RWM101 Study Guide

1a. Apply the commutative law of addition and multiplication

• Which operations does the commutative property apply to?
• Which operations does the commutative property NOT apply to?

The commutative property states that we can reverse the order of addition or multiplication without changing the outcome. For example: $2+3=3+2$. Therefore, the commutative property applies to addition. Since $3 \times 5=5 \times 3$, the commutative property also applies to multiplication. 

The commutative property does NOT apply to subtraction and division, since $3-5 ≠ 5-3$ and $4÷2 ≠ 2÷4$. 

To review, see:

• [1.1 Commutative Law of Addition]
• [1.1 Commutative Law of Multiplication]

1b. Apply the associative law of addition and multiplication

• Which operations does the associative property apply to?
• Which operations does the associative property NOT apply to?

The associative property states that we can do addition and multiplication in any order or with any grouping of numbers. For example, when adding $4+5+6$, you can add 4 and 6 first, or 5 and 6, or 4 and 5, and you will get the same answer. 

This is helpful when we are adding several numbers together, as grouping makes finding the answer easier. For example, when we add $4+7+16+3$, it is much easier to add $4+16=20$, then $7+3=10$, and finally $20+10=30$. 

Similarly, we can regroup multiplication problems to find the answer more easily. For example, with $5 \times 3 \times 20 \times 6$, it is much easier to calculate $5 \times 20=100$, and $3 \times 6=18$, then finish with $100 \times 18=1800$.

The associative property does NOT apply to subtraction or division. For example, $5-3-2 ≠ 2-3-5$ and $5 ÷ 2 ÷ 3 ≠ 2 ÷ 3 ÷ 5$.

To review, see:

• [1.2 Associative Law of Addition]
• [1.2 Associative Law of Multiplication]

1c. Apply the identity property of addition and multiplication

• What number is the identity for addition?
• What number is the identity for multiplication?

The identity property is based on the mathematical concept of an identity, which is a specific number that can be added to or multiplied by another number and not change its value. The identity for addition is 0, since 0 added to any number does not change its value. For example, $5+0=5$.

In the same way, there is an identity for multiplication, but it is 1, not 0. The idea is the same: 1 multiplied by any number does not change the value of that number. For example, $5 \times 1=5$.

To review, see:

• [1.4 Identity Property of Addition]
• [1.6 Identity Property of Multiplication]

1d. Apply the inverse property of addition and multiplication

• What does a number always equal when added to its inverse?
• What does a number always equal when multiplied by its inverse?

The inverse property of addition simply states that a number added to its "opposite" always equals 0. For example, $5+(-5)=0$. In this case, (-5) would be the additive inverse of 5. The additive inverse is often called the opposite. 

The inverse property of multiplication similarly states that a number multiplied by its reciprocal is always 1. If we have a number, a, then its multiplicative inverse is 1/a, also known as the reciprocal. For example, the multiplicative inverse of 2 is ½. 

To review, see:

• [1.4 Inverse Property of Addition]
• [1.6 Inverse Property of Multiplication]

1e. Apply the zero property of multiplication and division

• What happens when we multiply a number by 0?
• What happens when we divide a number by 0?

The zero property defines what happens when we multiply or divide by zero. In multiplication, any number multiplied by 0 is always 0. For example, $5 \times 0=0$. 

When dividing with 0, we have to consider both dividing 0 by another number and dividing a number by 0. When dividing 0 by another number, such as 0/5, the answer is always 0, since you can break 0 things into as many groups as you want and have 0 in each group. 

The issue comes when dividing by 0, such as 5/0. In this case, you cannot break 5 things into 0 groups. Therefore, the answer is undefined. Undefined means it cannot be computed, or has no value we can determine. Note that 0/0 is also undefined. 

To review, see:

• [1.7 Multiplication by Zero]
• [1.8 Why Dividing by Zero is Undefined]

1f. Apply the distributive property

• How do you apply the distributive property?

The distributive property shows us that we can multiply a number outside parentheses by every element inside the parentheses, provided that the elements inside the parentheses are connected with addition or subtraction. 

