1. Convert Rates and Measurements: Customary Units

Length (Distance) Conversions:

• \( 1 \text{ foot (ft)} = 12 \text{ inches (in)} \)
• \( 1 \text{ yard (yd)} = 3 \text{ feet} = 36 \text{ inches} \)
• \( 1 \text{ mile (mi)} = 5{,}280 \text{ feet} = 1{,}760 \text{ yards} \)

Weight (Mass) Conversions:

• \( 1 \text{ pound (lb)} = 16 \text{ ounces (oz)} \)
• \( 1 \text{ ton} = 2{,}000 \text{ pounds} \)

Capacity (Volume) Conversions:

• \( 1 \text{ tablespoon (tbsp)} = 3 \text{ teaspoons (tsp)} \)
• \( 1 \text{ fluid ounce (fl oz)} = 2 \text{ tablespoons} \)
• \( 1 \text{ cup (c)} = 8 \text{ fluid ounces} \)
• \( 1 \text{ pint (pt)} = 2 \text{ cups} \)
• \( 1 \text{ quart (qt)} = 2 \text{ pints} = 4 \text{ cups} \)
• \( 1 \text{ gallon (gal)} = 4 \text{ quarts} = 8 \text{ pints} = 16 \text{ cups} \)

Time Conversions:

• \( 1 \text{ minute (min)} = 60 \text{ seconds (sec)} \)
• \( 1 \text{ hour (hr)} = 60 \text{ minutes} = 3{,}600 \text{ seconds} \)
• \( 1 \text{ day} = 24 \text{ hours} \)
• \( 1 \text{ week} = 7 \text{ days} \)
• \( 1 \text{ year} = 365 \text{ days} \) (or 366 for leap year)

Rate Conversions:

Miles per Hour to Feet per Second:

\( 1 \text{ mph} = 1.467 \text{ ft/s} \) (approximately)

\( \text{ft/s} = \text{mph} \times \frac{5{,}280}{3{,}600} = \text{mph} \times 1.467 \)

Example: Convert 30 mph to feet per second

\( 30 \text{ mph} \times \frac{5{,}280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3{,}600 \text{ sec}} = \frac{158{,}400}{3{,}600} = 44 \text{ ft/s} \)

2. Convert Rates and Measurements: Metric Units

Metric System Prefixes:

Length Conversions:

• \( 1 \text{ kilometer (km)} = 1{,}000 \text{ meters (m)} \)
• \( 1 \text{ meter (m)} = 100 \text{ centimeters (cm)} \)
• \( 1 \text{ meter} = 1{,}000 \text{ millimeters (mm)} \)
• \( 1 \text{ centimeter} = 10 \text{ millimeters} \)

Mass (Weight) Conversions:

• \( 1 \text{ kilogram (kg)} = 1{,}000 \text{ grams (g)} \)
• \( 1 \text{ gram} = 1{,}000 \text{ milligrams (mg)} \)
• \( 1 \text{ metric ton (tonne)} = 1{,}000 \text{ kilograms} \)

Capacity (Volume) Conversions:

• \( 1 \text{ kiloliter (kL)} = 1{,}000 \text{ liters (L)} \)
• \( 1 \text{ liter} = 1{,}000 \text{ milliliters (mL)} \)
• \( 1 \text{ liter} = 100 \text{ centiliters (cL)} \)

Conversion Strategy:

Moving Up (to larger units): Divide or move decimal left

Moving Down (to smaller units): Multiply or move decimal right

Example 1: Convert 5 meters to centimeters

\( 5 \text{ m} = 5 \times 100 = 500 \text{ cm} \)

Example 2: Convert 3,500 grams to kilograms

\( 3{,}500 \text{ g} = 3{,}500 \div 1{,}000 = 3.5 \text{ kg} \)

3. Mixed Customary Units

Definition: Mixed units combine two different units together (e.g., 5 feet 8 inches, 2 hours 30 minutes)

Converting TO Mixed Units:

Step 1: Divide the larger value by the conversion factor

Step 2: The quotient is the larger unit

Step 3: The remainder is the smaller unit

Example 1: Convert 68 inches to feet and inches

\( 68 \div 12 = 5 \text{ R } 8 \)

Answer: 5 feet 8 inches

Example 2: Convert 150 minutes to hours and minutes

\( 150 \div 60 = 2 \text{ R } 30 \)

Answer: 2 hours 30 minutes

Converting FROM Mixed Units:

Step 1: Multiply the larger unit by the conversion factor

Step 2: Add the smaller unit

Example 3: Convert 3 feet 7 inches to inches

\( (3 \times 12) + 7 = 36 + 7 = 43 \text{ inches} \)

Example 4: Convert 4 pounds 12 ounces to ounces

\( (4 \times 16) + 12 = 64 + 12 = 76 \text{ ounces} \)

Operations with Mixed Units:

Addition/Subtraction: Add/subtract like units separately, then simplify

Example: Add 2 ft 9 in + 3 ft 8 in

Feet: \( 2 + 3 = 5 \text{ ft} \)

Inches: \( 9 + 8 = 17 \text{ in} = 1 \text{ ft } 5 \text{ in} \)

Total: \( 5 + 1 = 6 \text{ ft } 5 \text{ in} \)

4. Convert Between Customary and Metric Systems

Important Note: These conversions are approximate. The ≈ symbol means "approximately equal to."

