**Measurement Conversions**

Hi, and welcome to this video on **Measurement Conversions**! Converting between different measurements happens all the time, whether you’re measuring solutions in a chemistry lab or putting together a recipe in your kitchen. For example, let’s say you’re baking chocolate chip cookies and you only have measuring cups that measure in 1 cup, ½ cup, and ¼ cup but the recipe calls for 6 ounces of chocolate chips. How do you know how many chocolate chips to put in? In this video, we’ll learn how to use measurement conversions to solve these sorts of problems. Let’s get started!

Let’s start by figuring out our chocolate chip problem. Remember, we only have our cup measurements, but the recipe is asking for ounces. Well, this is where conversions come in handy. In order to figure this out, we need to set up conversion fractions. The first step in setting up conversion fractions is finding our conversion factor. A **conversion factor** relates one quantity to another. In our chocolate chip example, our conversion factor is 1 cup = 8 oz. Now that we know what our conversion factor is, we can set up our conversion fraction.

When setting up these fractions, always start with your known value as your first fraction by placing it over 1 like this:

\(\frac{6\text{ oz}}{1}\)

We then multiply by our conversion factor, which we also need to turn into a fraction. When we do this, we want to pay close attention to which value we place in the **numerator** and which value we place in the **denominator**. Since we have ounces and are looking for cups, we want to place our cup value in the numerator and our ounces value in the denominator so that our ounces units will cancel out and we will be left with cups.

There’s 1 cup has 8 ounces

\(\frac{6\text{ oz}}{1}\times \frac{1\text{ cup}}{8\text{ oz}}\)

We then multiply across and simplify.

So our ounces cancel out, and we’re left with 6, times 1 cup, is 6 cups, over 8. So there are six-eighths of a cup, or three-fourths cups is equal to 8 ounces.

\(\frac{6\text{ oz}}{1}\times \frac{1\text{ cup}}{8\text{ oz}}=\frac{6\text{ oz}×1\text{ cup}}{1×8\text{ oz}}=\frac{3}{4}\text{ cups}\)

Notice how our ounces cancel out on the top and bottom and we are able to simplify the fraction \(\frac{6}{8}\) by dividing both parts by 2. This gives us our final answer of 6 ounces = ¾ cup. So we need ¾ cups of chocolate chips for our cookie recipe and can use our ¼ cup measuring cup three times.

Let’s look at another example.

Sarah is driving down a road at 30 mph. How fast is she traveling in feet per second?

This problem is a little different from our last one because it will require two conversions. We need to find our conversion factors for miles to feet and for hours to minutes. There are 5,280 feet in a mile and 3,600 seconds in an hour, so our conversion factors are 1 mile = 5,280 feet and 1 hour = 3,600 seconds.

First, we write our known value as a fraction. Notice, this is in miles per hour, so we are going to write the miles part on top and the hour part on bottom. She travels 30 miles in one hour, so we will write it like this:

\(\frac{30\text{ miles}}{1\text{ hour}}\)

Now we want to multiply by our conversion factors. Remember, we have two different conversion factors so we need to be sure to include both in this step.

\(\frac{30\text{ miles}}{1\text{ hour}}\times \frac{5,280\text{ feet}}{1\text{ mile}} \times \frac{1\text{ hour}}{3,600\text{ seconds}}\)

There are 5,280 feet in 1 mile, and we want to put the mile on bottom because we want our units to cancel out. And in 1 hour there are 3,600 seconds. Again, we have our hour on top over here because we have it on bottom over here and we want them to cancel out. So when we multiply through, our hours cancel out, our miles cancel out, and then we can just multiply across. So we have 30 times 5,280 feet, over 3,600 seconds.

\(\frac{30\text{ miles}}{1\text{ hour}}\times \frac{5,280\text{ feet}}{1\text{ mile}} \times \frac{1\text{ hour}}{3,600\text{ seconds}}=\frac{30\times 5,280\times 1}{1\times 1\times 3,600}\text{ ft/sec}\)

When we simplify, this gives us: 44 feet per second

\(\frac{30\text{ miles}}{1\text{ hour}}\times \frac{5,280\text{ feet}}{1\text{ mile}} \times \frac{1\text{ hour}}{3,600\text{ seconds}}=\frac{30\times 5,280\times 1}{1\times 1\times 3,600}\frac{ft}{sec}=44\text{ ft/sec}\)

If Sarah is traveling 30 miles per hour, she is also traveling 44 feet per second.

Let’s take a look at one more example.

Caleb is painting a wall that is a total of 96 square feet. How big is the room in square inches?

In this example, we need to pay close attention to our units. We are measuring in square feet, which means our conversion fractions will look a little different. Our conversion factor for feet to inches is 1 foot = 12 inches. However, if you think ahead, we are going to run into a problem in our second step. Let’s go through our process as normal and see if we can figure out what to do once we get there.

We start by writing out our known value.

96 square feet. Remember, we use conversion fractions, so we want to turn this into a fraction, so we divide by 1.

\(\frac{96\text{ ft}^2}{1}\)

Then we multiply by our conversion factor.

There are 12 inches in 1 foot. Remember, feet needs to be diagonal to each other so that eventually they’ll cancel out.

\(\frac{96\text{ ft}^2}{1}\frac{12\text{ in}}{1\text{ ft}}\)

But what’s wrong with this conversion? Well, if you multiply the units out, you can see that we’ll be left with feet times inches, ’cause one of our feet will cancel out but the other one won’t. To fix this, we simply need to multiply once more by our conversion factor.