For example:

Here is an example involving subtraction:

To review, see:

• [1.9 Using the Distributive Property]

Unit 1 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• additive inverse
• associative property
• commutative property
• distributive property
• identity
• identity property
• inverse property
• multiplicative inverse
• reciprocal
• undefined
• zero property

2a. Find the greatest common factor and least common multiple of whole numbers

• What is the Greatest Common Factor of a set of numbers?
• What is the Least Common Multiple of a set of numbers?

A factor is a whole number that you can divide a given number by. When you have a set of 2 or more numbers, the greatest common factor is the largest number that you can divide all of the numbers in your set by. For example, the greatest common factor of 12 and 20 is 4, because 4 is the largest number that you can divide both 12 (12 /4 =3) and 20 (20/4=5) evenly by.

The least common multiple is the smallest number that a set of numbers is a factor of. If you have a set of numbers, the least common multiple is the smallest number that all the numbers in your set can be multiplied to get. For example, the least common multiple of 4 and 6 is 12, because $4 \times 3=12$ and $6 \times 2=12$. 12 is the smallest number that can be divided by both 4 and 6, making it the least common multiple.

To review, see:

• [2.1 The Greatest Common Factor]
• [2.2 The Least Common Multiple]

Unit 2 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• factor
• greatest common factor
• least common multiple

3a. Calculate problems using negative numbers 

• How do you add negative numbers?
• How do you subtract negative numbers?
• When adding positive and negative numbers, how do you know if the answer is positive or negative?

Negative numbers are the opposite of positive numbers – they are less than zero. Negative whole numbers, zero, and positive whole numbers together are called integers.

When adding a positive number and a negative number, the problem can be rewritten as subtraction (see 1a, where we review the commutative property). For example, $-2+5$ can be rewritten as $5-2$. When adding two negative numbers, you can add the numbers as you would with two positive numbers before making the answer negative. For example, $-5+(-3)= -8$.

When subtracting a negative number from a positive number, the double negative makes a positive. So when subtracting a negative number, change the subtraction to addition and make the number positive. For example, $5-(-3)=5+3=8$. When subtracting a negative number from a negative number, the double negative still becomes a positive. However, since the first number is also negative, the problem becomes like our first example. You can now rewrite the problem as subtraction. For example, $-5 - (-3)= -5+3 = 3-5 = -2$.

When multiplying or dividing negative numbers, if both numbers are positive or both numbers are negative, the answer will always be positive. If only one of the numbers is negative, the answer will be negative. For example $5(-3)= -15 and (-5)(-3)=15$.

To review, see:

• [3.1 Negative Numbers]
• [3.2 Adding Negative Numbers]
• [3.2 Subtracting Negative Numbers]
• [3.3 Multiplying and Dividing Positive and Negative Numbers]

3b. Calculate exponents

• What does an exponent represent?
• How do you calculate an exponent?

Exponents represent repeated multiplication. An exponent is written as a superscript (above a number) and tells you how many times to multiply the number by itself. For example, $4^3=4 \times 4 \times 4$.

To review, see:

• [3.4 Understanding Exponents]
• [3.4 Introduction to Exponents]
• [3.4 Exponent Examples]

3c. Apply the order of operations

• What is the order of operations?

The order of operations is often written as PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It is important to understand the correct order and details of the order of operations.

Parentheses are always first. Simplify within parentheses before moving on.

Exponents are always next. Evaluate any exponents in the problem after the parentheses.

The next step is very important: Although the acronym PEMDAS lists multiplication before division, multiplication is not necessarily done before division. Multiplication and division are done together, starting from left to right. Starting on the left-hand side of the problem, look for multiplication and/or division, and complete in the order that it appears.

Similarly, addition and subtraction are done together, also from left to right.
In the order of operations, think of it as four steps: Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction.

To review, see:

• [3.5 Use the Language of Algebra]

Unit 3 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• exponent
• integer
• order of operations
• PEMDAS

4a. Identify parts of a fraction

• What is the number on the top of a fraction called?
• What is the number on the bottom of the fraction called?

Fractions are made up of three parts: the numerator, the denominator, and the fraction bar.

The fraction bar is the line in the middle, with the numerator on top and the denominator on the bottom, like this: $\dfrac{\text{numerator}}{\text{denominator}}$.

