7.5 Systems of Measurement

U.S. Customary System

Conversions within the U.S. Customary System

The U.S. Customary system uses specific conversion ratios that you'll need to memorize. Unlike the metric system, these ratios aren't based on a single pattern, so each one is its own fact to know.

Length:

• 1 foot = 12 inches
• 1 yard = 3 feet
• 1 mile = 5,280 feet

Weight:

• 1 pound = 16 ounces
• 1 ton = 2,000 pounds

Volume:

• 1 cup = 8 fluid ounces
• 1 pint = 2 cups
• 1 quart = 2 pints
• 1 gallon = 4 quarts

A helpful way to remember the volume chain: there are 2 cups in a pint, 2 pints in a quart, and 4 quarts in a gallon. So a gallon contains $4 \times 2 \times 2 = 16$ cups total.

Calculations with Mixed U.S. Units

Before you can add or subtract measurements, they need to be in the same unit. If they aren't, convert first.

Example: You want to add 3 feet and 6 inches.

1. Convert 6 inches to feet: $6 \div 12 = 0.5$ feet
2. Add: $3 + 0.5 = 3.5$ feet

Multiplying and dividing can involve different unit types, because the result creates a combined unit. For instance, $1 \text{ gallon} \div 2 \text{ minutes} = 0.5$ gallons per minute. That's a rate, not a single measurement.

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 U.S. system. Every metric unit starts with a base unit: meters for length, grams for mass, and liters for volume. Prefixes are then attached to the base unit to show the size.

Example: Convert 3.5 kilometers to meters.

• Kilo- to the base unit is 3 steps to the right in the table, so move the decimal 3 places right: $3.5 \text{ km} = 3500 \text{ m}$

Example: Convert 275 centimeters to meters.

• Centi- to the base unit is 2 steps to the left, so move the decimal 2 places left: $275 \text{ cm} = 2.75 \text{ m}$

Problems with Mixed Metric Units

Just like with U.S. units, you need a common unit before adding or subtracting.

1. Identify which unit you want your answer in.
2. Convert all measurements to that unit.
3. Perform the calculation.

Example: Add 2 kilometers and 500 meters.

1. Convert 2 km to meters: $2 \times 1000 = 2000$ meters
2. Add: $2000 + 500 = 2500$ meters (or $2.5$ km)

Converting Between Systems

U.S. Customary to Metric Translations

These conversion factors bridge the two systems. You'll typically be given these on a test, but it helps to be familiar with them:

• 1 inch $\approx$ 2.54 centimeters
• 1 pound $\approx$ 0.454 kilograms
• 1 quart $\approx$ 0.946 liters

To convert, multiply by the appropriate factor. To convert back, divide by that same factor.

Example: Convert 5 inches to centimeters. $5 \times 2.54 = 12.7 \text{ cm}$

Example: Convert 12.7 cm back to inches. $12.7 \div 2.54 = 5 \text{ inches}$

Fahrenheit and Celsius Conversions

Two formulas connect these temperature scales:

Example: Convert 77°F to Celsius.

1. Subtract 32: $77 - 32 = 45$

2. Multiply by $\frac{5}{9}$: $45 \times \frac{5}{9} = 25$

3. Result: $25°C$

Two reference points worth memorizing:

• Water freezes at $32°F = 0°C$
• Water boils at $212°F = 100°C$

These are useful for checking whether your answer makes sense. If you convert a temperature and get something above 100°C for a warm day outside, something went wrong.

International System of Units (SI)

The International System of Units (SI) is the modern, standardized form of the metric system. It's the measurement system used by scientists worldwide and by most countries for everyday use.

SI uses the same base units and prefixes covered in the metric section above (meters, grams, liters, and the kilo-/centi-/milli- prefixes). For this course, the main takeaway is that SI and the metric system work the same way. When you see "SI units" on a problem, you can apply everything you already know about metric conversions.