\(\frac{96\text{ ft}^2}{1}\frac{12\text{ in}}{1\text{ ft}}\frac{12\text{ in}}{1\text{ ft}}\)

Since we are working with square units, it makes sense that we would multiply by our conversion factor twice, or in other words, multiply by our conversion factor squared.

Let’s finish out our problem by multiplying it all out.

Our feet squared will cancel out with both of our feet down here. So we’re left with 96 times 12, times 12 square inches. When we multiply this out, we will get an answer that’s equal to 13 thousand 824 square inches.

\(\frac{96\text{ ft}^2}{1}\frac{12\text{ in}}{1\text{ ft}}\frac{12\text{ in}}{1\text{ ft}}\)\(=13,824\text{ in}^2\)

And that’s all there is to it! This process will work for conversions between any two units. As long as you know your conversion factors, you can follow these steps and find your new measurement.

I hope this review was helpful. Thanks for watching and happy studying!

## Practice Questions

**Question #1:**

Solve the following: 47 hours = ? minutes

0.78

1,128

2,820

0.47

**Answer:**

The correct answer is C. We’re starting with a number measured in hours, and our goal is to output something measured in minutes. We know that there are 60 minutes in 1 hour (this is our conversion factor). So, we set up our conversion fraction:

\(\frac{47\text{ hours}}{1}\times\frac{60\text{ minutes}}{1\text{ hour}}\)

(Note that the “minutes” measurement is in the numerator and the “hours” measurement is in the denominator- because we want to cancel out the hours unit!)

\(\frac{47\text{ hours}}{1}\times\frac{60\text{ minutes}}{1\text{ hour}}=\frac{2,820\text{ minutes}}{1}=2{,}820\text{ minutes}\)

**Question #2:**

Suppose a water tank holds 28 gallons. How many pints of water could we fit in the same tank? (Note: 1 US gallon = 8 US pints)

3.5 pints

1.75 pints

448 pints

224 pints

**Answer:**

The correct answer is D. For this problem, we will only need to run through *one* conversion. Since we know our conversion factor (1 US gallon = 8 US pints), we can go ahead with our conversion fraction process:

\(\frac{28\text{ gallons}}{1}\times\frac{8\text{ pints}}{1\text{ gallon}}=224\text{ pints}\)

**Question #3:**

A local farmer, Jamie, just finished installing a running track somewhere on the land. Jamie wants to measure how long this 400 meter track is in feet. Given that there are approximately 3.28 feet in 1 meter, what measurement will Jamie end up with?

1,312 feet

121.9 feet

396.7 feet

1,463 feet

**Answer:**

The correct answer is A. The pieces of information that we need out of this word problem are straightforward to find: we have an input measuring 400 meters, we have our conversion factor (1 meter = 3.28 feet), and we know we want to end up with an output measured in feet. Luckily, we only have one step in our conversion process:

\(\frac{400\text{ meters}}{1}\times\frac{3.28\text{ feet}}{1\text{ meter}}=1,312\text{ feet}\)

**Question #4:**

Solve the following: 4 feet^{2} = ? centimeters^{2}

10.2 cm^{2}

25.8 cm^{2}

121.9 cm^{2}

3,716.1 cm^{2}

**Answer:**

The correct answer is D. In this question, not only are we dealing with “squared” units (meaning: we’ll apply conversion factors twice in order to cancel out this “squared” piece), but we’ll also need to use two different conversion factors: 1 ft = 12 in and 1 in = 2.54 cm. We’ll first convert feet^{2} into inches^{2}, and then we will convert inches^{2} into centimeters^{2}. Observe:

\(\frac{4\text{ ft}^2}{1}\times\frac{12\text{ in}}{1\text{ ft}}\times\frac{12\text{ in}}{1\text{ ft}}=576\text{ in}^2\)

and now

\(\frac{576\text{ in}^2}{1}\times\frac{2.54\text{ cm}}{1\text{ in}}\times\frac{2.54\text{ cm}}{1\text{ in}}=3{,}716.1\text{ cm}^2\)

**Question #5:**

The McPherson family is re-carpeting their 650 square-foot basement. If they can put down 50 sq. ft. of carpet per hour, how many in^{2} are the McPherson’s finishing every minute?

2 in^{2}/min

10 in^{2}/min

120 in^{2}/min

720 in^{2}/min

**Answer:**

The correct answer is C. This word problem may seem overwhelming at first, but let’s break everything down and figure out what our goal is and what information we need in order to meet this goal!

We are wanting to end up with a value measured in “inches^{2} per minute,” but we are given the measurement: 50 ft^{2}/hr. So, we will need to convert feet^{2} to inches^{2}, *and* we will need to convert hours to minutes.

(Notice that the “650 square-foot” value, given at the beginning of this problem, is a red-herring; we don’t need to pay *any* attention to this piece of information!)

For our calculations, we’ll use the conversion factors: 1 foot = 12 inches and 1 hour = 60 minutes. Let’s work through it:

\(\frac{50\text{ ft}^2}{1\text{ hr}}\times\frac{12\text{ in}}{1\text{ ft}}\times\frac{12\text{ in}}{1\text{ ft}}=\frac{7,200\text{ in}^2}{1\text{ hr}}\)

then (focusing on the hours-to-minutes conversion) we have

\(\frac{7,200\text{ in}^2}{1\text{ hr}}\times\frac{1\text{ hr}}{60\text{ min}}=\frac{120\text{ in}^2}{1\text{ min}}\)