3.1 Unit Rates, Proportions & Scale Factor

Why Unit Rates, Proportions & Scale Factor Matter on the GED

Official assessment target Q.3 — "Calculate and use ratios, percents and scale factors" — sits inside Quantitative Problem Solving, the content area that makes up roughly 45% of the GED Mathematical Reasoning test. Q.3 itself has four numbered indicators (Q.3.a through Q.3.d). This section teaches the first three — Q.3.a (compute unit rates), Q.3.b (use scale factors and convert between actual and scale drawings), and Q.3.c (solve multistep real-world problems using ratios or proportions, including unit conversions). Percent word problems (Q.3.d — tax, markup, commission, simple interest) get their own section next, because percent deserves its own set of formulas.

Proportional reasoning is one of the most-recycled skills on the entire exam. The GED Testing Service's own Assessment Guide lists unit rate again inside the Algebraic content area — indicator A.5.c, "interpret unit rate as the slope in a proportional relationship" — meaning the exact skill you build here reappears when you graph lines later in this study guide. Ratio and proportion problems also show up disguised inside geometry (scale drawings of floor plans, blueprints, and maps) and inside almost every percent problem in the next section. If you can set up a proportion correctly and solve it without a sign or unit error, you have unlocked a skill that pays off across both halves of the test, not just these four indicators.

Core Terms and Rules

• A ratio compares two quantities by division, written as $a:b$ or $\frac{a}{b}$. Example: a classroom with 8 boys and 12 girls has a boy-to-girl ratio of 8:12, which simplifies to 2:3.
• A rate is a ratio that compares two quantities measured in different units, such as dollars per pound or miles per hour.
• A unit rate is a rate simplified so the second quantity equals exactly 1 — "miles per one hour," "dollars per one item." You find it by dividing the first quantity by the second. The GED Assessment Guide names four official example categories for Q.3.a: unit pricing, constant speed, persons per square mile (population density), and BTUs per cubic foot (British Thermal Units, a heat-energy measurement used in appliance and fuel problems).
• A proportion is an equation stating that two ratios are equal, such as $\frac{3}{4} = \frac{x}{20}$. You solve a proportion by cross-multiplying: multiply diagonally, then divide to isolate the unknown.
• A scale factor is the ratio between a scaled measurement (on a map, blueprint, or model) and the actual measurement it represents, such as "1 inch = 25 miles" or "1 cm = 3 meters."

Reference Table: Official Q.3.a–c Indicators

Worked Example 1: Unit Rate + the Formula Sheet's "Total Cost" Formula

A bulk-foods bin sells rice at $2.88 for 4 pounds. What is the price per pound, and how much would 9 pounds cost?

1. Find the unit rate: $2.88 \div 4 = $0.72$ per pound.
2. Apply the GED formula sheet's total cost formula — total cost = (number of units) × (price per unit) — to scale up: $9 \times $0.72 = $6.48$.

This two-step pattern (find the unit rate, then multiply by the new quantity) solves nearly every "at this rate" question on the test, whether the units are pounds, gallons, or hours worked.

Worked Example 2: Cross-Multiplying a Proportion (Map Scale)

A road map uses a scale where 2 inches represents 15 actual miles. How many inches on the map represent 60 miles?

Set up the proportion with matching units in matching positions (map/actual = map/actual): $\frac{2}{15} = \frac{x}{60}$. Cross-multiply: $15x = 2 \times 60 = 120$, so $x = 8$ inches.

Worked Example 3: Scale Factor on a Blueprint (Q.3.b)

An architect's blueprint uses a scale of 1 cm = 3 meters. On the blueprint, a wall measures 5 cm. What is the wall's actual length?

Multiply by the scale factor: $5 \text{ cm} \times 3 \text{ m/cm} = 15$ meters. Going the other direction — converting an actual measurement back to a scale drawing — you divide instead of multiply, which is the reverse operation test-writers frequently swap to check whether you understand which direction the scale factor runs.

Worked Example 4: Multistep Proportion with a Unit Conversion (Q.3.c)

A cyclist rides at a constant rate of 15 miles per hour. How many feet does the cyclist travel in one minute? (1 mile = 5,280 feet)

This is a genuine two-conversion problem: first convert miles to feet (15 mph = $15 \times 5{,}280 = 79{,}200$ feet per hour), then convert hours to minutes (79,200 feet ÷ 60 minutes = 1,320 feet per minute). Skipping either conversion — say, dividing by 60 but forgetting the feet conversion — is the single most common wrong-answer trap on Q.3.c items, because the "almost right" answer (264, which is just 15,840 ÷ 60) is always sitting in the option list.

Common Traps

On test day, these items land in the calculator-allowed part of the test (items 6–46 use the onscreen TI-30XS Multiview), so you can check cross-multiplication arithmetic — but the calculator will not catch a setup error, which is why the unit-underlining habit above matters more than raw computation speed.

Practice Questions

Try these before checking the takeaways below.