularly those that involve multiple steps to resolve, your calculator has 10 memory cells. Your display is limited to 10 digits, but when you store a number into a memory cell the calculator retains all of the decimals associated with the number, not just those displaying on the screen. Your calculator can, in fact, carry up to 13 digit positions. It is strongly recommended that you take advantage of this feature where needed throughout this textbook.
Let’s say that you just finished keying in \(\dfrac{6}{17}\) on your calculator, and the resultant number is an interim solution that you need for another step. With 0.352941176 on your display, press STO followed by any numerical digit on the keypad of your calculator. STO stands for store. To store the number into memory cell 1, for example, press STO 1. The number with 13 digits is now in permanent memory. If you clear your calculator (press CE/C) and press RCL # (where # is the memory cell number), you will bring the stored number back. RCL stands for recall. Press RCL 1. The stored number 0.352941176 reappears on the screen.
Convert the following to decimals. Round each to four decimals or use the repeating decimal notation.
- \(\dfrac{6}{13}\)
- \(\dfrac{4}{9}\)
- \(\dfrac{4}{11}\)
- \(\dfrac{3}{22}\)
- \(5 \dfrac{1 / 7}{10 / 27}\)
Solution
Convert each of the fractions into decimal format, then round any final answer to four decimals or use repeating decimal notation.
What You Already Know
The fractions and clear instructions on how to round them have been provided.
How You Will Get There
To convert the fractions to decimals, you need to complete the division by obeying the rules of BEDMAS. Watch for hidden symbols and adhere to the rules of rounding.
Perform
- \(\dfrac{6}{13}=0.461538\). The fifth decimal is a 3, so round down. The answer is 0.4615.
- \(\dfrac{4}{9}=0.444444\). Note the repeating decimal of 4. Using the horizontal bar, write \(0 . \overline{4}\).
- \(\dfrac{4}{11}=0.363636\). Note the repeating decimals of 3 and 6. Using the horizontal bar, write \(0 . \overline{36}\).
- \(\dfrac{3}{22}=0.136363\). Note the repeating decimals of 3 and 6 after the 1. Using the horizontal bar, write \(0.1 \overline{36}\).
- \(5 \dfrac{1 / 7}{10 / 27}=5+\dfrac{(1 \div 7)}{(10 \div 27)}=5+\dfrac{0.142857}{0.370}=5+0.385714=5.385714\). Since the fifth digit is a 1, round down. The answer is 5.3857.
According to the rounding instructions, the solutions are 0.4615, \(0 . \overline{4}\), \(0 . \overline{36}\), \(0.1 \overline{36}\), and 5.3857, respectively.
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 6.340 as 6.34 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?
References
- Neal, Marcia. Candidate for the 3rd Congressional District Colorado State Board of Education, as quoted in Perez, Gayle. 2008. "Retired School Teacher Seeks State Board Seat." Pueblo Chieftain.