Ratios, rates, proportional relationships, and unit conversion fall under the Problem-Solving and Data Analysis category on the Digital SAT Math section (44 questions, 70 minutes). Expect roughly 3–5 questions on these concepts across the two math modules. Once you internalize the setup for each problem type, they become highly predictable.

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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 also appear 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 a constant, then $x$ and $y$ are proportional. The Digital SAT tests this by giving you a scenario and asking you to find an unknown value using the fact that the ratio between quantities doesn't change.

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

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

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

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

Cross-multiply: $84 \times 7 = 3x \implies 588 = 3x \implies x = 196$

Example 2 — Context with extra information: A biologist estimates the ratio of bluegill to bass in a lake is 5 to 2. A sample contains 120 bluegill. How many bass are estimated to be in the sample?

$\frac{5}{2} = \frac{120}{x} \implies 5x = 240 \implies x = 48$

The total number of fish in the lake is irrelevant. The Digital SAT often includes extra context to see whether you can identify what actually matters.

Scale Factor and Scaling

When two quantities are proportional, multiplying one by a scale factor multiplies the other by that same factor. If $y = kx$ and you multiply $x$ by 3, the new output is $k(3x) = 3y$ .

Example 3 — Scale factor shortcut: A factory uses 12 gallons of paint to coat 80 square feet. How many gallons are needed for 200 square feet at the same rate?

Scale factor for area: $\frac{200}{80} = 2.5$

Apply the same factor to paint: $12 \times 2.5 = 30 \text{ gallons}$

This is often faster than setting up a full proportion.

Example 4 — Scale drawings: A scale drawing uses a scale of 1 inch to 8 feet. A wall measures 3.5 inches in the drawing. What is the actual length of the wall in feet?

$3.5 \text{ in} \times \frac{8 \text{ ft}}{1 \text{ in}} = 28 \text{ ft}$

Derived Units

Derived units are built from combinations of other units through multiplication or division.

Quotient-based: population per square kilometer $\left(\frac{\text{people}}{\text{km}^2}\right)$ , miles per hour, cost per pound
Product-based: kilowatt-hours (kilowatts × hours = energy), worker-hours (workers × hours = total labor)

Example 5 — Quotient-based: A city has a population of 450,000 and an area of 180 km². What is the population density in people per km²?

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

Example 6 — Product-based: A data center uses 50 kilowatts continuously for 24 hours. How many kilowatt-hours does it consume?

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

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

Unit Conversion: One-Step and Multi-Step

Dimensional analysis is the standard technique: multiply by conversion fractions equal to 1, arranged so unwanted units cancel.

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

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

Example 8 — Multi-step with compound units: A car travels at 90 km/hr. What is its speed in meters per second? (1 km = 1,000 m; 1 hr = 3,600 s)

$\frac{90 \text{ km}}{1 \text{ hr}} \times \frac{1{,}000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3{,}600 \text{ s}} = \frac{90{,}000}{3{,}600} \text{ m/s} = 25 \text{ m/s}$

Example 9 — Chained conversion: A printer prints 8 pages per minute. How many pages does it print in 2.5 hours?

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

Write out units at every step. If they don't cancel to give you what the question asks for, something is set up wrong.

Proportional Reasoning in Context

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

Example 10 — Part-to-part vs. part-to-whole: The ratio of acid to water in a solution is 3 to 17. A chemist needs 60 liters of this solution. How many liters of acid are needed?

Total parts: $3 + 17 = 20$ . Acid is $\frac{3}{20}$ of the total.

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

Common trap: Using $\frac{3}{17}$ instead of $\frac{3}{20}$ . The ratio 3:17 is acid to water, not acid to total. Check whether the problem gives a part-to-part or part-to-whole ratio.

Example 11 — Survey scaling: 7 out of every 25 residents commute by public transit. The town has 14,000 residents. How many are expected to commute by public transit?

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

What to Watch For on Test Day

Match 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 errors.
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 $\frac{3}{5}$ when you need $\frac{3}{8}$ .
Write out units in every conversion step. Dimensional analysis is your error-checking system. If units don't cancel correctly, the setup is wrong.
Use the scale factor shortcut. When you can quickly see how one quantity scales, apply that same factor to the proportional quantity. This is faster than cross-multiplying and reduces arithmetic errors.
Understand product-based derived units. A job requiring 200 worker-hours could mean 10 workers for 20 hours or 25 workers for 8 hours. The Digital SAT tests whether you understand the structure of these units, not just whether you can compute with them.