Cheatsheet: Working With Units

1. Unit Fundamentals

1.1 What Are Units

1.2 Common Unit Systems

2. Unit Conversion

2.1 Conversion Factors

• A conversion factor is a ratio equal to 1 that relates two different units
• Example: 1 foot = 12 inches, so 12 in/1 ft = 1 or 1 ft/12 in = 1
• Multiply by conversion factors to change units while keeping the quantity the same

2.2 Common Length Conversions

2.3 Common Weight/Mass Conversions

2.4 Common Volume Conversions

2.5 Common Time Conversions

3. Conversion Process

3.1 Single-Step Conversion

• Set up the problem: Write the original quantity
• Choose the conversion factor: Pick the form where the old unit cancels
• Multiply and cancel: Multiply by the fraction and cross out matching units
• Calculate: Perform the arithmetic to find the answer
• Example: Convert 36 inches to feet: 36 in × (1 ft / 12 in) = 3 ft

3.2 Multi-Step Conversion

• Use multiple conversion factors in sequence
• Each step cancels one unit and introduces the next
• Continue until reaching the desired unit
• Example: Convert 2 miles to inches: 2 mi × (5,280 ft / 1 mi) × (12 in / 1 ft) = 126,720 in

3.3 Conversion Tips

• Always include units in calculations
• Check that unwanted units cancel out
• The final answer should have only the desired unit
• Set up fractions so units cancel diagonally
• Keep track of what you're multiplying and dividing

4. Unit Rates

4.1 Definition and Examples

4.2 Common Unit Rates

• Speed: miles per hour (mph), meters per second (m/s)
• Price: dollars per pound, cost per item
• Wage: dollars per hour
• Density: grams per cubic centimeter
• Fuel efficiency: miles per gallon

4.3 Calculating Unit Rates

• Divide the first quantity by the second quantity
• Example: 120 miles in 2 hours = 120 ÷ 2 = 60 miles per hour
• Example: $3.60 for 4 apples = $3.60 ÷ 4 = $0.90 per apple

5. Area and Volume Units

5.1 Area Unit Conversions

• Area units are squared: square inches (in²), square feet (ft²), square meters (m²)
• When converting area, square the conversion factor
• Example: 1 ft = 12 in, so 1 ft² = 144 in² (because 12² = 144)
• Example: Convert 3 ft² to in²: 3 ft² × (144 in² / 1 ft²) = 432 in²

5.2 Common Area Conversions

5.3 Volume Unit Conversions

• Volume units are cubed: cubic inches (in³), cubic feet (ft³), cubic meters (m³)
• When converting volume, cube the conversion factor
• Example: 1 ft = 12 in, so 1 ft³ = 1,728 in³ (because 12³ = 1,728)
• Example: Convert 2 ft³ to in³: 2 ft³ × (1,728 in³ / 1 ft³) = 3,456 in³

5.4 Common Volume Conversions

6. Metric Prefixes

6.1 Common Metric Prefixes

6.2 Using Metric Prefixes

• Each prefix represents a power of 10
• Example: 1 kilometer = 1,000 meters
• Example: 1 centimeter = 0.01 meters
• Example: 5 kg = 5,000 g (multiply by 1,000)
• Example: 250 mL = 0.25 L (divide by 1,000)

7. Problem-Solving Strategies

7.1 Setting Up Conversion Problems

• Identify the starting unit and the target unit
• Write down all known conversion factors between the units
• Arrange conversion factors so units cancel properly
• Check that the final answer has the correct unit

7.2 Common Mistakes to Avoid

• Forgetting to include units in the answer
• Using the conversion factor upside down
• Not canceling units properly
• Confusing area/volume units with linear units
• Mixing up similar-looking conversions (e.g., 1 yd = 3 ft, not 12 ft)

7.3 Checking Your Answer

• Does the answer make sense? (e.g., feet should be larger than inches for the same length)
• Are the units correct?
• Did all unwanted units cancel out?
• Is the magnitude reasonable?

8. Applications

8.1 Real-World Unit Conversions

• Cooking: converting recipes between metric and US customary units
• Travel: converting speed limits between mph and km/h
• Shopping: comparing prices using unit rates
• Construction: converting measurements for materials
• Science: converting between different measurement systems

8.2 Comparing Unit Rates

• Calculate unit rate for each option
• Compare unit rates to find the best value
• Example: $4.50 for 3 lb vs. $7.00 for 5 lb
• First option: $4.50 ÷ 3 = $1.50 per lb
• Second option: $7.00 ÷ 5 = $1.40 per lb (better value)