To review, see:

• [4.1 Fractions of Whole Numbers]

4b. Recognize fractions in lowest terms

• How can you tell if a fraction is in lowest terms?
• How can you rewrite a fraction in lowest terms?

Fractions are most often written in lowest terms. While it is common to say, "I will eat 1/2 of a piece" of something, you would not generally say, "I will eat 3/6ths of a piece", even though $\dfrac{1}{2}=\dfrac{3}{6}$.

A fraction is in lowest terms if the numerator and denominator have no factors in common. For example, 3/7 is in lowest terms because 3 and 7 have no factors in common besides 1. On the other hand, 4/8 is NOT in lowest terms, because 4 is a factor of both 4 and 8. Rewriting this fraction in lowest terms would give us ½.

To rewrite a fraction in lowest terms, which is also called reducing a fraction or simplifying a fraction, find a common factor between the numerator and the denominator. By breaking the numerator and denominator into factors, you can cancel the common factor and end up with a reduced fraction.

For example:

$\dfrac{12}{16}=\dfrac{4 \times 3}{4 \times 4}=\dfrac{4}{4} \times \dfrac{3}{4}=\dfrac{3}{4}$

As you can see, we found 4 was a common factor in 12 and 16. Once we rewrite the fraction, we can see that the 4s can be canceled, since $\dfrac{4}{4}=1$. Since 3 and 4 have no common factors, we can say the fraction is now written in lowest terms.

To review, see:

• [4.6 Fractions in Lowest Terms]
• [4.6 Reducing Fractions]

4c. Recognize equivalent fractions

• How can you verify that two fractions are equivalent?

Fractions can be written in many ways and still have the same value. For example, 1/3 and 2/6 are equivalent fractions because they both represent ⅓ of a whole.

You can verify that fractions are equivalent by writing them in their lowest terms and then checking that the lowest terms are identical or by dividing the numerator by the denominator and verifying that their decimal equivalents are the same.

To review, see:

• [4.2 Demonstrating Equivalent Fractions Part I]

4d. Identify improper fractions and mixed numbers

• What makes a fraction an improper fraction?
• What is a mixed number?

A fraction with a value of less than 1 can only be written one way, which is called a proper fraction. There are two ways, however, to write fractions with a value greater than 1: as a mixed number or as an improper fraction.

A fraction with a numerator greater than the denominator is called an improper fraction. Despite its name, improper fractions are commonplace and acceptable answers. For example, $\dfrac{12}{5}$, $\dfrac{11}{9}$, and $\dfrac{22}{2}$ are all improper fractions.

A mixed number is a number consisting of a whole number and a fraction. For example, $2\dfrac{1}{3}$ and $5\dfrac{1}{6}$ are mixed numbers.

To review, see:

• [4.3 Proper and Improper Fractions]
• [4.3 Proper Fractions, Improper Fractions, and Mixed Numbers]

4e.Convert between mixed numbers and improper fractions

• How can you convert from a mixed number to an improper fraction?
• How can you convert from an improper fraction to a mixed number?

To convert a mixed number to an improper fraction, convert the whole number into a fraction with the same denominator as the fraction in the mixed number. Then, add these two fractions together, resulting in an improper fraction. For example, consider $2\dfrac{1}{3}=2+\dfrac{1}{3}=\dfrac{6}{3}+\dfrac{1}{3}=\dfrac{7}{3}$. As shown, you first convert the whole number (2) into a fraction (6/3), and then add it to 1/3 to get the improper fraction 7/3.

To convert an improper fraction to a mixed number, start by dividing the numerator by the denominator. The whole number in the answer will be the whole number in your mixed number, and the remainder will be the numerator in the fraction part. The denominator will always stay the same. For example, when converting 13/5 to a mixed number, divide 13 by 5. You will find an answer of 2 with a remainder of 3. This means you were able to divide it 2 whole times, with 3 left over. This can now be written as a mixed number of $2\dfrac{3}{5}$.

To review, see:

• [4.5 Changing an Improper Fraction to a Mixed Number]
• [4.4 Visualize Improper Fractions]

4f. Determine the least common denominator between fractions

• When do you need a common denominator?
• How do you find the least common denominator?

