In this video lesson, you will learn how measurements are very much like variables when it comes to adding, subtracting, multiplying, and dividing them. You will also see examples of how they are done.
Measurements Are Like Variables
In this video lesson, we are going to talk about performing operations such as adding and multiplying, on measurements. Why is this important? It is important because in real life many problems deal with measurements where you will need to perform these operations to get your answer.
For example, restaurant owners need to add and subtract measurements to find out how many tables and chairs will fit inside their restaurant. They need to add the length and width of the tables and chairs together to make sure there is enough room for them.When you need to perform your basic operations on your measurements, treat them like variables, a symbol for something we don’t know. As we go in this video lesson, we will see how to subtract, add, multiply, and divide our measurements. You will see that it is a lot like adding, subtracting, multiplying, and dividing variables. So, let’s get going.
We first talk about adding measurements. So, say you are a restaurant owner. You want to put two tables together to create a longer table. But, you need to add the lengths of the tables to make sure that you have enough room in your restaurant to do this. One table measures 3 feet and the other table measures 4 feet. How do you add them together?First, we see that both measurements are in feet, so that tells us that we can go ahead with our addition without any changes.
We can go ahead and add 3 feet and 4 feet together. What do we get? We get 7 feet.Notice that I still have my measurement unit of feet attached to my number 7. If we thought about this as adding variables, we can change our feet to x. Our problem then becomes 3x + 4x.Because both terms have an x as the variable, they are like terms so we can go ahead and add to get 7x. We can switch out the x for feet again and we would have our answer of 7 feet as well.
We just talked about adding when our measurement units are the same. But what do we do when our measurement units are similar, but not exactly the same? For example, what if one table measured 3 feet and the other table measured 36 inches? What would we do then?Well, we see that our measurement units are not the same so that tells us that we can’t go ahead with addition. But we see that we can convert our measurement units. We need our measurement units to be the same before adding. We see that one measurement is in feet and the other is in inches. We can go about it two ways.
We can either change our feet measurement into inches or our inches measurement into feet.I’m going to change our inches measurement into feet. How do you choose? If you are doing multiple choice problems, pick the unit that is in the answers.
To change the inches measurement into feet, we need to use what we know about converting measurements. We know that there are 12 inches in a foot. So, I can divide my inches measurement by 12 to convert it into feet. Doing that, I get 36 inches / 12 = 3 feet. So, now I have 3 feet + 3 feet.
My measurement units are the same so I can go ahead with the addition to get 6 feet.
We also need to make sure our measurement units are the same when subtracting. If they are the same, we can go ahead with our subtraction.
If they are not, then we need to convert the measurement so that the measurement units are the same.For example, say you, as the restaurant owner, wanted to install a fish tank inside your restaurant. The fish tank measures 7 feet in length. The space where you want to install the fish tank is 10 feet in length. Will the fish tank fit? We can find out by subtracting the length of the fish tank from the length of the restaurant space.
We see that both measurement units are the same so we can go ahead with our subtraction of 10 feet – 7 feet. What do we get? We get 3 feet. That means that you will have 3 extra feet and the fish tank will fit.
Turning this into a variable problem, we have 10x – 7x, which equals 3x, or 3 feet.Again, if our measurement units are different, we need to convert them so that they are the same before proceeding. So, if our fish tank measured 120 inches and our restaurant space measured 10 feet, we need to either convert the 120 inches into feet or the 10 feet into inches.
This time we are going to convert the feet into inches. To do this, I multiply my 10 feet by 12 to get 120 inches. Hey, look at that. The measurements are exactly the same and 120 – 120 is 0. That means the fish tank just fits!
Now that you’ve got the fish tank covered, it’s time to look at the floor covering.
Your floor looks pretty bad. Customers would probably get grossed out looking at it. So, you decide to tile your restaurant floor.You call a flooring company and the company asks you how many square feet of tile you need for your restaurant. What they are asking you is the area of your restaurant. To find your area, you need to multiply the length of your restaurant by its width since your restaurant is a rectangle.
Your restaurant measures 7 feet 6 inches in width and 10 feet in length. So, we need to multiply these two numbers together.Since our problem wants our answer in square feet, I need to keep my measurement units in feet. I do notice that my width measurement has an inches component. I need to convert that part into the decimal part of my feet. I have to divide my inches by 12 to find the decimal part.
So, 6 divided by 12 is 0.5, so my decimal is 7.5 feet. Now, I can go ahead and multiply 7.5 feet by 10 feet. My answer is 75 square feet.
Notice that my units are now square feet instead of just feet. This is because we multiplied two feet measurements together.So, we have feet * feet = feet^2. If we translated this into variable form, we would have 7.5x * 10x, which equals 75x^2. See the squared part of the variable?Just like with addition and subtraction, if you had similar measurement units such as inches and feet, you would convert the measurement units so that they are the same.
Unlike addition and subtraction, though, you can multiply two measurement units that are completely different.For example, you can multiply 1 meter and 1 pound together to get 1 meter pound. Just like with variables, for adding and subtracting, your variables must be the same, but for multiplying, your variables can be different.All we did was multiply our measurement units together just like we would do if we had two different variables. If we were multiplying by a number, we would go ahead with the multiplication and our measurements would stay the same.
For example, 1 meter * 4 = 4 meters.
Dividing measurements follows similar rules to multiplying measurements. You can divide different measurement units. But, if you can convert your measurement units to the same, do so.If you had a measurement that consisted of both feet and inches, you would convert it into a decimal just like we did for multiplying. Also, just like with multiplication, if you are dividing by a number, you can go ahead with the division and your units will stay the same.For example, if you had a 6 inch lollipop stick that you wanted to cut into thirds, you can use division to find out how long each resulting piece would be.
You would divide 6 inches by 3 and you would get 2 inches as your answer. Note that your units stayed the same.But if you wanted to see how fast your friend’s car is going, you would actually keep your measurement units as a fraction. For example, say your friend’s car is going 70 miles every 2 hours. You would use division to find your car’s speed.
You would divide 70 miles by 2 hours. Your answer, then, is 35 miles per hour.Note that the measurement units of the answer are kept in fraction form. You can think of it in terms of algebra as 70x / 2y, which turns into 35x/y.
So, what have we learned? We learned that measurement units behave somewhat like variables, a symbol for something we don’t know. We can add and subtract measurements when they have the same measurement units just like we can add and subtract when our variables are the same. We convert our measurement units so they are alike whenever we can.We can multiply measurement units that are alike and those that are different. When we multiply two measurements that are the same, we end up with an answer whose units are squared. When we divide two measurement units that are different, we end up with an answer with our measurement units in fraction form.
After this lesson, you should have the ability to:
- Explain how to add, subtract, multiply and divide measurement units when thinking of them like variables
- Describe what to do when your measurement units are different