SAT Math – Problem Solving & Data Analysis
Unit Conversion
Mastering dimensional analysis and conversion factors for accurate calculations
Unit conversion is the essential skill of translating between measurement systems—converting miles to kilometers, hours to minutes, pounds to kilograms, or gallons to liters. On the SAT, these questions test whether you can work systematically with conversion factors and maintain dimensional consistency throughout calculations.
Success requires understanding the dimensional analysis method (also called factor-label method), knowing common conversion relationships, and carefully tracking units through multi-step problems. Unit conversion questions often appear embedded in larger problems—calculating speeds in different units, comparing quantities measured differently, or converting between metric and customary systems.
Understanding Unit Conversion
What is Unit Conversion?
Unit conversion is the process of changing a measurement from one unit to another while maintaining the same actual quantity. You're expressing the same amount using different units of measurement.
• 1 hour = 60 minutes = 3,600 seconds
• Same duration, different units
• The quantity doesn't change, only how we express it
Dimensional Analysis Method
The dimensional analysis (factor-label) method uses conversion factors written as fractions that equal 1. Multiply by these fractions so unwanted units cancel and desired units remain.
Since \(\frac{12 \text{ inches}}{1 \text{ foot}} = 1\), multiplying by this doesn't change the value
Units cancel like algebraic terms: \(\frac{\text{feet}}{\text{feet}} = 1\)
Conversion Factors
A conversion factor is a ratio expressing how many of one unit equals another. You can flip any conversion factor depending on which unit you want to cancel.
\(1 \text{ mile} = 5,280 \text{ feet}\) gives two conversion factors:
• \(\frac{1 \text{ mile}}{5,280 \text{ feet}}\) or \(\frac{5,280 \text{ feet}}{1 \text{ mile}}\)
Choose based on what you need to cancel
Essential Conversion Factors for SAT
Length/Distance
Customary: 1 foot = 12 inches; 1 yard = 3 feet; 1 mile = 5,280 feet
Metric: 1 meter = 100 cm; 1 kilometer = 1,000 meters
Between systems: 1 inch ≈ 2.54 cm; 1 mile ≈ 1.6 km
Time
1 minute = 60 seconds
1 hour = 60 minutes = 3,600 seconds
1 day = 24 hours
Weight/Mass
Customary: 1 pound = 16 ounces; 1 ton = 2,000 pounds
Metric: 1 kilogram = 1,000 grams
Between systems: 1 kg ≈ 2.2 pounds
Volume
Customary: 1 cup = 8 fluid ounces; 1 pint = 2 cups; 1 quart = 2 pints; 1 gallon = 4 quarts
Metric: 1 liter = 1,000 milliliters
Area and Volume Conversions
Area: Square both dimensions (1 ft² = 144 in²)
Volume: Cube all dimensions (1 ft³ = 1,728 in³)
The Dimensional Analysis Strategy
Step 1: Write the Starting Quantity with Units
Always include units. Example: 5 hours (not just 5)
Step 2: Choose Conversion Factor to Cancel Unwanted Unit
Place unwanted unit in denominator so it cancels with starting unit in numerator.
Step 3: Multiply Across, Cancel Units
Units cancel diagonally like algebraic terms. Continue until you reach desired unit.
Step 4: Calculate and Include Final Unit
Perform arithmetic and write the unit that remains. Check that it's what you wanted!
Common Pitfalls & Expert Tips
❌ Forgetting to square/cube units for area/volume
If converting ft² to in², you must square the linear conversion: (12 in/ft)² = 144 in²/ft². Don't just multiply by 12!
❌ Using conversion factors upside down
If converting feet to inches, use (12 in/1 ft), not (1 ft/12 in). The unit you want to cancel goes in the denominator!
❌ Mixing up similar conversions
Don't confuse 1 yard = 3 feet with 1 foot = 12 inches, or 1 quart = 2 pints with 1 pint = 2 cups. Write them out!
❌ Not tracking units through calculations
Write units with every number throughout the problem. This prevents errors and shows what you're calculating.
✓ Expert Tip: Draw a line for each conversion
Write out the conversion chain horizontally with multiplication symbols and fraction bars. This visual method prevents errors.
✓ Expert Tip: Check if answer magnitude makes sense
Converting 5 hours to seconds should give a large number (18,000). If you get 300, you made an error!
✓ Expert Tip: Memorize key conversions
Know by heart: 12 in/ft, 60 sec/min, 60 min/hr, 1000 m/km, 16 oz/lb. These appear constantly on the SAT.
Fully Worked SAT-Style Examples
Convert 2.5 hours to minutes.
Solution:
Step 1: Identify the conversion factor
1 hour = 60 minutes
Conversion factor: \(\frac{60 \text{ minutes}}{1 \text{ hour}}\)
Step 2: Set up dimensional analysis
\(2.5 \text{ hours} \times \frac{60 \text{ minutes}}{1 \text{ hour}}\)
Notice "hours" cancels out
Step 3: Calculate
\(2.5 \times 60 = 150\) minutes
Answer: 150 minutes
Convert 3 miles to inches.
