Customary Units and Conversions
You are making hot chocolate for the whole bus after a basketball game. The recipe says to use 2 quarts of milk, but the only measuring tool you have is a 1-cup scoop. How many scoops do you need? By the end of this page you will be able to answer that in seconds, and to handle conversions that are much bigger.
Customary units: length, weight, and capacity
The US customary system is what you see on road signs (miles), in recipes (cups), and at the grocery store (pounds). Every quantity belongs to a family: length (how long), weight (how heavy), or capacity (how much liquid fits).
Inside each family, units are just bigger or smaller "rulers" for the same thing. Here are the equivalences you need.
| Family | Equivalences |
|---|---|
| Length | 12 inches (in.) = 1 foot (ft) 3 feet = 1 yard (yd) 5,280 feet = 1 mile (mi) 1,760 yards = 1 mile |
| Weight | 16 ounces (oz) = 1 pound (lb) 2,000 pounds = 1 ton (T) |
| Capacity | 8 fluid ounces (fl oz) = 1 cup (c) 2 cups = 1 pint (pt) 2 pints = 1 quart (qt) 4 quarts = 1 gallon (gal) |
Capacity is the family people mix up most, so here it is as a picture. Each row covers the same amount of liquid, cut into smaller and smaller pieces.
Read the picture: 1 gallon = 4 quarts = 8 pints = 16 cups. Counting the 8 fl oz in each cup, 1 gallon = 128 fl oz.
Smaller units are for small things, bigger units for big things. You measure a pencil in inches, a football field in yards, and a road trip in miles.
During a summer camp, the counselors prepared $4.5$ gallons of lemonade for the afternoon activity. Convert this volume into quarts. Use the conversion rate…
During a summer camp, the counselors prepared $4.5$ gallons of lemonade for the afternoon activity. Convert this volume into quarts. Use the conversion rate $4 \text{ quarts} = 1 \text{ gallon}$.
A recipe for a large batch of pancakes calls for $32$ fluid ounces of milk. The measuring tools in the kitchen only measure in cups. Convert the amount of milk into cups…
A recipe for a large batch of pancakes calls for $32$ fluid ounces of milk. The measuring tools in the kitchen only measure in cups. Convert the amount of milk into cups to follow the recipe. Use the conversion rate $8 \text{ fluid ounces} = 1 \text{ cup}$.
Converting with a unit ratio
Take the fact $12 \text{ in.} = 1 \text{ ft}$. It says the same length has two names. So the fraction $\dfrac{12 \text{ in.}}{1 \text{ ft}}$ is equal to $1$. Multiplying by $1$ does not change the length. It only changes the name. This fraction is called a unit ratio (or conversion factor).
You can write it two ways: $\dfrac{12 \text{ in.}}{1 \text{ ft}}$ or $\dfrac{1 \text{ ft}}{12 \text{ in.}}$. Choose the one where the unit you want to get rid of ends up on the bottom, so it cancels.
The picture gives you a quick check. If you go from a bigger unit to a smaller unit, you need more of the small ones, so the number gets bigger: multiply. If you go from a smaller unit to a bigger unit, you need fewer, so the number gets smaller: divide.
A carpenter is building a custom bookshelf and needs a wooden board that is exactly $108$ inches long. Convert this length into feet to help him purchase the correct…
A carpenter is building a custom bookshelf and needs a wooden board that is exactly $108$ inches long. Convert this length into feet to help him purchase the correct board at the hardware store. Use the conversion rate $12 \text{ inches} = 1 \text{ foot}$.
A post office requires the weight of a package to be recorded in pounds. If a customer brings a box that weighs $80$ ounces, what is the weight of the box in pounds? Use…
A post office requires the weight of a package to be recorded in pounds. If a customer brings a box that weighs $80$ ounces, what is the weight of the box in pounds? Use the conversion rate $16 \text{ ounces} = 1 \text{ pound}$.
A restaurant chef is preparing a large soup that requires $3.5$ quarts of chicken broth. The supplier only sells the broth in $1$-pint containers. How many pint…
A restaurant chef is preparing a large soup that requires $3.5$ quarts of chicken broth. The supplier only sells the broth in $1$-pint containers. How many pint containers does the chef need to use to get exactly the required amount of broth?
Multi-step conversions
Sometimes there is no direct fact between your two units. How many inches are in a yard? The table gives yards to feet, and feet to inches. So you chain two conversions, one after the other. Each unit ratio cancels one unit.
Going down the ladder (big to small) you multiply. Going up you divide. And $1$ mile $= 5{,}280$ ft, since $1{,}760 \times 3 = 5{,}280$.
