Ratios, Rates, Proportional Relationships, and Units

Ratios, rates, proportional relationships, and unit conversion questions are among the most common problem types in the Problem-Solving and Data Analysis domain on the Digital SAT. You can expect roughly 3–5 questions on these concepts across the two math modules. The good news: once you internalize the core setup for these problems, they become very predictable. The questions test whether you can translate real-world scenarios into mathematical relationships, work with units carefully, and scale quantities up or down correctly.

Ratios, Rates, and Proportional Relationships

A ratio compares two quantities, typically written as $\frac{a}{b}$ or $a:b$. Ratios compare things measured in the same units (3 red marbles to 5 blue marbles).

A rate compares two quantities with different units. Speed is the classic example: $\frac{60 \text{ miles}}{1 \text{ hour}}$. Rates show up as prices per item, people per square mile, gallons per minute, and many other combinations.

Two quantities are in a proportional relationship when their ratio stays constant. If you can write $y = kx$ where $k$ is some constant, then $x$ and $y$ are proportional. The SAT tests this by giving you a scenario and asking you to find an unknown value using the fact that the ratio between the quantities doesn't change.

The standard technique is to set up a proportion (two equal ratios) and cross-multiply:

$\frac{a}{b} = \frac{c}{d} \implies ad = bc$

Worked Example 1 (Straightforward): A machine produces 84 parts in 3 hours. At this constant rate, how many parts does it produce in 7 hours?

Set up the proportion with parts on top and hours on the bottom on both sides:

$\frac{84}{3} = \frac{x}{7}$

Cross-multiply: $84 \times 7 = 3x$

$588 = 3x$

$x = 196$

Worked Example 2 (Context with extra information): A biologist studying a lake estimates that the ratio of bluegill to bass is 5 to 2. If a sample from the lake contains 120 bluegill, what is the estimated number of bass in the sample?

$\frac{5}{2} = \frac{120}{x}$

Cross-multiply: $5x = 240$

$x = 48$

The answer is 48 bass. Notice you don't need to know the total number of fish in the lake. The SAT often includes extra context to see if you can identify what matters.

Scale Factor and Scaling

When two quantities are proportional, changing one by a scale factor changes the other by that same scale factor. This is a specific property the SAT tests directly.

If $y = kx$ and you multiply $x$ by 3, then the new $y$ value is $k(3x) = 3kx = 3y$. The output triples too.

Worked Example 3: A factory uses 12 gallons of paint to coat 80 square feet of surface. How many gallons are needed to coat 200 square feet at the same rate?

Find the scale factor for the area change: $\frac{200}{80} = 2.5$

Since paint and area are proportional, multiply the paint by the same factor: $12 \times 2.5 = 30$ gallons.

This approach is often faster than setting up a full proportion and cross-multiplying.

Scale drawings are a common application. A map or blueprint uses a fixed ratio between drawn length and actual length.

Worked Example 4: A scale drawing of a building uses a scale of 1 inch to 8 feet. If a wall is 3.5 inches long in the drawing, what is the actual length of the wall, in feet?

The scale factor from drawing to real life is 8 feet per inch:

$3.5 \text{ inches} \times \frac{8 \text{ feet}}{1 \text{ inch}} = 28 \text{ feet}$

Derived Units

Derived units are units built from combinations of other units, either through multiplication or division. The SAT expects you to work with these and understand what they represent.

• Quotient-based derived units come from dividing: population per square kilometer ($\frac{\text{people}}{\text{km}^2}$), miles per hour ($\frac{\text{mi}}{\text{hr}}$), cost per pound.
• Product-based derived units come from multiplying: kilowatt-hours (kilowatts $\times$ hours = energy), person-hours (people $\times$ hours = total labor).

Worked Example 5: A city has a population of 450,000 and an area of 180 square kilometers. What is the population density, in people per square kilometer?

$\frac{450{,}000 \text{ people}}{180 \text{ km}^2} = 2{,}500 \text{ people per km}^2$

Worked Example 6: A data center uses an average of 50 kilowatts of power continuously for 24 hours. How many kilowatt-hours of energy does it consume?

