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Systems of Measurement

Make unit conversions in the U.S. system

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 gallons. 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 of measurement are listed in . The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System Units
Length	Volume
\(1\) foot (ft) = \(12\) inches (in) \(1\) yard (yd) = \(3\) feet (ft) \(1\) mile (mi) = \(5280\) feet (ft)	\(3\) teaspoons (t) = \(1\) tablespoon (T) \(16\) Tablespoons (T) = \(1\) cup (C) \(1\) cup (C) = \(8\) fluid ounces (fl oz) \(1\) pint (pt) = \(2\) cups (C) \(1\) quart (qt) = \(2\) pints (pt) \(1\) gallon (gal) = \(4\) quarts (qt)
Weight	Time
\(1\) pound (lb) = \(16\) ounces (oz) \(1\) ton = \(2000\) pounds (lb)	\(1\) minute (min) = \(60\) seconds (s) \(1\) hour (h) = \(60\) minutes (min) \(1\) day = \(24\) hours (h) \(1\) week (wk) = \(7\) days \(1\) year (yr) = \(365\) days

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. We’ll restate the Identity Property of Multiplication here for easy reference.

\[\text{For any real number}\ a,\ a\cdot 1=a\ 1\cdot a=a\]

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 \(\frac{\text{1 ft}}{\text{12 in}}.\) When we multiply by this fraction, we do not change the value but just change the units.

But \(\frac{\text{12 in}}{\text{1 ft}}\) also equals \(1.\) How do we decide whether to multiply by \(\frac{\text{1 ft}}{\text{12 in}}\) or \(\frac{\text{12 in}}{\text{1 ft}}?\) 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.

\[60\ \text{in}\cdot \frac{\text{1 ft}}{12\ \text{in}}=\text{5 ft}\]

On the other hand, if we wanted to convert \(5\) feet to inches, we would choose the fraction that has feet in the denominator.

\[\text{5 ft}\cdot \frac{\text{12 in}}{1\text{ft}}=\text{60 in}\]

Condensed — the full section is in OpenStax Prealgebra 2e.

Use Mixed Units of Measurement in the U.S. System

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

Example

Try it.

Charlie 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?

Solution

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds.
Convert 22 ounces to pounds and ounces.
Add the pounds.	2 pounds + 1 pound, 6 ounces 3 pounds, 6 ounces
	Charlie bought 3 pounds 6 ounces of steak.

Example

Try it.

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?

Solution

We will multiply the length of one plank by \(4\) to find the total length.

Multiply the inches and then the feet.
Convert 16 inches to feet.	24 feet + 1 foot 4 inches
Add the feet.	25 feet 4 inches
	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 \(\frac{1}{100}\) of a meter, because the prefix centi- means one one-hundredth (just like one cent is \(\frac{1}{100}\) of one dollar).

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

Metric Measurements
Length	Mass	Volume/Capacity
\(1\) kilometer (km) = \(1000\) m \(1\) hectometer (hm) = \(100\) m \(1\) dekameter (dam) = \(10\) m \(1\) meter (m) = \(1\) m \(1\) decimeter (dm) = \(0.1\) m \(1\) centimeter (cm) = \(0.01\) m \(1\) millimeter (mm) = \(0.001\) m	\(1\) kilogram (kg) = \(1000\) g \(1\) hectogram (hg) = \(100\) g \(1\) dekagram (dag) = \(10\) g \(1\) gram (g) = \(1\) g \(1\) decigram (dg) = \(0.1\) g \(1\) centigram (cg) = \(0.01\) g \(1\) milligram (mg) = \(0.001\) g	\(1\) kiloliter (kL) = \(1000\) L \(1\) hectoliter (hL) = \(100\) L \(1\) dekaliter (daL) = \(10\) L \(1\) liter (L) = \(1\) L \(1\) deciliter (dL) = \(0.1\) L \(1\) centiliter (cL) = \(0.01\) L \(1\) milliliter (mL) = \(0.001\) L
\(1\) meter = \(100\) centimeters \(1\) meter = \(1000\) millimeters	\(1\) gram = \(100\) centigrams \(1\) gram = \(1000\) milligrams	\(1\) liter = \(100\) centiliters \(1\) liter = \(1000\) milliliters

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

Have you ever run a \(\text{5 k}\) or \(\text{10 k}\) race? The lengths of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

Example

Try it.