As you know, the denominator of a fraction is the number on the bottom. So, finding a common denominator means making the denominators of two or more fractions the same. This involves finding the least common multiple of the denominators of the fractions. Common denominators are needed when adding or subtracting fractions and are also helpful if you want to compare two fractions.

For example, consider $\dfrac{1}{6}$ and $\dfrac{1}{8}$. To find a common denominator, think of the least common multiple of 6 and 8, which is 24. 24 is the smallest number that we can divide by both 6 and 8, therefore it is the least common multiple. Since we have to multiply 6 by 4 to get 24, we must also multiply its numerator by 4, giving us the equivalent fraction 4/24. Similarly, we multiply 8 by 3 to get 24, so we multiply its numerator by 3 and get the equivalent fraction 8/24. Now, we have found two equivalent fractions with the least common denominator.

To review, see:

• [4.7 Finding Common Denominators]

4g. Use equivalent fractions to add and subtract fractions and mixed numbers with like and unlike denominators

• What has to be done in order to add or subtract fractions?
• How do you add and subtract fractions?
• How do you add and subtract mixed numbers?

The steps that you take to add or subtract fractions will vary depending on the type of fractions that you are working with.

When adding or subtracting fractions with a common denominator (or like denominator), the answer is as simple as adding or subtracting the numerators while keeping the denominators the same. For example, $\dfrac{1}{5}+\dfrac{2}{5}=\dfrac{3}{5}$ or $\dfrac{7}{9}-\dfrac{5}{9}=\dfrac{2}{9}$.

In order to add or subtract fractions with different denominators (or unlike denominators), you must first convert them to fractions with a common denominator. Then add or subtract the numerators while keeping the denominators the same, such as $\dfrac{1}{3}+\dfrac{5}{6}=\dfrac{2}{6}+\dfrac{5}{6}=\dfrac{7}{6}$. As you can see, the $\dfrac{1}{3}$ is changed into the equivalent fraction $\dfrac{2}{6}$, so we have a common denominator. Then, we can add the fractions together.

To review, see:

• [4.8 Add and Subtract Fractions with Common Denominators]
• [4.8 Add and Subtract Fractions with Different Denominators]
• [4.8 More with Fractions]

4h. Solve multiplication and division problems with fractions and mixed numbers

• How do you multiply fractions?
• How do you divide fractions?
• How do you multiply or divide mixed numbers?

When multiplying fractions, finding a common denominator is not necessary-- simply multiply the numerators and the denominators. For example, $\dfrac{2}{3} \times \dfrac{5}{7}=\dfrac{2 \times 5}{3 \times 7}=\dfrac{10}{21}$. You can simplify the answer after multiplying or by simplifying any common factors between the numerator and denominator before multiplying (see 3c).

When dividing fractions, we can easily change the problem to multiplication. When dividing two fractions, keep the first fraction the same, change the division symbol to multiplication, then flip the second fraction. For example, $\dfrac{2}{3}÷ \dfrac{5}{7}=\dfrac{2}{3} \times \dfrac{7}{5}=\dfrac{14}{15}$.

When multiplying or dividing mixed numbers, simply convert them to improper fractions first, complete your multiplication or division, then, if necessary, convert the answer back to a mixed number.

To review, see:

• [4.11 Multiplying Fractions and Mixed Numbers]
• [4.12 Dividing Fractions]
• [4.12 Dividing Mixed Numbers]

4i. Solve real-world math problems involving fractions

• How do you solve word problems with fractions?

When solving a word problem involving fractions, it is important to first understand the problem and then transform the problem correctly into an equation that you can solve. Once you have transformed the problem into an equation, you can solve it just as you would solve any other equation involving fractions.

For example, if Bill eats ⅓ of a pizza every day for lunch, how much pizza does he eat in a full 7-day week?

First, we must write this problem as an equation. Since he eats ⅓ of a pizza each day for 7 days, our equations should be $\dfrac{1}{3} \times 7$. Therefore, the answer is $\dfrac{7}{3}$ or $2\dfrac{1}{3}$ pizzas.