Kilowatt-hours = kilowatts $\times$ hours:

$50 \times 24 = 1{,}200 \text{ kilowatt-hours}$

The key with derived units is understanding whether to multiply or divide based on what the unit represents.

Unit Conversion: One-Step and Multi-Step

Unit conversion problems ask you to express a quantity in different units. The technique is called dimensional analysis: multiply by conversion fractions that equal 1, arranged so unwanted units cancel.

One-Step Conversion

Worked Example 7: A rope is 15 feet long. What is its length in inches? (1 foot = 12 inches)

$15 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} = 180 \text{ in}$

The "ft" cancels, leaving inches.

Multi-Step and Multi-Dimensional Conversion

Some problems require chaining multiple conversion factors or converting units that appear in both the numerator and denominator.

Worked Example 8: A car travels at 90 kilometers per hour. What is its speed in meters per second? (1 km = 1,000 m; 1 hour = 3,600 seconds)

$\frac{90 \text{ km}}{1 \text{ hr}} \times \frac{1{,}000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3{,}600 \text{ s}}$

Cancel units step by step:

$= \frac{90 \times 1{,}000}{3{,}600} \text{ m/s}$

$= \frac{90{,}000}{3{,}600} \text{ m/s} = 25 \text{ m/s}$

Worked Example 9 (Tricky multi-step): A printer prints 8 pages per minute. How many pages does it print in 2.5 hours?

First convert hours to minutes: $2.5 \text{ hr} \times 60 \text{ min/hr} = 150 \text{ min}$

Then apply the rate: $8 \text{ pages/min} \times 150 \text{ min} = 1{,}200 \text{ pages}$

Or in one chain:

$8 \frac{\text{pages}}{\text{min}} \times 60 \frac{\text{min}}{\text{hr}} \times 2.5 \text{ hr} = 1{,}200 \text{ pages}$

The dimensional analysis approach keeps you from accidentally multiplying when you should divide (or vice versa). Write out the units every time and make sure they cancel properly.

Applying Proportional Reasoning in Science and Social Science Contexts

The SAT frequently wraps these concepts in real-world scenarios from natural and social sciences. The math is the same; the context just requires careful reading.

Worked Example 10: In a certain solution, the ratio of acid to water by volume is 3 to 17. If a chemist needs to prepare 60 liters of this solution, how many liters of acid are needed?

The total parts are $3 + 17 = 20$. Acid makes up $\frac{3}{20}$ of the solution.

$\frac{3}{20} \times 60 = 9 \text{ liters of acid}$

A common trap here: using $\frac{3}{17}$ instead of $\frac{3}{20}$. The ratio 3:17 is acid to water, not acid to total. Read carefully whether the problem gives a part-to-part ratio or a part-to-whole ratio.

Worked Example 11: A survey found that 7 out of every 25 residents in a town commute by public transit. If the town has 14,000 residents, how many are expected to commute by public transit?

$\frac{7}{25} = \frac{x}{14{,}000}$

$x = \frac{7 \times 14{,}000}{25} = \frac{98{,}000}{25} = 3{,}920$

What to Watch For on Test Day

1. Match your units before setting up a proportion. If one side uses inches and the other uses feet, convert first. Mismatched units are the most common source of wrong answers.

2. Part-to-part vs. part-to-whole. When a problem says "the ratio of A to B is 3 to 5," the total is 8 parts. Don't use 3/5 when you need 3/8. Read the question carefully to see whether it asks for a part or the whole.

3. Write out units in every conversion step. Dimensional analysis isn't just a technique; it's your error-checking system. If the units don't cancel to give you what the question asks for, something is set up wrong.

4. Use the scale factor shortcut. When you can quickly see how one quantity scales (doubles, triples, increases by a factor of 2.5), apply that same scale factor to the proportional quantity. This is faster than cross-multiplying and reduces arithmetic errors.

5. Watch for derived unit traps. If a question involves something like "worker-hours" or "kilowatt-hours," remember these are products. A job requiring 200 worker-hours could mean 10 workers for 20 hours or 25 workers for 8 hours. The SAT tests whether you understand the structure of these units, not just whether you can compute.