The same idea works for capacity: $1$ gal $\to$ $4$ qt $\to$ $8$ pt $\to$ $16$ cups $\to$ $128$ fl oz. For weight, one more step: $1$ ton $= 2{,}000$ lb $= 32{,}000$ oz. So $3$ tons is $3 \times 2{,}000 = 6{,}000$ lb.
A tailor is making curtains for a school auditorium and needs $4$ yards of special velvet fabric. The fabric store sells the material by the inch. How many inches of…
A tailor is making curtains for a school auditorium and needs $4$ yards of special velvet fabric. The fabric store sells the material by the inch. How many inches of fabric does the tailor need to order? Use the conversion rates $1 \text{ yard} = 3 \text{ feet}$ and $1 \text{ foot} = 12 \text{ inches}$.
A sports team brings a cooler containing $6$ gallons of water to a tournament. During the game, the players drink the water using paper cups that hold exactly $8$ fluid…
A sports team brings a cooler containing $6$ gallons of water to a tournament. During the game, the players drink the water using paper cups that hold exactly $8$ fluid ounces each. Assuming no water is spilled, how many full cups of water can be poured from the cooler?
A freight elevator in an office building has a maximum weight capacity of $2.5$ tons. A delivery crew needs to load $30$ identical crates into the elevator. If each…
A freight elevator in an office building has a maximum weight capacity of $2.5$ tons. A delivery crew needs to load $30$ identical crates into the elevator. If each crate weighs $160$ pounds, will the total weight exceed the elevator's capacity? Calculate the total weight of the crates in pounds and compare it to the capacity in pounds.
Rates and square units
Converting a rate
A rate like $6$ inches per minute has two units, one on top and one on the bottom. To change it, convert one unit at a time. Use a unit ratio for the top unit, then another for the bottom unit. To compare two rates (who is faster?), write both with the same units first.
Converting area
Area is measured in square units. A square yard is a square that is $3$ ft by $3$ ft, so it holds $3 \times 3 = 9$ square feet.
So you do not divide an area by $3$. The easiest and safest way is to convert each length first, then multiply.
In a science experiment observing animal speeds, a garden snail is recorded moving at a speed of $6$ inches per minute. A small box turtle is recorded moving at a speed…
In a science experiment observing animal speeds, a garden snail is recorded moving at a speed of $6$ inches per minute. A small box turtle is recorded moving at a speed of $25$ feet per hour. By converting both speeds to inches per minute, determine which animal is faster and by how many inches per minute.
A backyard swimming pool holds $2,400$ gallons of water. A garden hose is used to fill the pool, and water flows from the hose at a steady rate of $5$ quarts per minute.…
A backyard swimming pool holds $2,400$ gallons of water. A garden hose is used to fill the pool, and water flows from the hose at a steady rate of $5$ quarts per minute. How many hours will it take to completely fill the empty pool?
A family is replacing the carpet in their rectangular bedroom. The room's dimensions on the architectural plans are $180$ inches long and $144$ inches wide. The carpet…
A family is replacing the carpet in their rectangular bedroom. The room's dimensions on the architectural plans are $180$ inches long and $144$ inches wide. The carpet they want to buy is priced at $\$28.50$ per square yard. Calculate the total cost to carpet the room. (Hint: Convert the measurements to yards before finding the area).
Converting between customary and metric
Most of the world uses the metric system (kilometers, kilograms, liters), and science class does too. When you travel or read a foreign recipe, you convert to customary units. The facts are not exact, so you will see the symbol $\approx$, which means "is approximately equal to." Problems will give you the factor:
| Fact | Meaning |
|---|---|
| $1 \text{ km} \approx 0.62 \text{ mi}$ | a kilometer is shorter than a mile |
| $1 \text{ mi} \approx 1.61 \text{ km}$ | the same fact the other way |
| $1 \text{ kg} \approx 2.2 \text{ lb}$ | a kilogram is heavier than a pound |
| $1 \text{ gal} \approx 3.785 \text{ L}$ | a gallon is a bit less than 4 liters |
The rule is the same as before. The fact says "$1$ of this unit equals that many of the other unit." Going from the first unit to the second, multiply. Going back, divide.
While driving across the border from the United States to Canada, a family sees a sign indicating that their destination is $50$ kilometers away. Convert this distance…
While driving across the border from the United States to Canada, a family sees a sign indicating that their destination is $50$ kilometers away. Convert this distance into miles using the given conversion factor: $1 \text{ kilometer} \approx 0.62 \text{ miles}$.
