Of all of the topics in math and algebra, multiplying fractions is probably the single thing that has a reputation for being much harder than it actually is. If you want to learn how to multiply fractions, we’re going to show a simple and easy approach that you can use with normal fractions or if you want to multiply improper fractions. You’ll only have to learn this one idea to do them all.
Let’s start out with a sample problem:
A lot of people wouldn’t have any idea how to multiply these two fractions together, but this is one of the easiest things in the world to do. All you have to do is multiply the top two numbers together to get the top part of your answer. Since we know 5 times 3 is 15, we start off by getting 15 on the top of our answer:
And guess what? We multiply the bottom two numbers to get the bottom number of our answer. We know that 2 times 4 is 8, so the bottom number of our answer is 8 like so:
Isn’t that easy? There’s nothing special that you have to do if you want to multiply fractions. The top part of the answer is the top two numbers from your problem multiplied together. The bottom part of your answer is the bottom two numbers from your problem multiplied together. It doesn’t get any easier than that.
This is how to multiply fractions the easy way.
In the first part of this series on the foundations of algebra 1, we looked at expressions and terms. We showed exactly what they are and how they can be manipulated. In what follows, we take this information and look at how it can be used to make solving equations very easy.
Changing Around Equations
An equation is when we set two expressions equal to each other. For example, if we have 4 + 3 as an expression and 7 as another expression, we could make an equation by saying 4 + 3 = 7. We’ll often refer to the “left side” of an equation or the “right side” of the equation, and these names are relative to the equals sign. For example, 4 + 3 is on the left side of the equation above, and 7 is on the right side of the equation.
If we have an equation that has variables in it, then sometimes we can use logic to figure out what the values are for the variables. For example, suppose that we have the equation x + 1 = 4. It makes sense that x would be 3 since 3 + 1 is equal to 4. Whenever we have more complicated equations, however, we’ll need to make use of a trick that we’ll look at now.
The left and right sides of equation are balanced sort of like a scale. If we do something to the left side of an equation, then the equation will remain balanced if we do the same thing to the right side of an equation. Suppose we have a very simple equation like 5 = 5. If we add 3 to the left side of the equation and the right side of the equation, then we’ll have 5 + 3 = 5 + 3, and it’s obvious that it’s okay to do this.
Now suppose we have an equation like x – 14 = 33. If we could get rid of the – 14 portion of the left side of the equation, then we would know exactly what x is because we would have an equation that looks like x = with the value of x on the right side of the equation. We can do this by adding 14 to the left side of the equation so that the – 14 cancels out with the + 14. From our previous example, we know that we can add 14 to the left side of the equation as long as we add it to the right side as well.
So let’s add 14 to both sides to get x – 14 + 14 = 33 + 14. If we simplify each of the two expressions (x – 14 + 14 is the first expression and 33 + 14 is the second expression), then we’ll get x = 47. The equation is quickly solved with a small amount of logic.
For another example, suppose we have -3b = 42. We would like to get rid of the -3 on the left side of the equation, and we can do so by dividing it by -3. To keep the equation balanced, we have to divide the right side by -3 as well. We get -3b/-3 = 42/-3, and after simplifying both sides, we have b = -14.
In more complicated equations, we might have to make more than one change to the equation, but the general idea remains the same: we want to get the variable all by itself on one side of the equation. Suppose we have the equation 4a – 7 = 5. We can add 7 to both sides to get rid of the – 7 on the left side of the equation, and that will give us 4a = 12. Then we can divide both sides by 4 to get rid of the 4 on the left side of the equation, and we’ll get a = 3.
The last few paragraphs above show just about everything that there is to know about solving equations on the basic level. Everything else that you learn in algebra about solving equations is just learning new operations to add to the list you know already like addition, subtraction, multiplication and division. If you can understand the basic ideas presented in the above, then you’ll easily eliminate the vast majority of the hang-ups that you’ll have in any algebra class.
When dealing with students who are in math classes in general, I tend to find that the vast majority of them do not understand the basics. By basics, I don’t mean arithmetic, but the real basic inner workings of expressions and equations. In a previous blog post, I broke down the definitions of terms, expressions and equations because so many people do not really know what those three words mean without having to think about it. However, I did not cover how those parts work together, and that’s what I’m going to do here. If you’re like most students, then understanding what I’m going to cover in the following, will eliminate at least half of your problems understanding algebra in general.
Breaking Down Terms
A term is composed of two parts, one of which is optional. The first part is a constant, and a constant is just a number like 5, -6 or 1.2. The second part of a term is the variable portion, and it consists of one or more variables multiplied together. Note that these variables can have exponents as well. While all terms must have a constant portion, they do not have to have a variable portion.
You can add terms if they have the exact same variable portion. For example, you could add 3ab² and 9ab² because their variable portions are the same. You could not add 4ab² and 2a²b. Even though they use the same variables, they are set to different powers, so they are not exactly the same. When two terms have the exact same variable portion, they are called like terms.
To add terms, you make sure that their variable portions are exactly the same, and then you just add the constants. With our example above, we had 3ab² + 9ab². To add these, we just add 3 + 9 to get 12 for the constant portion, and then we just keep the variable portion the same to get 12ab². Subtraction works the exact same way; you could do 5a²b – 8a²b = –3a²b, for example.
It’s worth noting that 5 and -9 are also like terms because they have the same variable portion (ie: they have no variable portion). You add these numbers the same way, obviously, by adding the constants (5 + -9) and leaving the variable portion the same (there is no variable portion) to get -4.
Simplifying an Expression
We know from our previous blog post that an expression is a series of terms added (or subtracted) together. For example, 2a²b + 5ab² – 7ab is an expression. Sometimes you can simplify an expression by adding (or subtracting) like terms.
In the expression 6x² – 4y + 3x², there are two terms that have the exact same variable portion. We can combine them to get 9x² like we learned to do two paragraphs ago. The expression would then become 9x² – 4y. Since there aren’t any terms left that have the same variable portion, we know that the expression is simplified.
Sometimes we’ll need to use the distributive property to simplify an expression. Suppose we have the expression 3(2x – 4) + 9 – 4x. We’ll need to multiply the 3 by the (2x – 4) before we can make any progress. Once we do the multiplication, we’ll be left with 6x – 12 + 9 – 4x. At this point, we can combine like terms as before to get 2x – 3. Again, since we have no like terms, we know that we have simplified the expression as much as possible.