2.2: Fractions and Decimals
Introduction
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 ” (Neal, 2008). 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.
| Fraction | Terminology | Characteristics | Result of Division* |
|---|---|---|---|
| [latex]\frac25[/latex] | Proper | The numerator is smaller than the denominator. | Answer is between [latex]0[/latex] and [latex]1[/latex] |
| [latex]\frac52[/latex] | Improper | The numerator is larger than the denominator. | Answer is greater than [latex]1[/latex] |
| [latex]3\frac25[/latex] | Mixed | 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 |
| [latex]3\frac{\displaystyle\frac25}7[/latex] | 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 |
| [latex]\frac12\;\text{and}\;\frac24[/latex] | 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 [latex]0.5[/latex]. | 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.
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.
Example 2.2.1
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?
Solution
Step 1: Assign a meaningful variable to represent unknown.
[latex]s =[/latex] the number of slices to give out
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. There are a total of [latex]9[/latex] pieces, so we are looking for [latex]s/9[/latex].
[latex]\begin{align*}\frac13=\frac s9\end{align*}[/latex]
Step 2: Work with the denominators [latex]3[/latex] and [latex]9[/latex] since you know both of them.
Step 3: Take the larger number and divide it by the smaller number.
[latex]9\divsymbol3=3[/latex]
The denominator on the right is three times larger than the denominator on the left.
Step 4: Take the [latex]1[/latex] and multiply it by [latex]3[/latex] to get the [latex]s[/latex].
[latex]\begin{eqnarray*}\frac{{\color{red}{1\times3}}}{3\times3}&=&\frac{\color{red}{3}}9\\{\color{red}{s}}&=&{\color{red}{3}}\end{eqnarray*}[/latex]
Step 5: Write as a statement.
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.
-
- 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\divsymbol2}{12\divsymbol2}=\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.
Example 2.2.2
Reduce the following fraction: [latex]\begin{align*}\frac{12}{18}\end{align*}[/latex].
Solution
Step 1: Factor the numerator.
[latex]1\times12=12[/latex]
Step 2: Does it divide evenly?
[latex]12[/latex] does not divide evenly into the denominator.
Step 3: Try another factor.
[latex]2\times6=12[/latex]
[latex]6[/latex] does divide evenly into the denominator.
Step 4: Reduce the fraction into smaller terms.
Reduce the fraction to smaller terms by dividing by [latex]6[/latex].
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{\displaystyle\frac25}7\end{eqnarray*}[/latex] as [latex]\begin{eqnarray*}3+\left[\frac{\left({\displaystyle\frac25}\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].
Example 2.2.3
Identify the type of fraction represented by each of the following:
- [latex]\begin{align*}\frac23\end{align*}[/latex]
- [latex]\begin{align*}6\frac78\end{align*}[/latex]
- [latex]\begin{align*}12\frac{\displaystyle\frac43}{6{\displaystyle\frac45}}\end{align*}[/latex]
- [latex]\begin{align*}\frac{15}{11}\end{align*}[/latex]
- [latex]\begin{align*}\frac56\end{align*}[/latex]
- [latex]\begin{align*}\frac34\end{align*}[/latex] & [latex]\begin{align*}\frac9{12}\end{align*}[/latex]
Solution
Step 1: Identify what we are looking for.
For each of these six fractions, identify the type of fraction.
Step 2: State what we know.
There are five types of fractions, including proper, improper, compound, complex, or equivalent.
Step 3: Use the definition from the Types of Fractions table at the beginning of this section to identify the type.
a. [latex]\begin{align*}\frac23\end{align*}[/latex]
The numerator is smaller than the denominator. This matches the characteristics of a proper fraction.
b. [latex]\begin{align*}6\frac78\end{align*}[/latex]
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.
c. [latex]\begin{align*}12\frac{\displaystyle\frac43}{6{\displaystyle\frac45}}\end{align*}[/latex]
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 [latex]\left(\frac43\right)[/latex] and the denominator is a compound fraction containing an integer and a proper fraction [latex]\left(6\frac{\;4}5\right)[/latex]. This all matches the definition of a complex fraction: nested fractions combining elements of compound, proper, and improper fractions together.
d. [latex]\begin{align*}\frac{15}{11}\end{align*}[/latex]
The numerator is larger than the denominator. This matches the characteristics of an improper fraction.
e. [latex]\begin{align*}\frac56\end{align*}[/latex]
The numerator is smaller than the denominator. This matches the characteristics of a proper fraction.
f. [latex]\begin{align*}\frac34\&\frac9{12}\end{align*}[/latex]
There are two proper fractions here that are equal to each other. If you were to complete the division, both fractions calculate to [latex]0.75[/latex]. These are equivalent fractions.