To review, see:

• [4.13 Multiplying and Dividing Fractions Word Problems]

Unit 4 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• common denominator
• denominator
• equivalent fraction
• fraction bar
• improper fraction
• least common denominator
• like denominator
• lowest terms
• mixed number
• numerator
• proper fraction
• reducing a fraction
• simplifying a fraction
• unlike denominator

5a. Recognize the place value of decimals 

• What are the place values for decimals?

With whole numbers, we know that the location of a digit determines its value. The number 231 is really 200+30+1, because the 2 is in the hundreds place, the 3 is in the tens place, and the 1 is in the ones place. 

Decimals also have place values. The first position to the right of the decimal is the tenths place, then moving to the right, you have the hundredths place, the thousandths place, the ten-thousandths place, etc. For example, the decimal 0.247 is 2 tenths, 4 hundredths, and 7 thousandths. 

To review, see:

• [5.1 Introduction to Decimals]

5b. Apply rounding and estimation concepts with decimals 

• How do you round decimals?

When rounding decimals, there are two simple steps. First, determine the position that you want to round to and look at the digit to the right of that position. If it is a 5 or more, you round the number up, and if it is a 4 or less, you round the number down. For example, consider the number 23.4156. Let's round the number to the hundredths place. There is a 1 in the hundredths place, so we look at the digit to the right of the 1, which is a 5. Since it is 5, we round the 1 up to a 2. Therefore, the correctly rounded answer is 23.42. If, instead, the number was 23.4136, it would round down to 23.41, since the next digit is 3.

To review, see:

• [5.2 Rounding Decimals]

5c. Convert between fractions and decimals 

How do you convert a decimal to a fraction?
How do you convert a fraction to a decimal?

When converting a fraction to a decimal, simply remember that a fraction is just a division problem. Divide the fraction, and you will find the decimal value. For example, if you are trying to convert ¾ to a decimal, divide 3 by 4, and you will find the decimal value is 0.75. 

When converting a decimal to a fraction, you can use your knowledge of place value to quickly convert any decimal to a fraction. For example, the decimal 0.23 goes to the hundredths place, making it 23 hundredths. So, the decimal equivalent is 23/100. After converting to a fraction, you may need to simplify the fraction to write it in lowest terms.

To review, see:

• [5.2 More on Rounding Decimals]

5d. Perform operations with decimals 

• How do you add decimals?
• How do you subtract decimals?
• How do you multiply decimals?
• How do you divide decimals?

When adding or subtracting decimals, you must make sure to line up the decimal points when setting up the problem. Then, you can add or subtract as you would with any other numbers. The decimal will follow in the same place in the solution. 

For example:

When multiplying decimals, simply multiply the two numbers while ignoring the decimals. Once you're done, count the number of digits to the right of the decimals combined between the two numbers. Once you have multiplied, place the decimal with the same number of digits to the right as you counted. For example, when multiplying $2.4 \times 3.71$, simply multiply $24 \times 371$, then place the decimal with 3 digits to the right, since there are 3 total digits to the right of the decimal in the original problem. Therefore, the answer is 8.904.

When dividing by a decimal, you want to convert the problem to one without any decimals. So, when dividing $34.23 ÷ 0.25$, you can change the problem by moving the decimal on both numbers by the same number of spaces. The goal is to make the divisor, the number you are dividing by, into a whole number. So, in this case, since the divisor is 0.25, we want to move both the decimal 2 places to the right. This transforms our problem into $3423 ÷ 25$, and therefore the answer is 136.92.

To review, see:

• [5.5 Adding Decimals]
• [5.5 Subtracting Decimals]
• [5.6 Multiplying Decimals]
• [5.7 Dividing Decimals]

5e. Solve real-world and mathematical problems with decimals 

• How do you solve real-world problems with decimals?

When solving a word problem involving decimals, it is important to first understand the problem and then transform the problem correctly into an equation that you can solve. Once you have transformed the problem into an equation, you can solve it just as you would solve any other equation involving fractions.

For example, Josh has $3.45 and wants to buy some french fries for $1.09 including tax. How much will he have left over?

First, we must set up the subtraction problem, $3.45-1.09$. Then, we solve it like any subtraction problem with decimals, and the answer is $2.36.

