Units and rates

Carry the units through the calculation so they cancel into the unit the question asks for.

When to use it

The question converts between units (miles per hour to feet per second, square feet to square inches).
A rate is given in one time unit and asked in another.
An average speed, price or density combines two different rates.

Write the starting quantity with its units, then multiply by fractions that equal 1, such as (5,280 ft / 1 mi) and (1 hr / 3,600 s).
Arrange each fraction so the unit you want to remove is on the opposite side (top vs. bottom) from where it already stands.
When the unit is squared or cubed, square or cube the whole conversion factor.
For an average rate, use total amount ÷ total time, never the average of the two rates.

Watch out: Averaging two speeds. Going 30 mph one way and 60 mph back is not 45 mph on average, because you spend more time at the slower speed.

Time: Writing the unit chain takes ten seconds and avoids the classic ×/÷ mix-ups. If the final unit is not the one asked for, you are not done.

A two-step unit chain: miles to feet, then hours to seconds.

A car travels at 60 miles per hour. How many feet per second is that? (1 mile = 5,280 feet)

Write 60 mi/hr × (5,280 ft / 1 mi) × (1 hr / 3,600 s). Miles cancel and hours cancel, leaving ft/s.
60 × 5,280 = 316,800, and 316,800 ÷ 3,600 = 88.
The result is 88 feet per second.

Squared units need a squared conversion factor.

How many square inches are in 5 square feet?

An average speed is total distance over total time.

A driver goes to a town at 30 mph and returns along the same route at 60 mph. What is the average speed for the round trip?

Let the one-way distance be d. The trip out takes d/30 hours and the trip back takes d/60 hours.
Total time: d/30 + d/60 = 3d/60 = d/20. Total distance: 2d.
Average speed = 2d ÷ (d/20) = 40 mph. The average of the two speeds, 45, is the trap.