Example 2.2.4
- Solve for the unknown term: [latex]\begin{align*}x:\frac7{12}=\frac{49}x\end{align*}[/latex]
- Express this fraction in lower terms: [latex]\begin{align*}\frac5{50}\end{align*}[/latex]
Solution
a.
Step 1: Is the fraction in a format I can solve in?
Yes. You have both of the numerators, so work with that pair.
Step 2: Take the larger number and divide by the smaller number.
[latex]49\divsymbol7=7[/latex]
Step 3: Multiple the fraction on the left by 7 to get the fraction on the right. Applying the same relationship:
[latex]12\times7=84[/latex]
Step 4: Write as a statement.
The unknown denominator on the right is [latex]84[/latex], and therefore [latex]\frac7{12}=\frac{49}{84}[/latex].
b.
Step 1: Find a common divisor that divides into the numerator and denominator evenly.
As only [latex]1[/latex] and [latex]5[/latex] go into the number [latex]5[/latex], it makes sense that you should choose [latex]5[/latex] to divide into both the numerator and denominator.
Note that [latex]5[/latex] factors evenly into the denominator, [latex]50[/latex], meaning that no remainder or decimals are left over.
[latex]\begin{align*}\frac{5\divsymbol5}{50\divsymbol5}=\frac1{10}\end{align*}[/latex]
Step 2: Write as a statement.
In lower terms, [latex]\frac5{50}[/latex], is expressed as [latex]\frac1{10}[/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.
HOW TO
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\divsymbol4=0.75[/latex]. As well:
[latex]\begin{eqnarray*}&\;&\frac54\\[1ex]&=&5\divsymbol4\\&=&1.25\end{eqnarray*}[/latex]
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:
[latex]\begin{eqnarray*}&\;&3\frac45\\[1ex]&=&3+4\divsymbol5\\&=&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\divsymbol4)}{(1+1\divsymbol4)}\right]\\[1ex]&=&2+\left[\frac{(11\divsymbol4)}{(1+0.25)}\right]\\[1ex]&=&2+\left[\frac{2.75}{1.25}\right]\\[1ex]&=&2+2.2\\&=&4.2\end{eqnarray*}[/latex]
Example 2.2.5
Convert the following fractions into decimals:
- [latex]\begin{align*}\frac25\end{align*}[/latex]
- [latex]\begin{align*}6\frac78\end{align*}[/latex]
- [latex]\begin{align*}12\frac{\displaystyle\frac92}{1\;{\displaystyle\frac2{10}}}\end{align*}[/latex]
Solution
a.
Step 1: This is a proper fraction requiring you to complete the division.
[latex]\begin{eqnarray*}&\;&\frac{2}{5}\\[1ex]&=&2\divsymbol5\\[1ex]&=&0.4\end{eqnarray*}[/latex]
Step 2: Write as a statement.
The decimal form is [latex]0.4[/latex].
b.
Step 1: This is a compound fraction requiring you to reinsert the hidden addition symbol and then apply BEDMAS.
[latex]\begin{eqnarray*}&\;&6\frac78\\[1ex]&=&6+7\divsymbol8\\&=&6+0.875\\&=&6.875\end{eqnarray*}[/latex]
Step 2: Write as a statement.
The decimal form is [latex]6.875[/latex].
c.
Step 1: This is a complex fraction requiring you to reinsert all hidden symbols and apply BEDMAS.
[latex]\begin{eqnarray*}&\;&12\frac{\displaystyle\frac92}{1\;{\displaystyle\frac2{10}}}\\[1ex]&=&12+\left[\frac{(9\divsymbol2)}{(1+2\divsymbol10)}\right]\\[1ex]&=&12+\left[\frac{(9\divsymbol2)}{(1+0.2)}\right]\\[1ex]&=&12+\left[\frac{4.5}{1.2}\right]\\[1ex]&=&12+3.75\\&=&15.75\end{eqnarray*}[/latex]
Step 2: Write as a statement.
The decimal form is [latex]15.75[/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]6.875\%[/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?
- 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].
- 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.
Key Takeaways
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 [latex]\frac{6}{17}[/latex] on your calculator, and the resultant number is an interim solution that you need for another step. With [latex]0.352941176[/latex] 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 [latex]0.352941176[/latex] reappears on the screen.
Example 2.2.6
Convert the following to decimals. Round each to four decimals or use the repeating decimal notation.
- [latex]\begin{align*}\frac6{13}\end{align*}[/latex]
- [latex]\begin{align*}\frac49\end{align*}[/latex]
- [latex]\begin{align*}\frac4{11}\end{align*}[/latex]
- [latex]\begin{align*}\frac3{22}\end{align*}[/latex]
- [latex]\begin{align*}5\frac{\displaystyle\frac17}{\displaystyle\frac{10}{27}}\end{align*}[/latex]
Solution
a.