Nick ran a \(\text{10-kilometer}\) race. How many meters did he run?

Solution

We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in .

	10 kilometers
Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilometers and meters.
Simplify.
Multiply.	10,000 m
	Nick ran 10,000 meters.

Example

Try it.

Eleanor’s newborn baby weighed \(3200\) grams. How many kilograms did the baby weigh?

Solution

We will convert grams to kilograms.


Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilograms and grams.
Simplify.
Multiply.
Divide.	3.2 kilograms
	The baby weighed \(3.2\) kilograms.

We can apply this pattern when we make measurement conversions in the metric system.

Condensed — the full section is in OpenStax Prealgebra 2e.

Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures 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

Try it.

Ryland is \(1.6\) meters tall. His younger brother is \(85\) centimeters tall. How much taller is Ryland than his younger brother?

Solution

We will subtract the lengths in meters. 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 to find out how much taller Ryland is than his brother.

\[\begin{array}{l} \\ \\ \text{1.60 m} \\ \underset{\text{_______}}{\text{-0.85 m}} \\ \text{0.75 m}\end{array}\]

Ryland is \(0.75\) meters taller than his brother.

Example

Try it.

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?

Solution

We will find the amount of olive oil in milliliters then convert to liters.

	Triple 150 mL
Translate to algebra.	\(3\cdot 150\ \text{mL}\)
Multiply.	\(450\ \text{mL}\)
Convert to liters.	\(450\ \text{mL}\cdot \frac{0.001\ \text{L}}{1\ \text{mL}}\)
Simplify.	\(0.45\ \text{L}\)
	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 \(\text{2-liter}\) bottles, calcium may come in \(\text{500-mg}\) capsules, and we may run a \(\text{5-K}\) race. To work easily in both systems, we need to be able to convert between the two systems.

shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems
Length	Weight	Volume
\(1\) in = \(2.54\) cm \(1\) ft = \(0.305\) m \(1\) yd = \(0.914\) m \(1\) mi = \(1.61\) km \(1\) m = \(3.28\) ft	\(1\) lb = \(0.45\) kg \(1\) oz = \(28\) g \(1\) kg = \(2.2\) lb	\(1\) qt = \(0.95\) L \(1\) fl oz = \(30\) mL \(1\) L = \(1.06\) qt

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

Example

Try it.

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.

Solution

	500 mL
Multiply by a unit conversion factor relating mL and ounces.	\(500\ \text{mL}\cdot \frac{1\ \text{fl oz}}{30\ \text{mL}}\)
Simplify.	\(\frac{500\ \text{fl oz}}{30}\)
Divide.	\(16.7\ \text{fl. oz.}\)
	The water bottle holds 16.7 fluid ounces.

The conversion factors in are not exact, but the approximations they give are close enough for everyday purposes. In , we rounded the number of fluid ounces to the nearest tenth.

Example

Try it.

Soleil lives in Minnesota but often travels in Canada for work. While driving on a Canadian highway, she passes a sign 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.

Solution

	100 kilometers
Multiply by a unit conversion factor relating kilometers and miles.	\(100\ \text{kilometers}\cdot \frac{1\ \text{mile}}{1.61\ \text{kilometers}}\) \(100\cdot \frac{1\ \text{mi}}{1.61\ \text{km}}\)
Simplify.	\(\frac{100\ \text{mi}}{1.61}\)
Divide.	62 mi
	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 \(22\text{^{\circ}C}.\) What does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written \(\text{^{\circ}F}.\) The metric system uses degrees Celsius, written \(\text{^{\circ}C}.\) shows the relationship between the two systems.

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

Example

Try it.

Convert \(50\text{^{\circ}F}\) into degrees Celsius.

Solution

We will substitute \(50\text{^{\circ}F}\) into the formula to find \(\text{C}.\)

Use the formula for converting °F to °C	\(C=\frac{5}{9}(F-32)\)

Simplify in parentheses.	\(C=\frac{5}{9}(18)\)
Multiply.	\(C=10\)
	A temperature of 50°F is equivalent to 10°C.

