# 15 Converting Between Systems

You may use a calculator throughout this module.

Converting between the U.S. system and metric system is important in today’s global economy; like it or not, the metric system is infiltrating our lives.

The numbers in these conversion ratios are usually difficult to work with, so we will use a calculator whenever necessary and pay attention to rounding. If you discover other conversion ratios that aren’t provided here, write them down!

# Converting Measurements of Length

Some of these conversion ratios are exact, because a yard is defined to be exactly meters, which means that a foot is exactly meters and an inch is exactly centimeters.^{[1]} The conversions that are not exact are rounded to four significant figures.

in = cm

ft = cm

ft = m or m ft

yd = m or m yd

mi km or km mi

Let’s verify that is actually . We can start by writing as a fraction over 1 and then use the conversion ratio to cancel the units.

Rounding to three sig figs,^{[2]} that’s , which of course is .

Let’s verify that again using the approximate conversion ratio .

Okay, everything looks good; both conversions give us a result that rounds to . As long as we don’t try to keep more than four sig figs in our result, we can use either conversion ratio and get the same result.

Exercises

- In Canada, if a road in a city does not have a speed limit posted, the default speed limit is per hour.
^{[3]}Convert into miles. - A
*Star Wars*clone trooper is 6.00 feet tall. Convert this height to cm. - An Olympic-size swimming pool is meters in length.
^{[4]}How many feet is this? - A 4-inch paintbrush is labeled millimeters. Verify the accuracy of this conversion.
- An electric fan has an advertised diameter of inches, or centimeters. Verify the accuracy of this conversion.
- Is kilometers equivalent to miles? If not, what is the percent error?

# Converting Measurements of Weight or Mass

These conversions are approximate. (Technically, one pound is exactly 0.45359237 kilograms^{[5]}, but we’ll stick to four significant figures.)

oz g

kg lb or lb kg

Exercises

As shown in the picture, a shelving system is rated to hold a total weight of pounds, or pounds on each of its five shelves. The metric equivalents printed on the box are kilograms and kilograms.

- Convert pounds into kilograms. Does your answer agree with the number printed on the box?
- Convert pounds into kilograms. Does your answer agree with the number printed on the box?

- A hand weight weighs 5 kilograms. Convert 5 kilograms to pounds.
- How many grams is a half pound of ground beef?
- A gravy recipe calls for 4 ounces of flour. Convert 4 ounces into grams.
- A smoothie recipe has grams of protein. Convert grams to ounces.
- In around 2010, the National Collector’s Mint (not affiliated with the U.S. Mint) ran a TV commercial selling an imitation gold coin modeled after the U.S. “buffalo” nickel. The commercial made the following claims.
*This replica coin is coated in milligrams of pure gold! And the price of gold keeps going up; gold is worth about per ounce! But you can order these fake coins for only apiece!*What was the approximate dollar value of the gold in one of these coins? - In 2023, the National Collector’s Mint was still selling the imitation gold “buffalo” nickel, but had cut the price to . This version of the coin was coated in milligrams of pure gold.
^{[6]}The price of gold in 2023 was per ounce. What was the approximate dollar value of the gold in one of these coins?

# Converting Measurements of Volume or Capacity

These conversion ratios are approximations rounded to four significant figures.

fl oz mL

L fl oz

L qt

gal L

Exercises

- Stephanie lives in Vermont and buys her home renovation supplies at
*Réno-Dépôt*in Quebec. She buys a toilet that uses 4.8 L of water per flush. How many gallons is this? - How many milliliters of drink are in a 12-ounce can?
- TSA airline regulations limit liquids in carry-on luggage to 100 milliliters. How many fluid ounces is this? (Round your answer to the nearest tenth.)
- If you visit an oil change shop, you may notice large boxes of motor oil. (If you teach math, you may even take a picture of one so you can use it in your course materials.) Verify that 6 gallons of motor oil is equivalent to 22.7 liters.

# Converting Measurements: Extensions

Let’s finish up with some rates that require conversions.

Exercises

Maxine is driving across Canada. Her car has a 14.2-gallon gas tank and gets an average of 26 miles per gallon.

- Approximately how many kilometers—actually, the Canadian spelling is kilometres. Approximately how many kilometres can she travel on a full tank of gas?
- Of course, Canada measures gas in liters. Actually, litres. Convert Maxine’s mileage rate, 26 miles per gallon, to kilometres per litre.

- Source: https://en.wikipedia.org/wiki/International_yard_and_pound ↵
- We round to three sig figs because the measurement 1.98 meters has only three sig figs. ↵
- Source: https://niagarafalls.ca/city-hall/transportation-services/traffic/speed-limits.aspx ↵
- Source: https://swimswam.com/how-big-is-an-olympic-sized-swimming-pool/ ↵
- Source: https://en.wikipedia.org/wiki/International_yard_and_pound ↵
- Source: https://ncmint.com/2023-buffalo-tribute-proof/ ↵