Understanding and working with units is essential in everyday life, in science, in engineering, and in many professions. Measurements let us describe and compare objects, distances, time intervals, volumes, weights, and much more. This section introduces the U.S. customary system and the metric system, shows how to convert units within and between these systems, and demonstrates how units help you solve real-world problems.

Why do we have different measurement systems?

The U.S. customary system uses fixed unit relationships (e.g., 12 in = 1 ft); the metric system is built on powers of 10, so conversions are decimal shifts.

U.S. customary system

The U.S. customary system is mainly used in the United States. It includes units for length, volume, weight, and time. Because it isn’t based on powers of $10$, conversions depend on specific relationships between units.

Length units

Unit	Abbreviation
inch	in
foot	ft
yard	yd
mile	mi

Length relationships

Original	Converted
$12$ inches	$1$ foot
$3$ feet	$1$ yard
$36$ inches	$1$ yard
$5280$ feet	$1$ mile

Volume units

Unit	Abbreviation
cup	c
pint	pt
quart	qt
gallon	gal

Volume relationships

Original	Converted
$2$ cups	$1$ pint
$2$ pints	$1$ quart
$4$ quarts	$1$ gallon

Weight units

Unit	Abbreviation
ounce	oz
pound	lb
ton	ton

Weight relationships

Original	Converted
$16$ ounces	$1$ pound
$2000$ pounds	$1$ ton

Time units

Unit	Abbreviation
second	s
minute	min
hour	hr
day	d

Time relationships

Original	Converted
$60$ seconds	$1$ minute
$60$ minutes	$1$ hour
$24$ hours	$1$ day

Working within the customary system

Converting within the customary system means multiplying or dividing by the relationship between the two units. If you’re converting from a smaller unit to a larger one, you’ll divide. If you’re converting from a larger unit to a smaller one, you’ll multiply.

Example: Convert $72$ inches to feet

Since $12$ inches $=1$ foot, divide by $12$:

$$72\text{in}\times \frac{1\text{ft}}{12\text{in}}=\frac{72}{12}\text{ft}=6\text{ft}$$

Answer: $72$ inches $=6$ feet.

Estimation check: $72$ inches is about $6$ times the length of a ruler - $6$ feet sounds right.

Metric system

The metric system is used worldwide and is based on powers of ten. Prefixes tell you how many powers of $10$ separate a unit from the base unit. This structure makes conversions predictable and helps reduce calculation errors.

You’ll see metric units often in science experiments, medical dosing, engineering design, and international data reporting. Because the system scales in a consistent way, it’s well suited for precision and for comparing measurements across different fields.

Base units

Quantity	Base unit	Abbreviation
Length	meter	m
Volume	liter	L
Mass	gram	g
Time	second	s

Common metric prefixes

The prefixes you’ll encounter most often on the exam are kilo, centi, milli, and micro - these appear in everyday measurements like kilometers, centimeters, milligrams, and micrograms.

Prefix	Symbol	Multiplier
kilo	k	$10^3$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	µ	$10^{-6}$

Metric prefixes and conversions

In everyday situations - like reading medicine labels, interpreting scientific data, or converting distances on a map - you often need to convert between metric units.

Which direction do you move the decimal?

• Going to a larger unit → divide (the number gets smaller). For example, converting milligrams to grams means fewer grams than milligrams.
• Going to a smaller unit → multiply (the number gets bigger). For example, converting kilometers to meters means more meters than kilometers.

A quick check: if your answer is a bigger number than you started with, you moved to a smaller unit - and vice versa. If the direction seems off, flip multiply/divide and recalculate.

Use this prefix ladder to count steps and track decimal movement. Each step to the right (toward smaller units) multiplies by $10$; each step to the left (toward larger units) divides by $10$.

Metric prefix ladder

$$\text{kilo}\stackrel{\times 10}{\to }\text{hecto}\stackrel{\times 10}{\to }\text{deka}\stackrel{\times 10}{\to }\text{base}\stackrel{\times 10}{\to }\text{deci}\stackrel{\times 10}{\to }\text{centi}\stackrel{\times 10}{\to }\text{milli}\stackrel{\times 10^3}{\to }\text{micro}$$

Example: kilo to milli is $6$ steps to the right, so multiply by $10^6$. Milli to kilo is $6$ steps to the left, so divide by $10^6$. Micro is $3$ additional steps below milli ($1\mathrm{mg}=1000\mu \mathrm{g}$).

Example: Convert $5$ kilometers to meters

You might do this when interpreting a road race distance or a map scale.

(spoiler)

• $1\mathrm{km}=10^3\mathrm{m}$, so
• $5\mathrm{km}=5\times 10^3=5000\mathrm{m}$

Answer: $5000\mathrm{m}$

Example: Medicine dosage

Medication instructions are often written using very small metric units. A doctor prescribes $250$ µg of a medication. How many milligrams is that?

(spoiler)

• $1\mathrm{mg}=1000\mu \mathrm{g}$, so micrograms are a smaller unit than milligrams - we’re going to a larger unit, so we divide:
• $250÷1000=0.25$

Answer: $0.25\mathrm{mg}$

Watch out: A common mistake is multiplying by $1000$ to get $250,000$ - but we’re converting to a larger unit, so the number must get smaller. If your answer is bigger when you expected it to be smaller (or vice versa), check which direction you multiplied or divided.

U.S. customary to metric conversion factors

The table below shows conversion factors for going from a U.S. customary unit to its metric equivalent. Multiply when converting from the customary unit to the metric unit (left to right). Divide when going the other direction - from metric to customary.

Conversion factor	Equivalence
$2.54$	$1$ in = $2.54$ cm
$0.3048$	$1$ ft = $0.3048$ m
$1.609$	$1$ mi = $1.609$ km
$0.4536$	$1$ lb = $0.4536$ kg
$3.785$	$1$ gal = $3.785$ L

Sidenote

Mass vs. weight

The metric system measures mass (grams, kilograms), while the U.S. customary system measures weight (ounces, pounds). Technically, these are different physical quantities, but on Earth they are commonly interconverted - for example, $1$ lb $\approx 0.4536$ kg. For everyday and exam purposes, you can treat these conversions as equivalent.

Converting between systems

Converting between U.S. customary and metric units comes up often in travel, international communication, and scientific work. A reliable approach is to start with a known conversion factor and set up the calculation so the unwanted unit cancels.

Writing out the conversion as a fraction - called dimensional analysis - makes it clear which direction to multiply or divide. Place the unit you want to cancel in the denominator.

Example: Convert $10$ miles to kilometers

A fitness tracker or GPS device may report distances in kilometers.

(spoiler)

• $1$ mile $\approx 1.609$ km, so set up the fraction with miles in the denominator to cancel:

$$10\text{mi}\times \frac{1.609\text{km}}{1\text{mi}}=10\times 1.609\text{km}=16.09\text{km}$$

Answer: $16.09$ km

Estimation check: a mile is a bit longer than a kilometer, so $10$ miles should be a bit more than $10$ km - $16.09$ km ✓

Example: Convert $2$ liters to gallons

This is useful when comparing beverage sizes sold in different countries.

(spoiler)

• $1$ gallon $\approx 3.785$ L, so set up the fraction with liters in the denominator to cancel:

$$2\text{L}\times \frac{1\text{gal}}{3.785\text{L}}=\frac{2}{3.785}\text{gal}\approx 0.529\text{gal}$$

Answer: $0.529$ gallons

Estimation check: a gallon is nearly $4$ liters, so $2$ liters should be roughly half a gallon - $0.529$ ✓