4.1 Measurement & unit conversions
Key Takeaways
- Dimensional analysis multiplies by conversion factors equal to 1 so unwanted units cancel and only the target unit remains.
- Converting to a smaller unit gives a larger number; converting to a larger unit gives a smaller number.
- Key US bridges are 1 mi = 5,280 ft, 1 gal = 4 qt, and 1 lb = 16 oz; metric steps move by powers of ten.
- Convert rates one unit at a time: 60 mph equals 88 ft/s using 5,280 ft/mi and 3,600 s/hr.
- Always check reasonableness against anchors like 1 in ≈ 2.54 cm and a 2 m doorway ≈ 6.5 ft.
Moving Between Units Without Changing the Value
A measurement describes an amount using a number and a unit, such as 3 feet or 250 milliliters. A unit conversion rewrites that same amount using a different unit. The physical size never changes — only the label and the matching number. The most reliable method for the TSIA2 is dimensional analysis, also called the factor-label method: you multiply the starting quantity by fractions that each equal 1 so the unwanted units cancel and the unit you want survives.
Conversion factors are fractions equal to 1
Because 12 inches and 1 foot describe the same length, the fraction 12 in / 1 ft equals 1, and so does its reciprocal 1 ft / 12 in. Multiplying by 1 never changes a quantity's value, only how it looks. The skill is choosing the version of the factor that puts the unit you want to cancel on the opposite side of the fraction bar from where it starts, so those units divide out like common factors.
Reference table of common conversions
| Category | Relationships |
|---|---|
| US length | 1 ft = 12 in; 1 yd = 3 ft; 1 mi = 5,280 ft |
| US weight | 1 lb = 16 oz; 1 ton = 2,000 lb |
| US capacity | 1 cup = 8 fl oz; 1 pt = 2 cups; 1 qt = 2 pt; 1 gal = 4 qt |
| Metric length | 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm |
| Metric mass/volume | 1 kg = 1,000 g; 1 g = 1,000 mg; 1 L = 1,000 mL |
| US ↔ metric | 1 in ≈ 2.54 cm; 1 mi ≈ 1.61 km; 1 kg ≈ 2.20 lb; 1 L ≈ 1.06 qt |
Worked example 1: one step
Convert 4.5 feet to inches. Start with 4.5 ft and multiply by 12 in / 1 ft so that "ft" cancels:
4.5 ft × (12 in / 1 ft) = 4.5 × 12 = 54 inches.
The "ft" units cancel because one is on top and one is on the bottom, leaving the answer in inches.
Worked example 2: a multi-step chain
Convert 2 miles to inches. No single factor links miles directly to inches, so chain two factors together:
2 mi × (5,280 ft / 1 mi) × (12 in / 1 ft).
The "mi" cancels first, then "ft" cancels, leaving inches: 2 × 5,280 × 12 = 126,720 inches. Building a chain lets you connect any two units as long as you can link them through steps you know.
Worked example 3: metric
Convert 3.2 km to centimeters:
3.2 km × (1,000 m / 1 km) × (100 cm / 1 m) = 3.2 × 1,000 × 100 = 320,000 cm.
Because the metric system is built on powers of ten, each step simply shifts the decimal point.
Converting rates
A rate carries two units, such as miles per hour (mi/hr) or dollars per gallon. Convert one unit at a time, and remember that a unit in the denominator cancels when it appears in the numerator of a factor.
Worked example 4: change 60 miles per hour into feet per second.
60 mi/hr × (5,280 ft / 1 mi) × (1 hr / 3,600 s) = (60 × 5,280) / 3,600 = 316,800 / 3,600 = 88 ft/s.
Notice that "mi" cancels against the first factor's numerator and "hr" cancels against the second factor's denominator, leaving ft/s.
Worked example 5: gasoline costs $3.60 per gallon. Find the cost per quart.
$3.60 / gal × (1 gal / 4 qt) = 3.60 / 4 = $0.90 per quart.
Checking reasonableness
Every converted answer should pass a common-sense check. Converting to a smaller unit produces a larger number (54 in is more than 4.5 ft because inches are small). Converting to a larger unit produces a smaller number. If you convert 500 cm to meters and get 50,000, you multiplied when you should have divided — 500 cm is only 5 m.
Estimation anchors help too. One inch is about 2.54 cm, so a 30 cm ruler is roughly 12 in, about 1 foot — reasonable. A doorway near 2 m tall is about 6.5 ft — reasonable for a door. A 2-liter bottle is a little over 2 quarts, since 1 L ≈ 1.06 qt, which is close to half a gallon. When an answer clashes with such an anchor, recheck which direction you divided.
Working across the two systems
Many real problems mix US and metric units, so the bridge factors earn their place in the table. To convert 10 inches to centimeters, use 1 in ≈ 2.54 cm: 10 in × (2.54 cm / 1 in) = 25.4 cm. To go the other way, from 8 kilograms to pounds, use 1 kg ≈ 2.20 lb: 8 kg × (2.20 lb / 1 kg) = 17.6 lb. The procedure never changes between systems — you simply pick the factor that cancels the unit you are leaving and keeps the unit you want.
A dependable routine
- Write the starting quantity with its unit. 2. Multiply by conversion factors, each arranged so an unwanted unit cancels. 3. Keep chaining until only the target unit remains. 4. Multiply the numbers on top and divide by the numbers on the bottom. 5. Confirm the size is reasonable. Following these five steps turns every conversion — length, weight, capacity, metric, or rate — into the same predictable procedure, which is exactly what the TSIA2 rewards.
Convert 3 miles to feet using dimensional analysis.
A car travels 45 miles per hour. What is this speed in feet per second?
Convert 2,500 mL to liters.