To review, see:

• [5.8 Applications with Decimals]

Unit 5 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• divisor
• place value
• rounding decimals

6a. Use ratio concepts to solve problems

• What is a ratio?
• What does a ratio represent?

A ratio is essentially just a fraction. The only difference is how we think about the values of the numerator and denominator. In a fraction, we usually think of the numerator as the part and the denominator as the whole. In a ratio, the numerator and denominator represent a relationship between two quantities. 

For example, let's say that for every $2 you earn, your boss earns $3. That could be represented as a ratio of 2/3, also written as 2:3. In this case, the 3 isn't the total amount earned but rather the amount your boss earns in relation to the amount you earn. 

To review, see:

• [6.1 Introduction to Ratios]
• [6.1 Ratios]

6b. Calculate unit prices and rates 

• How do you calculate a price per unit?
• How do you calculate a rate per hour/day/etc.?

To calculate unit price (the price per unit), simply divide the total cost by the number of units. When you see the word per, the value that comes after it is always what you divide by, so price per unit means the total price divided by the number of units. For example, if you buy 6 cans of tomatoes for $3, the price per unit is $3/6=$0.50 per can.

To calculate a unit rate, first you must decide the unit that you want to calculate. For example, if a new roof for your house costs $10,000, and will take 4 days to complete, you could calculate the cost per day, by dividing 10,000 by 4. $10,000 ÷ 4=$2,500 per day. However, maybe the total working hours are 100 hours; then, you could calculate the cost per man-hour as $10,000 ÷ 100=$100 per working hour.

To review, see:

• [6.2 Finding Unit Rates and Prices]

6c. Apply the processes of solving a proportion 

• What is a proportion?
• How do you solve a proportion?

A proportion is an equation where two ratios are equal to each other, with one quantity missing. For example, continuing from the previous example, if the ratio of your income to your boss' income is 2:3, how much does your boss make when you earn $400? For a proportion like this, set up the known ratio on one side, and a fraction on the other side where the corresponding values match up. $\frac{2}{3}=\frac{400}{x}$ would be the correct setup. In this case, the x represents the amount your boss makes. To solve, simply cross-multiply to get $2x=1200$, and then by dividing, you will find the answer is $600.

To review, see:

• [6.3 Solving Proportions]

6d. Analyze proportional relationships to solve real-world and mathematical problems 

• How do you solve word problems with proportions?

When solving a word problem involving proportions, it is important to first understand the problem and then transform the problem correctly into an equation that you can solve. Once you have transformed the problem into an equation, you can solve it just as you would solve any other equation involving proportions.

For example, say you are driving your car at 75mph on the highway. How far will you travel in 5 hours?

First, set up the proportion. 75 mph is a ratio of $\dfrac{\text{75 miles}}{\text{1 hour}}$. Set up the proportion with 5 hours and the unknown distance, $\dfrac{\text{75 miles}}{\text{1 hour}} =\dfrac{\text{x miles}}{\text{5 hours}}$. Cross multiply to find $x=375 \text{ miles}$. 

To review, see:

• [6.3 Applications of Proportions]

Unit 6 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• per
• proportion
• rate
• ratio
• unit price

7a. Identify the components of a percent problem 

What does percent mean?

Percent has a literal meaning: "per" means "out of" and "cent" means 100. Think of the words century or centipede, where "cent" also means 100. When we say percent, it literally means "out of 100". For example, 35% means 35 out of 100. 

To review, see:

• [7.1 The Meaning of Percent]

7b. Convert between percent, decimal, and/or fraction notation 

• How do you convert a percentage to a fraction?
• How do you convert a percentage to a decimal?
• How do you convert a decimal to a percentage?
• How do you convert a fraction to a percentage?

Since percentages are basically just fractions, converting them to fractions or decimals is quite easy.

When converting a percentage to a decimal, simply rewrite the percentage as a fraction with a denominator of 100. Then, simplify as necessary. For example, 63% is 63/100 in fractional form.

