Scale & Rate
Problems
Master the art of solving scale drawings, unit rates, and proportional reasoning problems for the SSAT Upper Level.
What Is a Rate?
A rate is a ratio that compares two quantities with different units. When you say a car travels 60 miles per hour, you are stating a rate: distance per time.
What Is a Scale?
A scale is a special ratio that relates a measurement on a model (map, drawing, blueprint) to the actual measurement in real life. Both quantities measure the same thing: length.
Why Scale & Rate Skills Matter
Scale and rate problems appear throughout the SSAT Upper Level Quantitative section. They test your ability to set up proportions, convert units, and reason about relationships between quantities.
The Core Proportion Formula
Both scale and rate problems rely on the same fundamental idea: two equivalent ratios form a proportion. Setting up and solving proportions is the single most important skill.
The Rate Triangle
For distance-rate-time problems, the three quantities are connected by a simple formula. Covering any one variable in the triangle reveals the formula to find it.
Scale vs. Rate: Key Distinction
| Scale | Rate | |
|---|---|---|
| Units | Same type (length to length) | Different types (miles per hour) |
| Form | \(1 : n\) or \(\frac{1}{n}\) | \(\frac{\text{quantity A}}{\text{quantity B}}\) |
| Example | 1 inch = 5 miles | 60 miles per hour |
| Solving Method | Set up proportion, cross-multiply | Use \(D = R \times T\) or proportion |
| Common Context | Maps, models, blueprints | Travel, work, pricing |
Step-by-Step Approach
Example 1: Map Scale
On a map, 2 cm represents 15 km. Two cities are 7 cm apart on the map. What is the actual distance?
Example 2: Speed Problem
A train travels at 80 miles per hour. How long does it take to travel 340 miles?
Example 3: Unit Pricing
Store A sells 5 notebooks for $8.75. Store B sells 3 notebooks for $4.95. Which store offers the better price per notebook?
Example 4: Scale & Area
A blueprint uses a scale of \(1 \text{ inch} = 4 \text{ feet}\). A room measures \(3 \text{ in.} \times 5 \text{ in.}\) on the blueprint. What is the actual area of the room?
The Average Speed Trap
A car drives 30 mph for the first half of a trip and 60 mph for the second half of the same distance. What is the average speed for the entire trip?
Common Mistakes to Avoid
Summary of Problem Types
| Problem Type | Key Formula | Watch Out For |
|---|---|---|
| Map/Blueprint Scale | \(\frac{\text{model}}{\text{actual}} = \frac{\text{model}}{\text{actual}}\) | Unit conversion |
| Distance-Rate-Time | \(D = R \times T\) | Time in hours vs. minutes |
| Unit Pricing | \(\frac{\text{cost}}{\text{quantity}}\) | Compare per-unit, not totals |
| Average Speed | \(\frac{\text{Total } D}{\text{Total } T}\) | Don't average the speeds |
| Scale Area | Convert each dimension first | Scale factor applies to length, not area directly |
Question 1 of 5
On a map, 3 cm represents 24 km. If two towns are 10.5 cm apart on the map, what is the actual distance between them?
Question 2 of 5
A cyclist rides at a constant rate of 18 km per hour. How many minutes will it take to ride 12 km?
Question 3 of 5
A blueprint uses a scale of 1 inch = 6 feet. A hallway is 4.5 inches long on the blueprint. What is the actual length of the hallway?
Question 4 of 5
Brand X sells 8 pens for $6.00. Brand Y sells 12 pens for $8.40. How much less per pen does Brand Y cost compared to Brand X?
Question 5 of 5
A car drives 120 miles at 40 mph, then drives the same 120 miles at 60 mph. What is the average speed for the entire trip?