Length Conversions:

Mass/Weight Conversions:

Capacity/Volume Conversions:

Example 1: Convert 5 miles to kilometers

\( 5 \text{ mi} \times 1.61 = 8.05 \text{ km} \)

Example 2: Convert 10 kilograms to pounds

\( 10 \text{ kg} \times 2.20 = 22 \text{ lb} \)

5. Convert Between Celsius and Fahrenheit

Temperature Scale Background:

• Celsius (°C): Water freezes at 0°C, boils at 100°C
• Fahrenheit (°F): Water freezes at 32°F, boils at 212°F

Celsius to Fahrenheit:

\( F = \frac{9}{5}C + 32 \)

or \( F = 1.8C + 32 \)

Step-by-Step Method:

1. Multiply the Celsius temperature by \( \frac{9}{5} \) (or 1.8)
2. Add 32 to the result

Example 1: Convert 20°C to Fahrenheit

\( F = \frac{9}{5}(20) + 32 = 36 + 32 = 68°F \)

Example 2: Convert 100°C to Fahrenheit

\( F = 1.8(100) + 32 = 180 + 32 = 212°F \)

Fahrenheit to Celsius:

\( C = \frac{5}{9}(F - 32) \)

or \( C = \frac{F - 32}{1.8} \)

Step-by-Step Method:

1. Subtract 32 from the Fahrenheit temperature
2. Multiply the result by \( \frac{5}{9} \) (or divide by 1.8)

Example 3: Convert 68°F to Celsius

\( C = \frac{5}{9}(68 - 32) = \frac{5}{9}(36) = 20°C \)

Example 4: Convert 32°F to Celsius

\( C = \frac{5}{9}(32 - 32) = \frac{5}{9}(0) = 0°C \)

Common Temperature Conversions:

Memory Tip: To go from C to F, think "multiply and add" (×1.8, then +32). To go from F to C, think "subtract and divide" (-32, then ÷1.8).

Dimensional Analysis (Unit Conversion Method)

Definition: A systematic method for converting units by using conversion factors as fractions.

General Formula:

\( \text{Given Unit} \times \frac{\text{Desired Unit}}{\text{Given Unit}} = \text{Desired Unit} \)

Steps for Dimensional Analysis:

1. Identify the given measurement and unit
2. Identify the desired unit
3. Set up conversion factor(s) so given units cancel
4. Multiply and simplify
5. Check that units make sense

Key Rule:

Place the unit you want to cancel in the denominator of your conversion factor.

Example 1: Convert 5 feet to inches

\( 5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} = 60 \text{ in} \)

(feet cancel, leaving inches)

Example 2: Convert 3 miles to feet

\( 3 \text{ mi} \times \frac{5{,}280 \text{ ft}}{1 \text{ mi}} = 15{,}840 \text{ ft} \)

Multi-Step Conversions:

Chain multiple conversion factors together for complex conversions.

Example 3: Convert 2 miles to inches

\( 2 \text{ mi} \times \frac{5{,}280 \text{ ft}}{1 \text{ mi}} \times \frac{12 \text{ in}}{1 \text{ ft}} = 126{,}720 \text{ in} \)

Example 4: Convert 60 mph to feet per second

\( 60 \frac{\text{mi}}{\text{hr}} \times \frac{5{,}280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3{,}600 \text{ sec}} = \frac{316{,}800}{3{,}600} = 88 \text{ ft/s} \)

Quick Reference: Common Conversions

💡 Key Tips for Unit Conversions

• ✓ Memorize common conversion factors to save time
• ✓ Use dimensional analysis for complex multi-step conversions
• ✓ Check that units cancel properly in dimensional analysis
• ✓ Converting to larger units means dividing (numbers get smaller)
• ✓ Converting to smaller units means multiplying (numbers get larger)
• ✓ In metric system, move decimals based on prefix values
• ✓ Most cross-system conversions are approximate (use ≈)
• ✓ For temperature, remember the formulas and the order of operations
• ✓ Always include units in your work to catch errors
• ✓ Practice with real-world examples (cooking, travel, weather)