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Unit
Conversions

Master the art of converting between units of measurement using dimensional analysis: a skill tested on every SAT.

SAT Math

What Are Unit Conversions?

A unit conversion changes a measurement from one unit to another without changing the actual quantity. For example, \(12\) inches and \(1\) foot describe the same length.

Same Quantity
The physical amount stays identical: only the label changes.
Multiply by 1
Every conversion factor equals \(1\), so the value is preserved.

The Conversion Factor

A conversion factor is a fraction that equals \(1\). It's built from a known equivalence between two units.

Equivalence
\(1 \text{ foot} = 12 \text{ inches}\)
 Both describe the same length Two Factors
\(\frac{12 \text{ in}}{1 \text{ ft}}\)
or \(\frac{1 \text{ ft}}{12 \text{ in}}\)

Dimensional Analysis: The Core Method

Dimensional analysis is the process of multiplying by conversion factors so that unwanted units cancel and the desired unit remains.

Start
\(3 \text{ ft}\)
 × Factor
\(\frac{12 \text{ in}}{1 \text{ ft}}\)
 = Result
\(36 \text{ in}\)
 ft cancels ft

The General Rule

Every unit conversion follows one formula:

If you need to chain multiple conversions, simply multiply by additional factors:

Essential Conversion Factors

Length
\(1 \text{ ft} = 12 \text{ in}\)
\(1 \text{ yd} = 3 \text{ ft}\)
\(1 \text{ mi} = 5{,}280 \text{ ft}\)
\(1 \text{ in} = 2.54 \text{ cm}\)
Time
\(1 \text{ hr} = 60 \text{ min}\)
\(1 \text{ min} = 60 \text{ s}\)
\(1 \text{ day} = 24 \text{ hr}\)
\(1 \text{ yr} \approx 365 \text{ days}\)
Volume / Mass
\(1 \text{ gal} = 4 \text{ qt}\)
\(1 \text{ qt} \approx 0.946 \text{ L}\)
\(1 \text{ lb} = 16 \text{ oz}\)
\(1 \text{ kg} \approx 2.205 \text{ lb}\)

Single-Step vs. Multi-Step

Single-Step Multi-Step
Factors needed \(1\) \(2\) or more
Example feet → inches miles → inches
Strategy One multiplication Chain factors until old unit is gone
SAT frequency Common Very common

How to Apply It: Step by Step

1
Identify the starting value and its unit.
2
Identify the target (desired) unit.
3
Write the conversion factor(s) as fractions with the old unit in the denominator.
4
Cancel units diagonally: same unit top and bottom disappears.
5
Multiply across and simplify. Check that only the desired unit remains.

Example 1: Single-Step

Problem: A table is \(4.5\) feet long. How many inches is that?

\(4.5 \text{ ft}\)
 starting value ×
\(\frac{12 \text{ in}}{1 \text{ ft}}\)
 =
\(54 \text{ in}\)
 result

Example 2: Multi-Step

Problem: A car travels at \(60\) miles per hour. What is the speed in feet per second?

1
Miles cancel: \(60 \times 5{,}280 = 316{,}800\) ft/hr
2
Hours cancel: \(\frac{316{,}800}{60} = 5{,}280\) ft/min
3
Minutes cancel: \(\frac{5{,}280}{60} = \) 88 ft/s

Example 3: Rate Conversion

Problem: A printer prints \(8\) pages per minute. How many pages does it print in \(3\) hours?

1
Convert rate: \(8 \times 60 = 480\) pages per hour
2
Multiply by time: \(480 \times 3 = \) 1,440 pages

Squared and Cubed Units

When converting area or volume, the conversion factor must be raised to the same power as the unit.

Area (\(\text{ft}^2 \to \text{in}^2\))
$$1\;\text{ft}^2 = (12)^2\;\text{in}^2 = 144\;\text{in}^2$$
Volume (\(\text{ft}^3 \to \text{in}^3\))
$$1\;\text{ft}^3 = (12)^3\;\text{in}^3 = 1{,}728\;\text{in}^3$$

Example 4: Area Conversion

Problem: A room has an area of \(200\) square feet. Convert to square inches.

\(1 \text{ ft}^2\)
\(1 \text{ ft} \times 1 \text{ ft}\)
 =
\(144 \text{ in}^2\)
\(12 \text{ in} \times 12 \text{ in}\)
 Square the conversion factor!

Metric Prefixes on the SAT

The SAT sometimes uses metric units. Each prefix represents a power of 10.

Prefix Symbol Meaning Factor
kilo- k thousand \(10^3\)
centi- c hundredth \(10^{-2}\)
milli- m thousandth \(10^{-3}\)

Common Mistakes

✕ Flipping the Factor
Putting the wrong unit on top. If you want to cancel feet, feet must be in the denominator.
✕ Forgetting to Square
Using \(12\) instead of \(144\) when converting \(\text{ft}^2\) to \(\text{in}^2\). Always raise the factor to match the exponent.
✕ Mixing Up Direction
Converting to a larger unit should give a smaller number, and vice versa. If your answer goes the wrong way, the factor is flipped.

Summary: Quick Reference

Conversion Type Strategy Watch Out For
Single-step One factor Flipping the fraction
Multi-step Chain factors Missing a bridge unit
Rate conversion Convert denominator unit Which unit to convert
Area / Volume Square or cube the factor Forgetting the exponent
Metric Move the decimal Direction of movement

Question 1 of 5

A ribbon is 3 yards long. How many inches is that?
(\(1\;\text{yd} = 3\;\text{ft}\), \(1\;\text{ft} = 12\;\text{in}\))

Question 2 of 5

A car travels at 45 miles per hour. What is its speed in feet per minute?
(\(1\;\text{mi} = 5{,}280\;\text{ft}\))

Question 3 of 5

A square garden has a side length of 5 feet. What is its area in square inches?

Question 4 of 5

A factory produces 720 widgets per hour. At this rate, how many widgets are produced in 2.5 minutes?

Question 5 of 5

If \(1\;\text{inch} = 2.54\;\text{cm}\), approximately how many centimeters are in 2 feet?