Converting a percentage to a decimal is just as easy. First, convert the percentage to a fraction, then divide the fraction to get a decimal. Since the denominator of the fraction is always 100, you can simply move the decimal point in the numerator two places to the left. Continuing from our previous example, 63% is 63/100, which is 0.63 as a decimal. 

To convert a decimal to a percentage is even easier. Simply multiply the decimal by 100 (or move the decimal point two places to the right) and you are done. For example, the decimal 0.635 is equal to 63.5%. 

To convert a fraction to a percentage, simply convert the fraction to a decimal, then multiply by 100, as we just saw. For example, to convert $\dfrac{3}{5}$ to a percentage, first convert $\dfrac{3}{5}$ to 0.6, then convert to 60%. 

To review, see:

• [7.2 Converting Percents to Decimals and Fractions]

7c. Use proportional relationships to solve multi-step percent problems

• How do you calculate a percentage?

Calculating a percentage is all about correctly identifying the "part" and the "whole" in the situation, then dividing the part by the whole and finally multiplying by 100 to convert the decimal you found to a percentage. 

For example, say a tour company is giving one-hour tours. Out of 225 tours, they found that 37 were over 1 hour, and the rest were exactly 1 hour. What percentage of the tours finished on time? To solve this problem, first we need to find out how many tours ended on time. Since there were 225 total tours and 37 went over time, 225-37=188 tours finished on time. Now that we have the correct values for our problem, we can calculate the percentage. 188 tours out of 225 finished on time, so the percentage is $\dfrac{188}{225} \times 100 = 83.6\%$.

To review, see:

• [7.3 Solving Basic Percent Problems]

7d. Apply percent concepts in practical application 

• How do you calculate percent increase/decrease?
• How do you calculate sales tax or commission? 

To calculate percent of change (increase or decrease), divide the amount of the change by the original price/value. For example, if a pair of pants were originally $80 but are on sale for $60, divide the difference in price ($80-$20= $20) by the original price ($80). Since $\dfrac{20}{80}=0.25$, convert the decimal to a percentage, and you will find the percent decrease of 25%. 

To calculate tax (percent of increase) or commission (percent of what is sold), you must first convert the tax or commission percentage to a decimal, by dividing by 100. Then, multiply the decimal times the price to find the tax or commission. For example, if you buy some clothes for $150 plus 7% tax, first convert 7% to 0.07, then multiply $0.07 \times 150$ to find the tax is $10.50. 

To review, see:

• [7.4 Calculating Sales Tax and Commission

Unit 7 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• commission
• percent
• percent of change
• tax

8a. Determine the mean, medium, mode, and range from a given set of data

• What is the mean, and how do you find it?
• What is the median, and how do you find it?
• What is the mode, and how do you find it?
• What is the range, and how do you find it?

For this section, all examples will use the following data set: 3, 4, 5, 5, 7, 10, 13, 14, 15. The numbers in your set are called elements. 

The mean is more commonly called "the average". To find the mean, add all of the numbers in your data set, then divide by the number of elements in the set. For example, the sum of the elements in our sample data set is 76. Since there are 9 elements, we divide the sum by 9, making the mean 8.44. $\dfrac{3+4+5+5+7+10+13+14+15}{9}=8.44$.

The median is the "middle number" in the set, when you order the elements from least to greatest. To find the median, first arrange your data set from least to greatest, as in our sample. Then, cross off the numbers one at a time until you find the middle number. Since our data set has 9 elements, the 5th number is the median, which is 7. If your data set has an even number of elements, there isn't 1 middle number, so we find the 2 middle numbers, and the mean of those two numbers is the median. 

The mode is the element that appears the most number of times in a set. In our example, 5 is the mode because there are two 5s and every other element appears once. More than one number can be the mode if 2 or more elements appear the same number of times but more than any other elements. 

The range is the difference between the largest and smallest elements in a set. To find the range, subtract the smallest element from the largest. In our example, the range would be 15-3=12.

To review, see:

• [8.1 Averages and Probability]
• [8.1 Finding Mean, Median, and Mode]

8b. Represent and interpret data in a stem-and-leaf plot

• How do you read a stem-and-leaf plot?
• How does a stem-and-leaf plot help you interpret data?