A veterinarian in New York is treating a dog that was recently adopted from Europe. The dog's medical records state its weight as $25$ kilograms. Convert the dog's…
A veterinarian in New York is treating a dog that was recently adopted from Europe. The dog's medical records state its weight as $25$ kilograms. Convert the dog's weight into pounds using the given conversion factor: $1 \text{ kilogram} \approx 2.2 \text{ pounds}$.
While driving in Europe, an American tourist sees a speed limit sign that reads $90 \text{ km/h}$. The rental car's speedometer shows the speed in miles per hour (mph).…
While driving in Europe, an American tourist sees a speed limit sign that reads $90 \text{ km/h}$. The rental car's speedometer shows the speed in miles per hour (mph). Using the conversion factor $1 \text{ mile} \approx 1.61 \text{ kilometers}$, calculate the speed limit in mph. Round your answer to the nearest whole number.
Converting temperature
Temperature is different. A degree Celsius ($^\circ$C) is a different size than a degree Fahrenheit ($^\circ$F), and the two scales also start at different places: water freezes at $0^\circ$C but at $32^\circ$F. So a single multiplication is not enough. We use a formula.
$$F = \tfrac{9}{5}C + 32 \qquad\qquad C = \tfrac{5}{9}(F - 32)$$The first changes Celsius to Fahrenheit. The second changes Fahrenheit to Celsius. Use the order of operations: in the second formula, do the parentheses first.
Good landmarks to remember: water freezes at $0^\circ$C $= 32^\circ$F, normal body temperature is $37^\circ$C $= 98.6^\circ$F, and water boils at $100^\circ$C $= 212^\circ$F.
A temperature change is different from a temperature. If a problem says the temperature drops $3.6^\circ$F, that is already in Fahrenheit. Convert only the starting temperature, then subtract the drop.
The local weather forecast in Chicago predicts a high temperature of $77^\circ\text{F}$ for the day. Convert this temperature to degrees Celsius using the formula…
The local weather forecast in Chicago predicts a high temperature of $77^\circ\text{F}$ for the day. Convert this temperature to degrees Celsius using the formula $C = \frac{5}{9}(F - 32)$.
An American baker finds a French recipe for croissants that requires the oven to be preheated to $200^\circ\text{C}$. The baker's oven only displays temperatures in…
An American baker finds a French recipe for croissants that requires the oven to be preheated to $200^\circ\text{C}$. The baker's oven only displays temperatures in Fahrenheit. Using the formula $F = \frac{9}{5}C + 32$, calculate the exact temperature the baker should set on the oven dial.
A hiker starts climbing a mountain early in the morning when the temperature at the base is $15^\circ\text{C}$. For every $1,000$ feet of elevation gained, the…
A hiker starts climbing a mountain early in the morning when the temperature at the base is $15^\circ\text{C}$. For every $1,000$ feet of elevation gained, the temperature drops by $3.6^\circ\text{F}$. If the hiker climbs $4,000$ feet, what is the expected temperature at that altitude in degrees Fahrenheit? Use the formula $F = \frac{9}{5}C + 32$ to convert the initial temperature first.
Solving real-world measurement problems
Real problems often hide a conversion in the middle of the story: a price per pound but an order in ounces, a limit in tons but weights in pounds, or a map scale and a distance in feet. Follow the same four steps every time.
The key rule: before you add, subtract, multiply, or compare, put all quantities in the same unit.
At a local deli counter, premium roasted turkey is priced at $\$8.40$ per pound. A customer orders exactly $24…
At a local deli counter, premium roasted turkey is priced at $\$8.40$ per pound. A customer orders exactly $24$ ounces of the turkey. Calculate the total cost of the customer's order.
An athlete is running a standard marathon, which has a distance of $26.2$ miles. During the race, the athlete's average stride length (the distance covered in one step)…
An athlete is running a standard marathon, which has a distance of $26.2$ miles. During the race, the athlete's average stride length (the distance covered in one step) is $32$ inches. Calculate the total number of strides the athlete takes to complete the marathon.
A driver has a car with an empty gas tank that holds exactly $15$ gallons. In the United States, gasoline costs $\$3.40…
A driver has a car with an empty gas tank that holds exactly $15$ gallons. In the United States, gasoline costs $\$3.40$ per gallon. Right across the border in Canada, gasoline costs $\$1.25$ per liter. Using the conversion factor $1 \text{ gallon} \approx 3.785 \text{ liters}$, determine where it is cheaper to completely fill the tank and by how many US dollars.