Systems of Measurement

Make unit conversions in the U.S. system

There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.

The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length, and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallon. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.

The equivalencies of measurements are shown below, with the common abbreviation for each measurement in parentheses.

U.S. System of Measurement

In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, and so on. We use the identity property of multiplication to do these conversions. We’ll restate it here for easy reference.

To use the identity property of multiplication, we write $1$ in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that $1$ foot is equal to $12$ inches, so we can write $1$ as the fraction $\tfrac{1\ \text{foot}}{12\ \text{inches}}$. When we multiply by this fraction we do not change the value, but just change the units.

But $\tfrac{12\ \text{inches}}{1\ \text{foot}}$ also equals $1$. How do we decide whether to multiply by $\tfrac{1\ \text{foot}}{12\ \text{inches}}$ or $\tfrac{12\ \text{inches}}{1\ \text{foot}}$? We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert $66$ inches to feet, which multiplication will eliminate the inches?

The first form works, since $66\ \text{inches} \cdot \tfrac{1\ \text{foot}}{12\ \text{inches}}$ lets the inches divide out and leaves only feet. The second form does not have any units that will divide out, so it will not help us.

Example. MaryAnne is $66$ inches tall. Convert her height into feet.

When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from divide out. Usually this means we want the conversion fraction to have those units in the denominator.

Example. Ndula, an elephant at the San Diego Safari Park, weighs almost $3.2$ tons. Convert her weight to pounds.

We will convert $3.2$ tons into pounds, using the identity property of multiplication and writing $1$ as the fraction $\tfrac{2{,}000\ \text{pounds}}{1\ \text{ton}}$.

Ndula weighs almost $6{,}400$ pounds.

Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

Example. Juliet is going with her family to their summer home. She will be away from her boyfriend for $9$ weeks. Convert the time to minutes.

To convert weeks into minutes we will convert weeks into days, days into hours, and then hours into minutes. To do this we multiply by conversion factors of $1$.

The weeks divide out with the “per week,” the days divide out with the “per day,” and the hours divide out with the “per hour,” leaving only minutes:

Juliet and her boyfriend will be apart for $90{,}720$ minutes (although it may seem like an eternity)!

Example. How many ounces are in $1$ gallon?

We will convert gallons to ounces by multiplying by several conversion factors, chaining gallons to quarts, quarts to pints, pints to cups, and cups to ounces:

Each unit divides out with the matching unit in the next fraction, leaving only ounces:

There are $128$ ounces in a gallon.

Use mixed units of measurement in the U.S. system

We often use mixed units of measurement in everyday situations. Suppose Joe is $5$ feet $10$ inches tall, stays at work for $7$ hours and $45$ minutes, and then eats a $1$ pound $2$ ounce steak for dinner — all these measurements have mixed units.

Performing arithmetic operations on measurements with mixed units of measure requires care. Be sure to add or subtract like units!

Example. Seymour bought three steaks for a barbecue. Their weights were $14$ ounces, $1$ pound $2$ ounces, and $1$ pound $6$ ounces. How many total pounds of steak did he buy?

We will add the weights of the steaks to find the total weight. First add the ounces column, then the pounds column:

Since $22$ ounces is more than a pound, we convert it to $1$ pound $6$ ounces and add the pounds:

Seymour bought $3$ pounds $6$ ounces of steak.

Example. Anthony bought four planks of wood that were each $6$ feet $4$ inches long. What is the total length of the wood he purchased?

We multiply the length of one plank by $4$, multiplying the inches and then the feet separately:

We convert the $16$ inches to $1$ foot $4$ inches and add the feet:

Anthony bought $25$ feet and $4$ inches of wood.

Make unit conversions in the metric system

In the metric system, units are related by powers of $10$. The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is $1{,}000$ meters; the prefix kilo means thousand. One centimeter is $\tfrac{1}{100}$ of a meter, just like one cent is $\tfrac{1}{100}$ of one dollar.

The equivalencies of measurements in the metric system are shown below. The common abbreviations for each measurement are given in parentheses.

Metric System of Measurement

To make conversions in the metric system, we use the same technique we did in the U.S. system. Using the identity property of multiplication, we multiply by a conversion factor of one to get to the correct units.

Have you ever run a 5K or 10K race? The length of those races is measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

Example. Nick ran a $10\text{K}$ race. How many meters did he run?

We convert kilometers to meters using the identity property of multiplication, writing $1$ as $\tfrac{1{,}000\ \text{meters}}{1\ \text{kilometer}}$:

Nick ran $10{,}000$ meters.