Example

Try it.

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

Solution

We will substitute \(20\text{^{\circ}C}\) into the formula to find \(\text{F}.\)

Use the formula for converting °F to °C	\(F=\frac{9}{5}C+32\)

Multiply.	\(F=36+32\)
Add.	\(F=68\)
	So 20°C is equivalent to 68°F.

Chapter Practice Test

In the following exercises, simplify the given expression.

In the following exercises, solve using the appropriate unit conversions.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Multiply: \(4.29(1000).\)
If you missed this problem, review .

Vastuse avalikustamine

\(4,290\)
Simplify: \(\frac{30}{54}.\)
If you missed this problem, review .

Vastuse avalikustamine

\(\frac{5}{9}\)
Multiply: \(\frac{7}{15}\cdot \frac{25}{28}.\)
If you missed this problem, review .

Vastuse avalikustamine

\(\frac{5}{12}\)

Mary Anne is \(66\) inches tall. What is her height in feet?

Vastuse avalikustamine

Convert 66 inches into feet.
Multiply the measurement to be converted by 1.	\(66\) inches \(\cdot 1\)
Write 1 as a fraction relating the units given and the units needed.	\(\text{66 inches}\cdot \frac{\text{1 foot}}{\text{12 inches}}\)
Multiply.	\(\frac{\text{66 inches}\cdot \text{1 foot}}{\text{12 inches}}\)
Simplify the fraction.	\(\frac{66\ \text{inches}\cdot \text{1 foot}}{12\ \text{inches}}\)
	\(\frac{\text{66 feet}}{12\ }\)
	\(\text{5.5 feet}\)

Notice that the when we simplified the fraction, we first divided out the inches.

Mary Anne is \(5.5\) feet tall.

Lexie is \(30\) inches tall. Convert her height to feet.

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2.5 feet
Rene bought a hose that is \(18\) yards long. Convert the length to feet.

Vastuse avalikustamine

54 feet

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

Vastuse avalikustamine

We will convert \(3.2\) tons into pounds, using the equivalencies in . We will use the Identity Property of Multiplication, writing \(1\) as the fraction \(\frac{\text{2000 pounds}}{\text{1 ton}}.\)

	\(\text{3.2 tons}\)
Multiply the measurement to be converted by 1.	\(\text{3.2 tons}\cdot 1\)
Write 1 as a fraction relating tons and pounds.	\(\text{3.2 tons}\cdot \frac{\text{2000 lbs}}{\text{1 ton}}\)
Simplify.	\(\frac{3.2\ \text{tons}\cdot \text{2000 lbs}}{1\ \text{ton}}\)
Multiply.	\(\text{6400 lbs}\)
	Ndula weighs almost 6,400 pounds.

Arnold’s SUV weighs about \(4.3\) tons. Convert the weight to pounds.

Vastuse avalikustamine

8600 pounds
A cruise ship weighs \(51,000\) tons. Convert the weight to pounds.

Vastuse avalikustamine

102,000,000 pounds

Juliet is going with her family to their summer home. She will be away for \(9\) weeks. Convert the time to minutes.

Vastuse avalikustamine

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

	\(\text{9 weeks}\)
Write 1 as \(\frac{7\ \text{days}}{1\ \text{week}},\frac{24\ \text{hours}}{1\ \text{day}},\frac{60\ \text{minutes}}{1\ \text{hour}}\).
Cancel common units.
Multiply.	\(\frac{9\cdot 7\cdot 24\cdot 60\ \text{min}}{1\cdot 1\cdot 1\cdot 1}=90,720\ \text{min}\)
	Juliet will be away for 90,720 minutes.

The distance between Earth and the moon is about \(250,000\) miles. Convert this length to yards.

Vastuse avalikustamine

440,000,000 yards
A team of astronauts spends \(15\) weeks in space. Convert the time to minutes.

Vastuse avalikustamine

151,200 minutes

How many fluid ounces are in \(1\) gallon of milk?

