6.2 Ratios, Rates, Proportions, and Unit Conversion
Key Takeaways
- A ratio compares two quantities; a rate compares quantities with different units such as miles per hour or dollars per pound.
- A unit rate rewrites a comparison with 1 in the denominator, making prices, speeds, and densities directly comparable.
- A proportion is valid only when the relationship is purely multiplicative; a fixed starting fee makes the relationship nonproportional.
- Unit conversion is safest when each factor equals 1 and the unwanted units cancel before the arithmetic is finished.
- GED ratio items are usually multi-step, so labeling units throughout is as important as the calculation.
Ratios and Rates on the GED
The GED Mathematical Reasoning targets include unit rates, scale factors, proportions, and unit conversions because they drive everyday decisions: comparing two prices, scaling a recipe, converting miles to feet, or reading an actual length off a drawing. The math is straightforward once the setup is clear and the units are labeled.
A ratio compares quantities of the same kind. A rate is a ratio of quantities with different units. A unit rate is a rate with a denominator of 1.
| Situation | Setup | Meaning |
|---|---|---|
| 12 red tiles, 8 blue tiles | 12:8 = 3:2 | 3 red for every 2 blue |
| $18 for 6 pounds | 18 / 6 = 3 | $3 per pound |
| 150 miles in 3 hours | 150 / 3 = 50 | 50 miles per hour |
| 1 inch represents 5 feet | 1 in : 60 in | Scale-drawing relationship |
Unit Rates
A unit rate turns an awkward comparison into a one-number answer.
Worked example: Store A sells 3 notebooks for $8.25. Store B sells 5 for $13.50. Which is cheaper per notebook?
Store A: 8.25 / 3 = $2.75 each. Store B: 13.50 / 5 = $2.70 each. Store B is cheaper by $0.05 per notebook. The GED trap is comparing the two totals; $13.50 is more money than $8.25, yet it is the better per-unit deal. Always ask, per what? Per pound, per hour, per square mile, and per serving are different units, and the requested answer choice must match the unit asked for.
Proportions
A proportion states that two ratios are equal. Solve by scaling or by cross multiplication.
Worked example: A recipe uses 2 cups of rice for 5 servings. How much rice for 12 servings?
Write cups/servings: 2/5 = x/12. Cross multiply: 5x = 24, so x = 4.8 cups. Reasonableness check: 12 servings is a bit more than double 5 servings, so the rice should be a bit more than double 2 cups, and 4.8 fits.
Use a proportion only when the relationship is strictly multiplicative. A taxi fare of a $4 starting fee plus $2 per mile is not proportional, because the fixed $4 means doubling the miles does not double the fare. A correct model is fare = 4 + 2(miles), not a single ratio. Spotting a fixed fee, flat rate, or starting amount is the clue to abandon a proportion.
Unit Conversion With Canceling
A conversion factor is a fraction equal to 1, such as 12 inches / 1 foot or 1 hour / 60 minutes. Arrange each factor so the unwanted unit cancels.
Worked example: A car travels 54 miles in 1.5 hours. Find the speed in feet per second.
First, 54 / 1.5 = 36 miles per hour. Then chain factors: 36 mi/hr x 5280 ft / 1 mi x 1 hr / 3600 s. Miles cancel, hours cancel, leaving feet per second. Compute 36 x 5280 / 3600 = 52.8 feet per second.
If a conversion sends the number the wrong way, stop. Converting miles to feet should produce a larger number (feet are smaller). Converting seconds to hours should produce a smaller number (hours are larger).
Scale Factors
Scale factors relate a drawing or model to the real object. If 0.75 inch represents 6 feet, then 1 inch represents 6 / 0.75 = 8 feet, and a 2.5-inch length represents 2.5 x 8 = 20 feet. Write the unit beside every number; if the units do not cancel down to the unit requested, fix the setup before you calculate.
Common U.S. and Metric Conversions to Know
The GED supplies a formula sheet, but it does not list every basic length, weight, and time conversion, so keep these ready.
| Category | Conversion |
|---|---|
| Length | 1 foot = 12 inches; 1 yard = 3 feet; 1 mile = 5,280 feet |
| Weight | 1 pound = 16 ounces; 1 ton = 2,000 pounds |
| Volume | 1 gallon = 4 quarts; 1 quart = 2 pints; 1 cup = 8 fluid ounces |
| Time | 1 hour = 60 minutes; 1 minute = 60 seconds |
| Metric | 1 kilometer = 1,000 meters; 1 meter = 100 centimeters; 1 kilogram = 1,000 grams |
Three-Step Proportion Setup
Many GED ratio items hide more than one step. A clean three-step habit prevents lost units. First, write the known ratio as a labeled fraction such as miles/gallons. Second, set it equal to the unknown ratio with the same labels in the same positions. Third, cross multiply and solve, then re-read the question to confirm you reported the requested quantity, not an intermediate value.
Worked example: A car uses 9 gallons to travel 261 miles. At that rate, how many gallons are needed to travel 435 miles?
Write gallons/miles consistently: 9/261 = x/435. Cross multiply: 261x = 9 x 435 = 3,915, so x = 15 gallons. The label check matters: had you written the second ratio as miles/gallons by accident, the cross product would solve for miles, not gallons, and the answer would be meaningless. Keeping identical labels top and bottom on both sides is the safeguard that turns a multi-step ratio item into routine arithmetic.
A store sells 3 cans of soup for $7.50, and another package sells 5 cans for $11.75. Which statement is correct?
A taxi charges a $4.00 flat fee plus $2.00 per mile. A rider is told a 3-mile trip costs $10. Why can you NOT find the cost of a 6-mile trip by simply doubling $10?
A recipe uses 2.5 cups of flour for 4 servings. Scaled proportionally, how much flour is needed for 10 servings?