A stem-and-leaf plot is one way of representing a set of data. Each data point is broken into two parts: the stem and the leaf. The stems are to the left of the vertical line, and the leaves are on the right. Each stem-and-leaf plot has a key that explains how each stem and leaf should be interpreted. 

For example:

In this stem-and-leaf plot, the key shows that the stem is the tens digit, and the leaf is the ones. That means the data in the above set is 7, 11, 14, 18, 25, 25, 25, 26, 27, 27, 29. 

A stem-and-leaf plot is an easy way to visualize the spread of the data, and see where the data is concentrated, particularly in a large set of data. 

To review, see:

• [8.2 Stem and Leaf Plots]

8c. Represent and interpret data in a line graph

• What is a line graph?
• How do you interpret a line graph?

A line graph is a way of representing data as points above the number line. You use one dot for each corresponding data point. This gives an easy way to view both the values in the data and the spread of the data. For example, consider a data set of 15, 15, 16, 16, 17, 17, 19, 20. The corresponding line graph would be:

Line graphs can also be drawn with a vertical axis, where instead of putting multiple dots vertically to represent the quantity of a single data point, the vertical axis can represent the quantities, and a single dot can be placed to correspond with the quantity of data. 

For example:

In this line graph, you can see that the price is represented on the vertical axis, and then a point is placed to correspond with the correct value for each month. 

To review, see:

• [8.3 Reading Line Graphs]

8d. Represent and interpret data in a bar graph

• What is a bar graph?
• How do you interpret a bar graph?

A bar graph is a graph where categories of data appear on the horizontal axis, and the quantity appears on the vertical axis. For each category, you draw a bar vertically with the height corresponding to the quantity for that category. 

In this example, you can see the four teams represented on the horizontal axis, and the bar above each represents the points they scored. The data can be read as approximately Team 1=24, Team 2=36, Team 3=12, and Team 4=38. 

To review, see:

• [8.4 Reading Bar Graphs]

8e. Represent and interpret data in a box-and-whisker plot

• What is a box-and-whisker plot?
• How do I interpret a box-and-whisker plot?

A box-and-whisker plot is an excellent way of representing a lot of data in a way that can clearly show the spread of the data. The spread of the data refers to how much of the data is grouped where. For example, two sets of data could have the same minimum and maximum value, but one set could have many values close to the minimum, while the other may have many values close to the maximum. This can be easily deciphered through a box-and-whisker plot. 

In the box-and-whisker plot above, the thin lines extending outward are called the whiskers. The end of the left whisker is the smallest value, called the minimum, and the end of the right whisker is the largest value, the maximum. The line inside the box represents the median, and the ends of the box represent the median of the lower half of the data, also called Quartile 1, and the median of the upper half of the data, also called Quartile 3. In the example above, the minimum would be 41, Quartile 1 is 43, the median is 45, Quartile 3 is 48, and the maximum is 50. 

To review, see:

• [8.5 Box-and-Whisker Plots]

8f. Represent and interpret data in a circle graph

• What is a circle graph?
• How do you interpret a circle graph?

A circle graph (also called a pie graph) is a graph that is a circle broken into sections representing the proportional parts of a whole. 

For example:

In this example, you can see that four teams are playing against each other. The relative frequency of how often they win is their percentage, and the size of their section is relative to the percentage. The largest section represents the most wins. 

To review, see:

• [8.6 Reading Pie Graphs (Circle Graphs)]

8g. Represent and interpret data in a pictograph

• What is a pictograph?
• How do you interpret a pictograph?

A pictograph is a graph similar to a bar graph, but the quantity in each category is represented by pictures rather than the height of a bar. There is usually a key, which tells you how many people/items are represented by each picture. For example:

In this pictograph, each pony image represents 3 ponies, and therefore the Fancy Farm has 15 ponies, Pony Pasture has 6, etc. 

To review, see:

• [8.7 Reading Pictographs]

Unit 8 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• bar graph
• box-and-whisker plot
• circle graph
• element
• horizontal axis
• key
• leaf
• line graph
• mean
• median
• mode
• pictograph
• Quartile 1
• Quartile 3
• range
• spread
• stem
• stem-and-leaf plot
• vertical axis
• whisker