Vastuse avalikustamine

Use conversion factors to get the right units: convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces.

	1 gallon
Multiply the measurement to be converted by 1.	\(\frac{\text{1 gal}}{1}\cdot \frac{\text{4 qt}}{\text{1 gal}}\cdot \frac{\text{2 pt}}{\text{1 qt}}\cdot \frac{\text{2 C}}{\text{1 pt}}\cdot \frac{\text{8 fl oz}}{\text{1 C}}\)
Simplify.	\(\frac{1\ \text{gal}}{1}\cdot \frac{4\ \text{qt}}{1\ \text{gal}}\cdot \frac{2\ \text{pt}}{1\ \text{qt}}\cdot \frac{2\ \text{C}}{1\ \text{pt}}\cdot \frac{\text{8 fl oz}}{1\ \text{C}}\)
Multiply.	\(\frac{1\cdot 4\cdot 2\cdot 2\cdot \text{8 fl oz}}{1\cdot 1\cdot 1\cdot 1\cdot 1}\)
Simplify.	128 fluid ounces
	There are 128 fluid ounces in a gallon.

How many cups are in \(1\) gallon?

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16 cups
How many teaspoons are in \(1\) cup?

Vastuse avalikustamine

48 teaspoons
Charlie 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?

Vastuse avalikustamine

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds.
Convert 22 ounces to pounds and ounces.
Add the pounds. 2 pounds + 1 pound, 6 ounces
3 pounds, 6 ounces
Charlie bought 3 pounds 6 ounces of steak.
Laura gave birth to triplets weighing \(3\) pounds \(12\) ounces, \(3\) pounds \(3\) ounces, and \(2\) pounds \(9\) ounces. What was the total birth weight of the three babies?

Vastuse avalikustamine

9 lbs. 8 oz
Seymour cut two pieces of crown molding for his family room that were \(8\) feet \(7\) inches and \(12\) feet \(11\) inches. What was the total length of the molding?

Vastuse avalikustamine

21 ft. 6 in.
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?

Vastuse avalikustamine

We will multiply the length of one plank by \(4\) to find the total length.

Multiply the inches and then the feet.
Convert 16 inches to feet. 24 feet + 1 foot 4 inches
Add the feet. 25 feet 4 inches
Anthony bought 25 feet 4 inches of wood.
Henri wants to triple his spaghetti sauce recipe, which calls for \(1\) pound \(8\) ounces of ground turkey. How many pounds of ground turkey will he need?

Vastuse avalikustamine

4 lbs. 8 oz.
Joellen wants to double a solution of \(5\) gallons \(3\) quarts. How many gallons of solution will she have in all?

Vastuse avalikustamine

11 gal. 2 qts.
Nick ran a \(\text{10-kilometer}\) race. How many meters did he run?

Vastuse avalikustamine

We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in .

10 kilometers
Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilometers and meters.
Simplify.
Multiply. 10,000 m
Nick ran 10,000 meters.
Sandy completed her first \(\text{5-km}\) race. How many meters did she run?

Vastuse avalikustamine

5000 m
Herman bought a rug \(2.5\) meters in length. How many centimeters is the length?

Vastuse avalikustamine

250 cm
Eleanor’s newborn baby weighed \(3200\) grams. How many kilograms did the baby weigh?

Vastuse avalikustamine

We will convert grams to kilograms.

Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilograms and grams.
Simplify.
Multiply.
Divide. 3.2 kilograms
The baby weighed \(3.2\) kilograms.
Kari’s newborn baby weighed \(2800\) grams. How many kilograms did the baby weigh?

Vastuse avalikustamine

2.8 kilograms
Anderson received a package that was marked \(4500\) grams. How many kilograms did this package weigh?

