1.10 Systems of Measurement

US Customary System

Conversions within the US Customary System

The US Customary system organizes length, weight, and volume into a set of related units. The conversion factors between them aren't based on powers of 10, so you'll need to memorize the key ones.

Length

• 12 inches (in) = 1 foot (ft)
• 3 feet = 1 yard (yd)
• 5,280 feet = 1 mile (mi)

Weight

• 16 ounces (oz) = 1 pound (lb)
• 2,000 pounds = 1 ton (t)

Volume

• 8 fluid ounces (fl oz) = 1 cup (c)
• 2 cups = 1 pint (pt)
• 2 pints = 1 quart (qt)
• 4 quarts = 1 gallon (gal)

A helpful way to remember the volume chain: think of it as doubling up. 8 fl oz makes a cup, 2 cups make a pint, 2 pints make a quart, and 4 quarts make a gallon. That means 1 gallon = 128 fluid ounces.

Calculations with Mixed US Units

When you're adding or subtracting measurements that use more than one unit (like feet and inches), convert everything to the smallest unit first. Then do the math, and convert back at the end.

Example: $2 \text{ ft } 5 \text{ in} + 3 \text{ ft } 8 \text{ in}$

1. Convert both to inches: $(2 \times 12 + 5) + (3 \times 12 + 8) = 29 \text{ in} + 44 \text{ in}$
2. Add: $29 + 44 = 73 \text{ in}$
3. Convert back: $73 \div 12 = 6 \text{ ft}$ with a remainder of $1 \text{ in}$, so the answer is $6 \text{ ft } 1 \text{ in}$

For multiplication or division, the same idea applies: convert to a single unit, do the operation, then convert back.

Example: $2 \text{ yd } 1 \text{ ft} \times 4$

1. Convert to feet: $2 \times 3 + 1 = 7 \text{ ft}$
2. Multiply: $7 \times 4 = 28 \text{ ft}$
3. Convert back: $28 \div 3 = 9 \text{ yd}$ with a remainder of $1 \text{ ft}$, so the answer is $9 \text{ yd } 1 \text{ ft}$

Metric System

Conversions within the Metric System

The metric system is built on powers of 10, which makes converting between units much more straightforward than in the US system. Every metric unit uses the same set of prefixes attached to a base unit (meter for length, gram for mass, liter for volume).

Here are the prefixes from largest to smallest:

• Larger unit → smaller unit: Move the decimal to the right (the number gets bigger because you're counting in smaller pieces).
  • $3.5 \text{ km} = 3{,}500 \text{ m}$ (move 3 places right, since kilo- is 3 steps from the base)
• Smaller unit → larger unit: Move the decimal to the left (the number gets smaller).
  • $450 \text{ mL} = 0.45 \text{ L}$ (move 3 places left, since milli- is 3 steps from the base)

Count the number of steps between the two prefixes in the table above. That tells you how many places to move the decimal.

Calculations with Mixed Metric Units

Just like with US units, convert everything to the same unit before calculating.

Example: $2.3 \text{ m} + 45 \text{ cm}$

1. Convert cm to m: $45 \text{ cm} = 0.45 \text{ m}$
2. Add: $2.3 + 0.45 = 2.75 \text{ m}$

Converting Between Systems

US Customary to Metric Translations

These approximate conversion factors show up frequently, so it's worth committing them to memory:

Length

• 1 inch ≈ 2.54 centimeters
• 1 foot ≈ 0.305 meters
• 1 mile ≈ 1.609 kilometers

Volume

• 1 quart ≈ 0.946 liters
• 1 gallon ≈ 3.785 liters

To convert from US to metric, multiply by the conversion factor. To go from metric to US, divide by it.

Example: How many centimeters is 5 inches?

$5 \times 2.54 = 12.7 \text{ cm}$

Example: How many quarts is 2 liters?

$2 \div 0.946 \approx 2.11 \text{ qt}$

Fahrenheit and Celsius Conversions

Temperature is the one conversion that uses a formula instead of a simple multiplication factor, because the two scales have different starting points (water freezes at 32°F but 0°C).

Fahrenheit to Celsius:

$°C = (°F - 32) \times \frac{5}{9}$

1. Subtract 32 from the Fahrenheit temperature.
2. Multiply the result by $\frac{5}{9}$.

Example: Convert 77°F to Celsius.

$°C = (77 - 32) \times \frac{5}{9} = 45 \times \frac{5}{9} = 25°C$

Celsius to Fahrenheit:

$°F = °C \times \frac{9}{5} + 32$

1. Multiply the Celsius temperature by $\frac{9}{5}$.
2. Add 32.

Example: Convert 30°C to Fahrenheit.

$°F = 30 \times \frac{9}{5} + 32 = 54 + 32 = 86°F$

Measurement Precision and Accuracy

These two terms sound similar but mean different things:

• Precision refers to the level of detail in a measurement, often shown by the number of decimal places. A ruler marked in millimeters is more precise than one marked only in centimeters.
• Accuracy is how close a measurement is to the true or accepted value. You can be very precise but still inaccurate if your measuring tool is off.
• Significant figures are the meaningful digits in a measurement that reflect its precision. For example, 2.50 m has three significant figures, while 2.5 m has only two.