Step 1: Divide to convert to decimal.
[latex]\begin{align*}\frac6{13}=0.461538\end{align*}[/latex]
Step 2: Round and write as a statement.
The fifth decimal is a [latex]3[/latex], so round down.
Step 3: Write as a statement.
The answer is [latex]0.4615[/latex].
b.
Step 1: Divide to convert to decimal.
[latex]\begin{align*}\frac49=0.444444\end{align*}[/latex]
Step 2: Round and write as a statement.
Note the repeating decimal of [latex]4[/latex].
Step 3: Write as a statement.
Using the horizontal bar, write [latex]0.\overline4[/latex].
c.
Step 1: Divide to convert to decimal.
[latex]\begin{align*}\frac4{11}=0.363636\end{align*}[/latex]
Step 2: Round and write as a statement.
Note the repeating decimals of [latex]3[/latex] and [latex]6[/latex].
Step 3: Write as a statement.
Using the horizontal bar, write [latex]0.\overline{36}[/latex].
d.
Step 1: Divide to convert to decimal.
[latex]\begin{align*}\frac3{22}=0.136363\end{align*}[/latex]
Step 2: Round and write as a statement.
Note the repeating decimals of [latex]3[/latex] and [latex]6[/latex] after the [latex]1[/latex].
Step 3: Write as a statement.
Using the horizontal bar, write [latex]0.1\overline{36}[/latex].
e.
Step 1: Divide to convert to decimal.
[latex]\begin{eqnarray*}&\;&5\frac{\displaystyle\frac17}{\displaystyle\frac{10}{27}}\\[1ex]&=&5+\frac{(1\divsymbol7)}{(10\divsymbol27)}\\[1ex]&=&5+\frac{0.142857}{0.\overline{370}}\\&=&5+0.385714\\&=&5.385714\end{eqnarray*}[/latex]
Step 2: Round and write as a statement.
Since the fifth digit is a [latex]1[/latex], round down.
Step 3: Write as a statement.
The answer is [latex]5.3857[/latex].
Section 2.2 Exercises
Mechanics
- For each of the following, identify the type of fraction presented.
- [latex]\begin{align*}\frac18\end{align*}[/latex]
- [latex]\begin{align*}3\frac34\end{align*}[/latex]
- [latex]\begin{align*}\frac{10}9\end{align*}[/latex]
- [latex]\begin{align*}\frac{34}{49}\end{align*}[/latex]
- [latex]\begin{align*}1\frac{\displaystyle\frac23}{9\;{\displaystyle\frac53}}\end{align*}[/latex]
- [latex]\begin{align*}\frac{56}{27}\end{align*}[/latex]
- [latex]\begin{align*}\frac{10\;{\displaystyle\frac15}}9\end{align*}[/latex]
- [latex]\begin{align*}\frac6{11}\end{align*}[/latex]
- In each of the following equations, identify the value of the unknown term.
- [latex]\begin{align*}\frac{3}{4}=\frac{x}{36}\end{align*}[/latex]
- [latex]\begin{align*}\frac{y}{8}=\frac{16}{64}\end{align*}[/latex]
- [latex]\begin{align*}\frac{2}{z}=\frac{18}{45}\end{align*}[/latex]
- [latex]\begin{align*}\frac{5}{6}=\frac{75}{p}\end{align*}[/latex]
- Take each of the following fractions and provide one example of the fraction expressed in both higher and lower terms.
- [latex]\begin{align*}\frac{5}{10}\end{align*}[/latex]
- [latex]\begin{align*}\frac{6}{8}\end{align*}[/latex]
- Convert each of the following fractions into decimal format.
- [latex]\begin{align*}\frac{7}{8}\end{align*}[/latex]
- [latex]\begin{align*}15\frac{5}{4}\end{align*}[/latex]
- [latex]\begin{align*}\frac{13}{5}\end{align*}[/latex]
- [latex]\begin{align*}133\frac{\frac{17}{2}}{3\frac{2}{5}}\end{align*}[/latex]
- Convert each of the following fractions into decimal format and round to three decimals.
- [latex]\begin{align*}\frac{7}{8}\end{align*}[/latex]
- [latex]\begin{align*}15\frac{3}{4}\end{align*}[/latex]
- [latex]\begin{align*}\frac{10}{9}\end{align*}[/latex]
- [latex]\begin{align*}\frac{15}{32}\end{align*}[/latex]
- Convert each of the following fractions into decimal format and express in repeating decimal notation.