Vastuse avalikustamine

4.5 kilograms
Convert:
1. ⓐ \(\ 350\) liters to kiloliters
2. ⓑ \(\ 4.1\) liters to milliliters.
Vastuse avalikustamine

ⓐ We will convert liters to kiloliters. In , we see that \(\text{1 kiloliter}=\text{1000 liters}.\)

350 L
Multiply by 1, writing 1 as a fraction relating liters to kiloliters.
Simplify.
Move the decimal 3 units to the left.
0.35 kL

ⓑ We will convert liters to milliliters. In , we see that \(\text{1 liter}=1000\ \text{milliliters.}\)

4.1 L
Multiply by 1, writing 1 as a fraction relating milliliters to liters.
Simplify.
Move the decimal 3 units to the left.
4100 mL
Convert: ⓐ \(\ 7.25\) L to kL ⓑ \(\ 6.3\) L to mL.
Vastuse avalikustamine
1. ⓐ 0.00725 kL
2. ⓑ 6300 mL
Convert: ⓐ \(\ 350\) hL to L ⓑ \(\ 4.1\) L to cL.
Vastuse avalikustamine
1. ⓐ 35,000 L
2. ⓑ 410 cL
Ryland is \(1.6\) meters tall. His younger brother is \(85\) centimeters tall. How much taller is Ryland than his younger brother?

Vastuse avalikustamine

We will subtract the lengths in meters. 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 to find out how much taller Ryland is than his brother.
\[\begin{array}{l} \\ \\ \text{1.60 m} \\ \underset{\text{_______}}{\text{-0.85 m}} \\ \text{0.75 m}\end{array}\]
Ryland is \(0.75\) meters taller than his brother.
Mariella is \(1.58\) meters tall. Her daughter is \(75\) centimeters tall. How much taller is Mariella than her daughter? Write the answer in centimeters.

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83 cm
The fence around Hank’s yard is \(2\) meters high. Hank is \(96\) centimeters tall. How much shorter than the fence is Hank? Write the answer in meters.

Vastuse avalikustamine

1.04 m

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?

Vastuse avalikustamine

We will find the amount of olive oil in milliliters then convert to liters.

	Triple 150 mL
Translate to algebra.	\(3\cdot 150\ \text{mL}\)
Multiply.	\(450\ \text{mL}\)
Convert to liters.	\(450\ \text{mL}\cdot \frac{0.001\ \text{L}}{1\ \text{mL}}\)
Simplify.	\(0.45\ \text{L}\)
	Dena needs 0.45 liter of olive oil.

A recipe for Alfredo sauce calls for \(250\) milliliters of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by \(8.\) How many liters of milk will she need?

Vastuse avalikustamine

2 L
To make one pan of baklava, Dorothea needs \(400\) grams of filo pastry. If Dorothea plans to make \(6\) pans of baklava, how many kilograms of filo pastry will she need?

Vastuse avalikustamine

2.4 kg

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.

Vastuse avalikustamine

	500 mL
Multiply by a unit conversion factor relating mL and ounces.	\(500\ \text{mL}\cdot \frac{1\ \text{fl oz}}{30\ \text{mL}}\)
Simplify.	\(\frac{500\ \text{fl oz}}{30}\)
Divide.	\(16.7\ \text{fl. oz.}\)
	The water bottle holds 16.7 fluid ounces.

How many quarts of soda are in a \(\text{2-liter}\) bottle?

Vastuse avalikustamine

2.12 quarts
How many liters are in \(4\) quarts of milk?

Vastuse avalikustamine

3.8 liters

Vastuse avalikustamine

	100 kilometers
Multiply by a unit conversion factor relating kilometers and miles.	\(100\ \text{kilometers}\cdot \frac{1\ \text{mile}}{1.61\ \text{kilometers}}\) \(100\cdot \frac{1\ \text{mi}}{1.61\ \text{km}}\)
Simplify.	\(\frac{100\ \text{mi}}{1.61}\)
Divide.	62 mi
	It is about 62 miles to the next rest stop.

Symbols used here

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Systems of Measurement

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
Multiply the measurement to be converted by
Multiply.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Proovi ise.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Rohkem Arithmetic

Order of operations Fractions Percentages Powers and roots

	350 L
Multiply by 1, writing 1 as a fraction relating liters to kiloliters.
Simplify.
Move the decimal 3 units to the left.
	0.35 kL

	4.1 L
Multiply by 1, writing 1 as a fraction relating milliliters to liters.
Simplify.
Move the decimal 3 units to the left.
	4100 mL