We know 5 city blocks and 5 miles are very different lengths to walk; $5 and 5¢ are very different values of money; 5 minutes and 5 years are very different amounts of time to wait - even though all of these quantities are represented by the number 5. Every variable is measured in terms of some unit. Since there are often several different units available to use, it is important when naming a variable to state which units we are choosing to measure it in.
Since 1 meter and 3.281 feet are just two different ways of saying (approximately) the same distance, here are two more math sentences we can write down:
A fraction where the top and bottom are equal quantities expressed in different units is sometimes called a unit conversion fraction. Because it’s equal to 1 (or at least very close to 1), multiplying by the unit conversion fraction doesn’t change the value, just the units.
You might wonder how we knew to choose the unit conversion fraction that has the feet on the top and the meters on the bottom. One reminder for how this works is to think fractions. It’s like the meters on the top and bottom cancel, leaving the units as feet.
One more idea to keep in mind when converting units: a few large things equals a lot of small things. Instead of buying a lot of small bags of ice to fill your cooler, you can buy a few larger bags of ice instead. In our example, a meter is much bigger than a foot. So it makes sense that a small number of meters (3 meters) equalled a larger number of feet (9.8 feet). That might seem backwards, but that’s how it works.
When we converted 3 meters into feet, we got 9.8 feet, which might sound like a funny answer. We’re much more used to a whole number of feet and then the fraction in inches. It’s 9 feet and some number of inches. To figure out the inches we look at the decimal part:
The board is about 9 feet and 10 inches high. The common shorthand for this answer is 9′ 10″. (That’s pronounced 9 foot 10, as in our team’s new center is 6 foot 7.) The ′ symbol indicates feet and ″ indicates inches.
You might have guessed that 9.8 feet would be 9 feet and 8 inches, or 9′ 8″. I mean, that sort of looks obvious. The reason it’s not is because decimal numbers are based on 10, but inches are based on 12. So, the 0.8 feet really means \(\frac{8}{10}\) feet, and 8 inches would mean
So, the diver jumped from a board that was about 9′ 10″ high. What was the highest height she reached? The highest height we had recorded for the diver was 4.48 meters. Now we know that’s
Quick caution here. When we converted 3 meters into feet, we got 9.843 feet, which we rounded off to get 9.8 feet, and then we just used the 0.8 feet to find the extra inches. Maybe we should have used the more precise 0.843 feet instead. Here’s what happens.
Similarly, when we converted 4.48 meters into feet, we got \(14.69888\ldots\) feet, which we rounded to 14.7 feet, and then we just used the 0.7 feet to find the extra inches. Maybe we should have used the more precise 0.69888 feet instead:
Good, that one also worked out the same. You should be careful, though, because rounding too much too early might produce a big change in your final result.
Some units, like the ones we use for length or time, are pretty straightforward. Other units, like the ones we use for speed, are made up of several other units put together. We can use the same thinking to convert these more complicated units: we just have to do them one at a time.
During the first 0.2 seconds of her dive, we calculated the diver’s speed as 4.4 meters per second. How fast is that? We can certainly convert to feet per second, using the fact that \(3.281 \text{ feet} \approx 1 \text{ meter}\text{:}\)
Does that help us understand how fast she’s going? Maybe a little. But, we’re probably most familiar with speeds measured in miles per hour, because that’s what mph stands for.
Let’s convert 14.4364 feet per second to miles per hour. To accomplish this, we’ll need to convert seconds into hours, and we’ll need to convert feet into miles. It doesn’t matter which order you do this, or how many steps you need to take in between. Our plan in this example is to go from seconds to minutes, and then from minutes to hours, and to go from feet to miles.
This time we got a smaller number because she can go a lot fewer miles in an hour compared to feet in an hour. Notice how we needed to divide by 5,280. Numbers on top of the fraction multiply. Those on the bottom divide.
If you’re having trouble setting up unit conversions, remember to write down the units so you can see how they cancel. If you can’t remember a number for a unit conversion, like feet for one mile, try searching online.
Jenna is studying in Finland this term and rented an older car to drive. Her car gets 7.6 kilometers per liter; she’d like to convert that to miles per gallon (mpg).
Kamari is shopping at an internationally-based retail store. She is looking at a curtain rod that projects 10 cm from the wall. What is that in inches?
Ephraim works at a plant that produces very delicate electronic switches. He measured the lifetime for one switch at 4.18 hours. Another had lifetime 19.75 hours. Convert each time into hours and minutes. That means H:MM format.
Phillip measured his office using a digital measure. One wall is 21.8 feet. The other is 10.2 feet. How long is each wall measured in the more usual feet and inches?
Abdi volunteers at a food bank. He noticed that the shelf on the back wall was bending so he measured its length at 12′5″. The formula for load needs the length written as a decimal. Convert the length to a decimal number of feet.
Siri carries around a insulated water bottle that holds 0.6 liters. How many of those bottles full of water does she need to drink each day? Use that 1 liter \(\approx\) 1.057 quarts and 1 quart = 32 (liquid) ounces.
She learns that no matter what the road signs might say, the maximum speed limit in Finland in winter is never more than 100 km/hr. How fast is that in miles per hour (mph)? Use 1 mile \(\approx\) 1.609 km.
In August 2012, Phelps improved his time and won Olympic gold, but failed to break the world record his teammate Ryan Lochte had set a year earlier of 1 minute, 54 seconds.
The typical weight limit for a suitcase on flights within Africa is 20 kg. How many pounds is that? Use \(1 \text{ kilogram} \approx 2.2 \text{ pounds}\text{.}\)
How many servings are in a 20 ounce package of cookies where a serving size is 3 cookies and each cookie weighs 11 grams? Use \(1 \text{ ounce} = 28.3 \text{ grams}\text{.}\)
My corner convenience store sells a “thirst quencher” size of soft drink; it holds 64 (fluid) ounces. If a can of soft drink is 12 (fluid) ounces, how many cans are in the “thirst quencher”?
Authorities are tracking down the source of a pollution spill on a nearby river. They suspect that the local plant is inadvertently leaking waste water. Last week they found 35 minutes of waste water flow on Monday, 1 hour and 11 minutes on Tuesday, 1/4 hour on Wednesday (that’s 0.25 hours in decimal), none on Thursday, and then 98 minutes Friday.
Convert units as needed to complete the following table showing each time in minutes, each time in hours, and each time in hours and minutes (H:MM format).
Harold’s History textbook mentions that in 1800 the city encompassed about 6,000 acres. How many square miles is that? Use \(1 \text{ square mile} = 640 \text{ acres}\text{.}\)
Harold’s Economics textbooks lists the recent high price of crude oil at $100 per barrel. He’d like to know what that means in $/gallon of gasoline. It turns out that 1 barrel of crude oil produces about 19.4 gallons of gasoline.
