Rates and Unit Conversion

A Rate Is a Ratio With Units

A rate compares two quantities measured in different units — miles per hour, dollars per pound, words per minute. A unit rate is a rate whose denominator is exactly $1$: $60$ miles in $2$ hours is a rate, but $30$ miles per one hour is the unit rate. To find a unit rate, divide the top quantity by the bottom quantity; the answer's units are "top per one bottom."

Unit rates are powerful because they behave like a constant of proportionality. Once you know a car travels $30$ miles per hour, you can find distance for any time by multiplying and time for any distance by dividing. The relationship $\text{distance} = \text{rate} \times \text{time}$ — and its cousins for cost, work, and density — is really just $y = kx$ wearing units.

Dimensional Analysis — Converting Units

The most reliable way to convert units is dimensional analysis: multiply by conversion factors written as fractions equal to $1$, arranged so the unwanted units cancel diagonally. Because each factor equals $1$, you never change the quantity's value — only the way it is expressed.

For instance, convert a cyclist's speed of $12$ meters per second to kilometers per hour:

$\begin{aligned} \frac{12 \text{ m}}{1 \text{ s}} \times \frac{3600 \text{ s}}{1 \text{ hr}} \times \frac{1 \text{ km}}{1000 \text{ m}} &= \frac{12 \cdot 3600}{1000} \; \frac{\text{km}}{\text{hr}} \\ &= 43.2 \; \frac{\text{km}}{\text{hr}} \end{aligned}$

Every unit except km and hr cancels. That cancellation is your safety check: if a unit you wanted to keep disappears, or an unwanted one survives, you flipped a factor and should rewrite it. A squared or cubed unit needs its conversion factor applied that many times — converting $\text{cm}^2$ to $\text{m}^2$ uses the length factor twice.

Combining Rates

Some SAT problems combine two rates. If one printer makes $5$ pages per minute and a second makes $3$ pages per minute, together they produce $5 + 3 = 8$ pages per minute, so $200$ pages take $\tfrac{200}{8} = 25$ minutes. Decide first whether the rates should be added (working together), subtracted (opposing, like a tank filled and drained at once), or handled separately.

Average Speed Is Not the Average of Speeds

The subtlest rate idea on the SAT is average speed. If you drive $60$ miles at $30$ mph and $60$ miles at $60$ mph, the average speed is not $\tfrac{30 + 60}{2} = 45$ mph. Average speed is always total distance over total time:

$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$

The first leg takes $\tfrac{60}{30} = 2$ hours and the second takes $\tfrac{60}{60} = 1$ hour, so the trip covers $120$ miles in $3$ hours — an average of $40$ mph, not $45$. The slower leg eats up more time, so it pulls the average below the midpoint. You can only average two speeds directly when the times spent at each speed are equal.

Choosing an Approach

• "Per one" of something → compute the unit rate by dividing.
• Changing units → chain conversion factors so unwanted units cancel.
• Two machines/pipes/people at once → add (or subtract) their rates.
• A whole-trip average → use total distance over total time, never the average of the speeds.

Rates reward slow, careful setup. Write the units on every quantity, make them cancel, and let the arithmetic follow.