Systems of Measurement

In this section we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of $1$, the multiplicative identity, to change the units but not the value of a quantity.

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 United States 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, or hours. The equivalencies among the basic units of the U.S. system are:

In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions, restated here for easy reference: for any real number $a$, $a \cdot 1 = a$ and $1 \cdot a = a$.

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 convert inches to feet. We know that $1$ foot is equal to $12$ inches, so we can write $1$ as the fraction $\tfrac{1\text{ ft}}{12\text{ in}}$. When we multiply by this fraction, we do not change the value but just change the units.

But $\tfrac{12\text{ in}}{1\text{ ft}}$ also equals $1$. How do we decide whether to multiply by $\tfrac{1\text{ ft}}{12\text{ in}}$ or $\tfrac{12\text{ in}}{1\text{ ft}}$? We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert $60$ inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches:

On the other hand, if we wanted to convert $5$ feet to inches, we would choose the fraction that has feet in the denominator:

We treat the unit words like factors and “divide out” common units like we do common factors.

Example. Mary Anne is $66$ inches tall. What is her height in feet?

Multiply by $1$, writing $1$ as a fraction relating inches and feet, then divide out the inches:

Notice that when we simplified the fraction, we first divided out the inches. Mary Anne is $5.5$ feet tall.

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 away for $9$ weeks. Convert the time to minutes.

To convert weeks into minutes, convert weeks to days, days to hours, and then hours to minutes, multiplying by conversion factors of $1$ the whole way:

Juliet will be away for $90{,}720$ minutes.

Example. How many fluid ounces are in $1$ gallon of milk?

Convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces:

There are $128$ fluid ounces in a gallon.

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

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

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

Add the ounces, then add the pounds:

Convert $22$ ounces to $1$ pound $6$ ounces, then add the pounds:

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

Example. Anthony bought four planks of wood that were each $6$ feet $4$ inches long. If the four planks are placed end-to-end, what is the total length of the wood?

Multiply the inches, then the feet; convert the extra inches to feet, then add:

Anthony bought $25$ feet $4$ 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 $1000$ meters — the prefix kilo- means thousand. One centimeter is $\tfrac{1}{100}$ of a meter, because the prefix centi- means one one-hundredth (just like one cent is $\tfrac{1}{100}$ of one dollar). The equivalencies of measurements in the metric system are:

To make conversions in the metric system, we use the same technique we did in the U.S. system — multiplying by a conversion factor of one, using the Identity Property of Multiplication.

Example. Nick ran a $10$-kilometer race. How many meters did he run?

Multiply by $1$, writing $1$ as a fraction relating kilometers and meters:

Nick ran $10{,}000$ meters.

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

Multiply by $1$, writing $1$ as a fraction relating grams and kilograms:

The baby weighed $3.2$ kilograms.

Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. We learned how to simplify these calculations by just moving the decimal: to multiply by $10$, $100$, or $1000$, move the decimal to the right $1$, $2$, or $3$ places respectively; to multiply by $0.1$, $0.01$, or $0.001$, move the decimal to the left $1$, $2$, or $3$ places respectively. We can apply this pattern when we make measurement conversions in the metric system — in the example above, we changed $3{,}200$ grams to kilograms by moving the decimal $3$ places to the left.

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

(a) Multiply by $1$, writing $1$ as a fraction relating liters to kiloliters, and move the decimal $3$ units to the left:

(b) Multiply by $1$, writing $1$ as a fraction relating milliliters to liters, and move the decimal $3$ units to the right:

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 $10$. We still must 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?

Convert $85$ centimeters to meters by moving the decimal $2$ places to the left: $85$ cm is the same as $0.85$ m. Now that both measurements are in meters, subtract:

Ryland is $0.75$ meters 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?

Triple $150$ mL, then convert to liters:

Dena needs $0.45$ liter of olive oil.

Convert between U.S. and metric systems of measurement

Many measurements in the United States are made in metric units. A drink may come in $2$-liter bottles, calcium may come in $500$-mg capsules, and we may run a $5$-K race. To work easily in both systems, we need to be able to convert between the two. Some common conversion factors are:

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 fluid ounces are in the bottle? Round to the nearest tenth of an ounce.

Multiply by a unit conversion factor relating mL and ounces:

The water bottle holds about $16.7$ fluid ounces.

The conversion factors in the table above are not exact, but the approximations they give are close enough for everyday purposes.

Example. Soleil passes a sign on a Canadian highway that says the next rest stop is in $100$ kilometers. How many miles until the next rest stop? Round your answer to the nearest mile.

Multiply by a unit conversion factor relating kilometers and miles:

It is about $62$ miles to the next rest stop.

Convert between Fahrenheit and Celsius temperatures

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

The U.S. system uses degrees Fahrenheit, written $^\circ\text{F}$. The metric system uses degrees Celsius, written $^\circ\text{C}$. On the Celsius scale, water freezes at $0^\circ\text{C}$ and boils at $100^\circ\text{C}$. On the Fahrenheit scale, water freezes at $32^\circ\text{F}$ and boils at $212^\circ\text{F}$. Normal body temperature is about $37^\circ\text{C}$, which is equivalent to $98.6^\circ\text{F}$.

If we know the temperature in one system, we can use a formula to convert it to the other system.

Example. Convert $50^\circ\text{F}$ into degrees Celsius.

Substitute $50$ for $F$ in the conversion formula, then simplify:

A temperature of $50^\circ\text{F}$ is equivalent to $10^\circ\text{C}$.

Example. The weather forecast for Paris predicts a high of $20^\circ\text{C}$. Convert the temperature into degrees Fahrenheit.

Substitute $20$ for $C$ in the conversion formula, then simplify:

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

Key terms

U.S. system — the system of measurement used in the United States, with units like inch, foot, mile, pound, ton, cup, and gallon. metric system — the system of measurement used in most of the world, with units related by powers of $10$ (kilo-, centi-, milli-, and so on). Unit conversions multiply a measurement by a cleverly chosen form of $1$ — a fraction equal to $1$ relating two units — so the value stays the same while the units change. Fahrenheit and Celsius — the two common temperature scales, related by $C = \tfrac{5}{9}(F - 32)$ and $F = \tfrac{9}{5}C + 32$.

Practice

Make unit conversions in the U.S. system

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

Make unit conversions in the metric system

Use mixed units of measurement in the metric system

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

Convert between Fahrenheit and Celsius temperatures

This section is adapted from Prealgebra 2e, Section 7.5: 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., metric, and U.S.-to-metric conversion tables (Tables 7.2–7.4) as markdown tables; omitted the decorative photographs, the Celsius/Fahrenheit thermometer figure (its key reference points are given in prose instead), the Be Prepared quiz, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, restating two mixed-unit answers as single-unit quantities.