Example. Eleanor’s newborn baby weighed $3{,}200$ grams. How many kilograms did the baby weigh?

We convert grams into kilograms, writing $1$ as $\tfrac{1\ \text{kg}}{1{,}000\ \text{grams}}$:

The baby weighed $3.2$ kilograms.

As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.

To multiply by $10$, $100$, or $1{,}000$, we move the decimal to the right one, two, or three places, respectively. To multiply by $0.1$, $0.01$, or $0.001$, we move the decimal to the left one, two, or three places, respectively.

We can apply this pattern when we make measurement conversions in the metric system. When we changed $3{,}200$ grams to kilograms above, we changed $3{,}200$ grams to kilograms by multiplying by $\tfrac{1}{1{,}000}$ (or $0.001$). This is the same as moving the decimal three places to the left: $3{,}200. \to 3.2$.

Example. Convert (a) $350$ L to kiloliters (b) $4.1$ L to milliliters.

(a) We will convert liters to kiloliters. We know that $1\ \text{kiloliter} = 1{,}000\ \text{liters}$.

(b) We will convert liters to milliliters. We know that $1\ \text{liter} = 1{,}000\ \text{milliliters}$.

Use mixed units of measurement in the metric system

Performing arithmetic operations on measurements with mixed units of measure in the metric system requires the same care we used in the U.S. system. But it may be easier, because of the relation of the units to the powers of $10$. Make sure to add or subtract like units.

Example. Ryland is $1.6$ meters tall. His younger brother is $85$ centimeters tall. How much taller is Ryland than his younger brother?

We can convert both measurements to either centimeters or meters. Since meters is the larger unit, we will subtract the lengths in meters. We convert $85$ centimeters to meters by moving the decimal $2$ places to the left:

Ryland is $0.75$ m taller than his brother.

Example. Dena’s recipe for lentil soup calls for $150$ milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?

We first find the amount of olive oil in milliliters, then convert to liters:

Dena needs $0.45$ liters of olive oil.

Convert between the U.S. and the metric systems of measurement

Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.

The table below shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems

A ruler that shows both inches and centimeters, or a measuring cup marked with both ounces and milliliters, or a bathroom scale marked with both pounds and kilograms, lets us read a measurement directly in either system.

We make conversions between the systems just as we do within the systems — by multiplying by unit conversion factors.

Example. Lee’s water bottle holds $500$ mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.

The water bottle has $16.7$ ounces.

Example. Soleil was on a road trip and saw a sign that said the next rest stop was in $100$ kilometers. How many miles until the next rest stop?

Soleil will travel $62$ miles.

Convert between Fahrenheit and Celsius temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $22\degree\text{C}$, what does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written $\degree\text{F}$. The metric system uses degrees Celsius, written $\degree\text{C}$. On the Fahrenheit scale, water freezes at $32\degree\text{F}$ and boils at $212\degree\text{F}$. On the Celsius scale, water freezes at $0\degree\text{C}$ and boils at $100\degree\text{C}$. Normal body temperature is about $98.6\degree\text{F}$, or $37\degree\text{C}$.

Example. Convert $50\degree$ Fahrenheit into degrees Celsius.

We substitute $50$ for $F$ into the formula to find $C$:

So $50\degree\text{F}$ is equivalent to $10\degree\text{C}$.

Example. While visiting Paris, Woody saw the temperature was $20\degree$ Celsius. Convert the temperature into degrees Fahrenheit.

We substitute $20$ for $C$ into the formula to find $F$:

So $20\degree\text{C}$ is equivalent to $68\degree\text{F}$.

Key terms

U.S. system — the system of measurement commonly used in the United States, using units such as inches, feet, yards, and miles for length; pounds and tons for weight; and cups, pints, quarts, and gallons for capacity. metric system — the system of measurement used by most countries, where units of length, mass, and capacity are related by powers of $10$ (kilo-, hecto-, deka-, deci-, centi-, milli-). identity property of multiplication — for any real number $a$, $a \cdot 1 = a$ and $1 \cdot a = a$; used to build unit conversion factors that equal $1$. Fahrenheit — the temperature scale used in the U.S. system, where water freezes at $32\degree\text{F}$ and boils at $212\degree\text{F}$. Celsius — the temperature scale used in the metric system, where water freezes at $0\degree\text{C}$ and boils at $100\degree\text{C}$.

This section is adapted from Elementary Algebra 2e, Section 1.10: Systems of Measurement by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the U.S.-system, metric-system, and U.S.-to-metric conversion-factor tables as markdown tables; omitted the Be Prepared quiz, figure images (ruler, measuring cup, and scale), Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.