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Your aunt has an excellent recipe for a breakfast sandwich that her customers love, but the cafe is all out of eggs and buns. She needs you to run to the grocery store for supplies. Exactly one egg and one bun are required to make each sandwich. Unfortunately, you can only purchase eggs in packages of 12 and buns in packages of eight. If you return with one dozen eggs and one eight-package of buns, four eggs will go unused. If you return with one dozen eggs and two packages of buns, four buns will be left over. What is the smallest number of eggs and buns you can purchase so there are no leftovers of eggs or buns? (Feel free to take out a sheet of paper and work the numbers! It will not take long to see that the answer is 24).
In this brief unit, we explore the concepts of least common multiple (LCM) and greatest common factor (GCF). Our run-of-the-mill questions, such as our egg-and-bun problem, indicate how these concepts are relevant to our everyday lives, but they also show up in lots of other mathematical questions. Here we are talking about sets of whole numbers: in Unit 1, our operations applied to whole numbers, fractions, and every other type of real number. Consequently, notions of least common multiple and greatest common factor belong to the part of mathematics that studies patterns among whole numbers, an area called number theory.
Completing this unit should take you approximately 1 hour.
We can use our four basic operations to solve so many types of mathematical and real-world problems, but we need to be careful when we combine or repeat them. The commutative and associative laws apply to certain operations but not others. We use the distributive law to guide calculations when we combine different operations. But we need to follow an order of operations, which we apply to any combination of operations we need.
In this unit, we learn how to use exponents and exponential notation to represent repeated multiplication. For example, repeatedly multiplying two against itself five times results in 
. We will pay careful attention to how negative numbers interact with our operations and clarify how to group different symbols (subjected to different operations) in the correct and clear order.
Completing this unit should take you approximately 4 hours.
Fractions are amazingly helpful; we can use them to tell so many different kinds of stories, but this can also make them seem overwhelming or confusing. How likely will a coin flip result in heads? We use a fraction to express the answer to this probability 
: our outcome (a heads up) should occur 1 out of 2 times or one-half of the time.
If you serve 10 pizzas to 40 students, how much pizza should each student receive? We use fractions to determine that each student should receive 
or one-fourth of a single pizza. We also use fractions to discuss ratios and rates, such as this car gets 40 miles per gallon.
Combining fractions using addition, subtraction, multiplication, or division is a bit more complicated than combining integers, but these combinations are useful in many situations and problems.
For example, if your friend Andy has one-half of a chocolate bar and Tai has one-third of a bar, how much do they have together? To answer this question, we need to add the rational numbers 
(see Section 4.8). In this unit, we explore these kinds of topics, beginning with a basic review of how to notate a fraction and ending with a discussion on how to use them in multi-step problems.
Completing this unit should take you approximately 8 hours.
In this unit, we study decimals, which are simply another way to express fractions. We use this way of notating fractions (including improper ones and mixed numbers), most notably when dealing with money or currencies.
For example, if the price of your aunt's amazing vanilla latte is $3.50, we understand this to mean it costs three whole dollar bills and fifty cents. The period or decimal point, as mathematicians call it, separates whole numbers from the fractions in this expression – it serves this purpose in this price and in all other decimal expressions. Our whole numbers (in this case, 3) sit to the left of the decimal point, and fractional quantities sit to the right of the decimal point.
Since the digit 5 appears immediately to the right of the decimal, it really means five-tenths. The zero means zero hundredths. Of course, we can reduce the fraction five tenths as
.
Altogether this means we can translate $3.50 literally or mathematically to three whole dollars and one-half of a dollar. That is
.
Because decimals are just a different way to express factions, they are just as useful and omnipresent. In this unit, we explore how to convert decimals into explicit fractions and how to add, subtract, multiply, and divide them.
Completing this unit should take you approximately 6 hours.
In this unit, we study another application of fractions, ratios, and proportions. These are mathematical concepts we use all the time, probably without even realizing it. Have you ever compared unit prices for different packages of the same type of food at the grocery store? That is a ratio.
When driving 65 mph (miles per hour) on the highway, have you ever determined how long it will take you to get to your destination? That is a proportion. In sports, statisticians use proportions to predict an athlete's performance based on what they have accomplished in the past. In this unit, we explore how to write ratios, set up and solve proportions, and apply these skills to real-world experiences.
