Mastering Rates

Learn to solve every type of rate problem on the SAT: unit rates, combined rates, and conversions using the fundamental relationship \(\text{Rate} = \frac{\text{Amount}}{\text{Time}}\).

SAT Math

What Is a Rate?

A rate describes how one quantity changes relative to another. Any time you see "per" in a problem: miles per hour, dollars per item, words per minute: you are dealing with a rate.

The Rate Triangle

Why Rates Matter on the SAT

High Frequency

Rate problems appear in 3 to 5 questions per test across both modules.

Real-World Contexts

Speed, production, pricing, population growth: expect word-problem wrappers.

Quick Points

Most rate questions are medium difficulty: high reward for a solid method.

The Unit Rate Formula

A unit rate expresses how much of one quantity corresponds to exactly one unit of another. It is the simplest building block of every rate problem.

Combined Rates

When two workers, machines, or pipes work together, add their individual rates:

Rate vs. Ratio vs. Proportion

	Rate	Ratio	Proportion
Definition	Comparison of two quantities with different units	Comparison of two quantities with the same unit	An equation stating two ratios or rates are equal
Example	\(60\) miles per hour	\(3 : 5\) boys to girls	\(\frac{3}{5} = \frac{x}{20}\)
Key Word	"per"	"to"	"equal" or "same rate"

Unit Conversions in Rate Problems

Many SAT rate problems require converting between units. Use dimensional analysis: multiply by conversion fractions so unwanted units cancel.

How to Solve Any Rate Problem

1

Identify the rate, amount, and time. Label what you know and what you need.

2

Check units. Are all quantities in compatible units? If not, convert first.

3

Choose your formula. Use \(R = \frac{A}{T}\), or for combined work: \(R_1 + R_2\).

4

Set up a proportion if needed. Cross-multiply and solve for the unknown.

5

Verify your answer. Plug back in: does \(\text{Rate} \times \text{Time} = \text{Amount}\)?

Example 1: Basic Rate

A printer produces 240 pages in 8 minutes. At this rate, how many pages will it produce in 35 minutes?

1

Find the unit rate: \(\frac{240}{8} = \) 30 pages per minute.

2

Multiply: \(30 \times 35 = \) 1,050 pages.

Example 2: Combined Work

Machine A can package 600 boxes in 4 hours. Machine B can package the same 600 boxes in 6 hours. Working together, how long will it take them?

1

Rate A: \(\frac{600}{4} = 150\) boxes/hr. Rate B: \(\frac{600}{6} = 100\) boxes/hr.

2

Combined rate: \(150 + 100 = 250\) boxes/hr.

3

Time: \(\frac{600}{250} = 2.4\) hours = 2 hours 24 minutes.

Example 3: Average Speed Trap

A car drives 120 miles to a city at 40 mph and returns the same distance at 60 mph. What is the average speed for the entire trip?

1

Time going: \(\frac{120}{40} = 3\) hours. Time returning: \(\frac{120}{60} = 2\) hours.

2

Total distance: \(120 + 120 = 240\) miles. Total time: \(3 + 2 = 5\) hours.

3

Average speed: \(\frac{240}{5} = \) 48 mph. Not 50 mph!

Rates on a Graph

Average Speed: The Harmonic Mean

When you travel the same distance at two different speeds, the average speed is the harmonic mean, not the arithmetic mean:

Common Mistakes to Avoid

✕ Averaging Speeds

Taking \(\frac{v_1 + v_2}{2}\) instead of computing total distance ÷ total time. This gives the wrong answer when time spent at each speed differs.

✕ Adding Times for Combined Work

Adding individual completion times instead of individual rates. Always convert to rates first, then add.

✕ Ignoring Unit Mismatches

Mixing hours and minutes, or miles and kilometers, without converting. Always check that all units are consistent before calculating.

Rate Problem Cheat Sheet

Problem Type	Formula	Watch Out For
Basic Rate	\(R = \frac{A}{T}\)	Unit mismatches
Combined Work	\(R_1 + R_2\)	Adding times instead of rates
Average Speed (same distance)	\(\frac{2v_1 v_2}{v_1 + v_2}\)	Using arithmetic mean
Unit Conversion	Dimensional analysis	Flipping conversion factors
Proportion	\(\frac{a}{b} = \frac{c}{d}\)	Cross-multiply carefully

Question 1 of 5

A car travels 180 miles in 3 hours. At this rate, how far will the car travel in 7 hours?

The unit rate is \(\frac{180}{3} = 60\) mph. In 7 hours: \(60 \times 7 = 420\) miles.

Question 2 of 5

Pipe A fills a pool in 10 hours. Pipe B fills the same pool in 15 hours. Working together, how long does it take to fill the pool?

\(R_A = \frac{1}{10}\), \(R_B = \frac{1}{15}\). Combined: \(\frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}\). Time: \(\frac{1}{1/6} = 6\) hours.

Question 3 of 5

A runner jogs 5 km at 10 km/h and then sprints 5 km at 15 km/h. What is the runner's average speed for the entire 10 km?

Same distance at each speed, so use the harmonic mean: \(\frac{2 \times 10 \times 15}{10 + 15} = \frac{300}{25} = 12\) km/h. Note: 12.5 is the arithmetic mean, which is the trap answer.

Question 4 of 5

A factory produces 1,200 widgets in 8 hours with 6 machines. At the same rate per machine, how many widgets would 9 machines produce in 12 hours?

Rate per machine: \(\frac{1200}{6 \times 8} = 25\) widgets per machine-hour. With 9 machines for 12 hours: \(25 \times 9 \times 12 = 2{,}700\) widgets.

Question 5 of 5

A train travels at 90 km/h. How many meters does the train travel in 20 seconds?

Convert: \(90\) km/h \(= \frac{90}{3.6} = 25\) m/s. In 20 seconds: \(25 \times 20 = 500\) meters.