- [latex]\begin{align*}\frac{1}{12}\end{align*}[/latex]
- [latex]\begin{align*}5\frac{8}{33}\end{align*}[/latex]
- [latex]\begin{align*}\frac{4}{3}\end{align*}[/latex]
- [latex]\begin{align*}\frac{-34}{110}\end{align*}[/latex]
Solutions
1a. proper
1b.compound
1c.improper
1d. proper
1e. complex
1f. improper
1g. complex
1h.proper
2a. [latex]\begin{align*}\frac{3}{4}=\frac{27}{36}\end{align*}[/latex]
2b. [latex]\begin{align*}\frac{2}{8}=\frac{16}{64}\end{align*}[/latex]
2c. [latex]\begin{align*}\frac{2}{5}=\frac{18}{45}\end{align*}[/latex]
2d. [latex]\begin{align*}\frac{5}{6}=\frac{75}{90}\end{align*}[/latex]
3a. [latex]\begin{align*}\frac{5\times 2}{10\times 2}=\frac{10}{20}\;\;\;\frac{5\divsymbol 5}{10\divsymbol 5}=\frac{1}{2}\end{align*}[/latex]
3b. [latex]\begin{align*}\frac{6\times 5}{8\times 5}=\frac{30}{40}\;\;\;\frac{6\divsymbol 2}{8\divsymbol 2}=\frac{3}{4}\end{align*}[/latex]
4a. [latex]0.875[/latex]
4b. [latex]16.25[/latex]
4c. [latex]2.6[/latex]
4d. [latex]137.7\overline{2}[/latex]
5a. [latex]0.875[/latex]
5b. [latex]15.750[/latex]
5c. [latex]1.111[/latex]
5d. [latex]0.469[/latex]
6a. [latex]0.08\overline{3}[/latex]
6b. [latex]5.\overline{24}[/latex]
6c. [latex]1.\overline{3}[/latex]
6d. [latex]−0.3\overline{09}[/latex]
Applications
- Calculate the solution to each of the following expressions. Express your answer in decimal format.
- [latex]\begin{align*}\frac{1}{5}+3\frac{1}{4}+\frac{5}{2}\end{align*}[/latex]
- [latex]\begin{align*}1\frac{\frac{3}{8}}{2}-\frac{11}{40}+19\frac{1}{2}\times\frac{3}{4}\end{align*}[/latex]
- Calculate the solution to each of the following expressions. Express your answer in decimal format with two decimals.
- [latex]\begin{align*}\left(1+\frac{0.11}{12}\right)^4\end{align*}[/latex]
- [latex]\begin{align*}1-0.05\times\frac{263}{365}\end{align*}[/latex]
- [latex]\begin{align*}200\left[1-\frac{1}{\left(1+\frac{0.10}{4}\right)^2}\right]\end{align*}[/latex]
- Calculate the solution to each of the following expressions. Express your answer in repeating decimal notation as needed.
- [latex]\begin{align*}\frac{1}{11}+3\frac{1}{9}\end{align*}[/latex]
- [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.
- [latex]\begin{align*}\$134,000(1+0.14\times 23/365)\end{align*}[/latex]
- [latex]\begin{align*}\$10,000\left(1+\frac{0.0525}{2}\right)^{13}\end{align*}[/latex]
- [latex]\begin{align*}\frac{\$535,000}{\left(1+\frac{0.07}{12}\right)^3}\end{align*}[/latex]
- [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]
- [latex]\begin{align*}\frac{\$155,600}{\left(1+\frac{0.06}{12}\right)^8}\end{align*}[/latex]
Solutions
7a. [latex]5.95[/latex]
7b. [latex]15.0375[/latex]
8a. [latex]1.04[/latex]
8b. [latex]0.96[/latex]
8c. [latex]9.64[/latex]
9a. [latex]3.\overline{20}[/latex]
9b. [latex]0.5[/latex]
10. [latex]\$135,182.14[/latex]
11. [latex]\$14,005.26[/latex]
12. [latex]\$525,745.68[/latex]
13. [latex]\$2,693.13[/latex]
14. [latex]\$149,513.74[/latex]
Challenge, Critical Thinking, & Other Applications
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.
- [latex]\begin{align*}\frac{\$648}{0.0575/12}\left[1-\frac{1}{\left(1+\frac{0.0575}{12}\right)^7}\right]\end{align*}[/latex]
- [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]
- [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]
- [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]
- [latex]\begin{align*}\$15,000\left[\frac{\left(1+\frac{0.058}{4}\right)^{16}-1}{\frac{0.058}{4}}\right]\end{align*}[/latex]
- [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]
Solutions
- [latex]\$4,450.29[/latex]
- [latex]\$9,579.23[/latex]
- [latex]\$138,934.38[/latex]
- [latex]\$12,252.25[/latex]
- [latex]\$267,952.30[/latex]
- [latex]\$1,041.58[/latex]