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Cooking is probably where we use ratios most.
Imagine your aunt's coffee shop is hosting a party. She and her staff are being handsomely rewarded for providing the venue and all of the food. She knows her special chocolate raspberry cake feeds 10 people and uses the following ingredients: four cups of flour, two cups of sugar, three sticks of butter, six eggs, and lots of chocolate. However, 25 people will be attending the gathering, and she needs to make a larger cake! Exactly how much flour and sugar does she need to pull this off?
Working through the next few sections will help you use rates and ratios to answer these questions so your aunt can complete this delicious task (and others, too!).
Completing this unit should take you approximately 2 hours.
We use percentages every day. For example, how about when your aunt's coffee shop advertised pumpkin spice lattes for 50% off – you and your aunt knew this meant half off, and so did the individuals in the ensuing rush of new customers. In this unit, we explore how to compute percentages. What does this mean, and how do we think about percentages more generally?
In this unit, we answer this question and lots of others that involve percentage computations and applications. For example, we will convert between percents and fractions or decimals and learn about percentage increases and decreases. We will also explore how to calculate some real-world uses, such as restaurant tips and sale prices.
Completing this unit should take you approximately 2 hours.
After you perform a series of calculations or gather data, you may need or want to visualize your information so it is easier to make sense of. It is usually easier to make sense of a graph or chart than study a long list of numbers. Visual presentations and summaries can make certain trends or patterns apparent and clear. Long lists of numbers can hide these data trends.
For example, we call this image a bar graph. This one shows the results of a survey your aunt asked the customers at her coffee shop to complete. She wanted to know what flavors she should offer next month to increase her sales, so she asked them to choose their favorite latte flavor. The image is pretty clear, even from a distance: you do not need to sort through a long list of numbers to see that pumpkin spice is the most popular flavor.
We use graphs and charts in almost every field, not just with coffee shop customers. Politicians use them to explain demographics and voting trends in campaigns and elections. Businesses use graphs and charts to show growth trends and areas where they are not doing as well.
It is worth noting that unscrupulous politicians and businesses can, and often do, misuse or distort their performance charts so they display information in a way that supports their interests or goals. For example, suppose a chai tea salesman conducted your aunt's coffee shop survey and used this image to present the data results:
Their presentation significantly reduces the visual gap between pumpkin spice and chai tea lattes. They are trying to convince your aunt to buy more chai tea ingredients, leaving her with less money to buy more pumpkin spice flavoring.
Because charts are so ubiquitous and because they are so easy to manipulate, it is important to know how to read and interpret graphs and charts. Learning this skill will also help you create your own visual presentations. In this unit, we discuss how to use different types of graphs and charts in mathematics. We will interpret the results for each type of graph or chart and learn to create them. We will also explore topics related to long lists of numbers (or data sets), including the measures of central tendency.
Completing this unit should take you approximately 3 hours.
This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary. It is not meant to replace the course materials!
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Take this exam if you want to earn a free Course Completion Certificate.
To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7-day waiting period between each attempt. Once you pass this final exam, you will be awarded a free Course Completion Certificate.
In this unit, we discuss properties and patterns for adding, subtracting, multiplying, or dividing numbers. You are probably so familiar with these facts that it may seem strange to point them out at all. But there are benefits to naming and using these properties. Some are purely mathematical, but plenty are common, everyday ones too.
First, let's talk about the commutative properties of addition and multiplication. These tell us that it does not matter if you rearrange your numbers when you add or multiply them. So 
and 
both equal five and 
and 
both equal 60.
But we cannot swap all of our actions around like this. For example, what about when we bathe our dog, Harry, first and then dry him? Subtracting provides a mathematical example. Subtracting is not commutative: 
is not the same as 
. Division is not commutative, either. You get a different result if you divide a 60-dollar dinner bill among 10 diners than if you divide a 10-dollar dinner bill among 60 diners! The order is important.
In other words, some of our basic operations enjoy certain properties (like commutativity), while others do not. You need to keep track of these rules to avoid confusion, incorrect dinner bills, and wet dogs. These properties will not only help you calculate more complicated equations later on, but they will also help you as you consider less intuitive or obvious properties, such as What happens if I divide 60 by 0? Paying careful attention to all of these rules will help you on your mathematical journey and in the real world.
Completing this unit should take you approximately 2 hours.