Solution:
Step 1: Plan the conversion path
Miles → Feet → Inches (two steps)
Need: 1 mile = 5,280 feet and 1 foot = 12 inches
Step 2: Set up dimensional analysis chain
\(3 \text{ miles} \times \frac{5,280 \text{ feet}}{1 \text{ mile}} \times \frac{12 \text{ inches}}{1 \text{ foot}}\)
Miles cancels, then feet cancels, leaving inches
Step 3: Calculate
\(3 \times 5,280 \times 12 = 189,840\) inches
Reality Check:
3 miles is a long distance, so nearly 190,000 inches makes sense!
Answer: 189,840 inches
A car travels at 60 miles per hour. What is this speed in feet per second?
Solution:
Step 1: Identify needed conversions
Need to convert miles → feet AND hours → seconds
1 mile = 5,280 feet; 1 hour = 3,600 seconds
Step 2: Set up dimensional analysis
\(60 \frac{\text{miles}}{\text{hour}} \times \frac{5,280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3,600 \text{ seconds}}\)
Miles cancels, hours cancels
Step 3: Calculate
\(\frac{60 \times 5,280}{3,600} = \frac{316,800}{3,600} = 88\) feet per second
Quick Check:
88 ft/s means the car travels the length of a basketball court every second—sounds right for highway speed!
Answer: 88 feet per second
A room has an area of 2 square yards. What is the area in square feet?
Solution:
Step 1: Recognize area needs squared conversion
1 yard = 3 feet (linear)
For area: 1 yd² = (3 ft)² = 9 ft²
Step 2: Set up conversion
\(2 \text{ yd}^2 \times \frac{9 \text{ ft}^2}{1 \text{ yd}^2}\)
Step 3: Calculate
\(2 \times 9 = 18\) ft²
Common Error:
Don't just multiply by 3! That would give 6 ft², which is wrong.
Remember: Area uses squared conversions!
Answer: 18 square feet
A water tank fills at a rate of 3 gallons per minute. How many quarts does it fill in 5 minutes?
Solution:
Step 1: Find total gallons filled
\(3 \frac{\text{gallons}}{\text{minute}} \times 5 \text{ minutes} = 15\) gallons
Step 2: Convert gallons to quarts
1 gallon = 4 quarts
\(15 \text{ gallons} \times \frac{4 \text{ quarts}}{1 \text{ gallon}} = 60\) quarts
Alternative: Single Chain
\(3 \frac{\text{gal}}{\text{min}} \times 5 \text{ min} \times \frac{4 \text{ qt}}{1 \text{ gal}} = 60\) quarts
Answer: 60 quarts
A package weighs 30 kilograms. Approximately how many pounds is this? (Use 1 kg ≈ 2.2 lbs)
Solution:
Step 1: Set up conversion
Given: 1 kg ≈ 2.2 pounds
\(30 \text{ kg} \times \frac{2.2 \text{ lbs}}{1 \text{ kg}}\)
Step 2: Calculate
\(30 \times 2.2 = 66\) pounds
Note on Approximation:
The ≈ symbol means "approximately equal to"
The exact conversion is 1 kg = 2.20462 lbs, but 2.2 is commonly used
Answer: Approximately 66 pounds
A factory produces 1,200 widgets per hour. How many widgets does it produce per second?
Solution:
Step 1: Identify conversion needed
Convert hours to seconds in denominator
1 hour = 60 minutes = 3,600 seconds
Step 2: Set up conversion
\(1200 \frac{\text{widgets}}{\text{hour}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}\)
Hours cancel, leaving widgets per second
Step 3: Calculate
\(\frac{1200}{3600} = \frac{1}{3} = 0.\overline{3}\) widgets per second
Answer: \(\frac{1}{3}\) or approximately 0.33 widgets per second
Gasoline costs $3.60 per gallon. What is the cost per quart?
Solution:
Step 1: Identify the relationship
1 gallon = 4 quarts
So 1 quart is smaller than 1 gallon
Step 2: Set up conversion
\(\frac{\$3.60}{1 \text{ gallon}} \times \frac{1 \text{ gallon}}{4 \text{ quarts}}\)
Step 3: Calculate
\(\frac{3.60}{4} = \$0.90\) per quart
Logic Check:
A quart is 1/4 of a gallon, so it should cost 1/4 as much ✓
$0.90 × 4 = $3.60 ✓
Answer: $0.90 per quart
Quick Conversion Reference Table
SAT Unit Conversion Checklist
Before You Start
- Identify starting and target units
- Write down needed conversion factors
- Plan conversion path (if multi-step)
- Include units with every number
During Calculation
- Arrange factors so units cancel
- Double-check if area/volume (squared/cubed)
- Show unit cancellation clearly
- Calculate carefully
After Solving
- Check final unit matches question
- Verify magnitude makes sense
- Round appropriately if needed
- Include units in your answer
Common Mistakes to Avoid
- Don't flip conversion factors accidentally
- Don't forget to square/cube for area/volume
- Don't drop units mid-calculation
- Don't skip the reality check
Unit Conversion: The Language of Measurement
Unit conversion is more than arithmetic—it's dimensional reasoning, the ability to maintain consistency while changing perspectives. Whether you're a scientist converting experimental data, an engineer working with specifications across measurement systems, a traveler calculating distances in foreign units, or a consumer comparing prices, unit conversion is essential. The dimensional analysis method you've learned here is the universal approach used across technical fields because it's systematic, visual, and self-checking through unit cancellation. Master this technique and you'll not only succeed on SAT questions, but you'll possess a fundamental tool for quantitative work in any discipline. The units tell the story—learn to read them fluently.