6.2 Ratios, Rates, Proportions, and Unit Conversion
Key Takeaways
- A ratio compares two quantities, while 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 easier to compare.
- A proportion works when two ratios describe the same multiplicative relationship, but it fails when a fixed fee or nonproportional pattern is present.
- Unit conversion is safest when each conversion factor equals 1 and unwanted units cancel before arithmetic is finished.
- GED ratio questions often combine multiple steps, so labeling units is as important as calculating the numbers.
Ratios and Rates on the GED
The GED assessment targets include unit rates, scale factors, proportions, and unit conversions because these skills show up in everyday decisions. You may compare two prices, scale a recipe, convert miles to feet, or use a drawing to find an actual length. The math is not hard when the setup is clear.
A ratio compares quantities. A rate is a ratio with different units. A unit rate has a denominator of 1.
| Situation | Setup | Meaning |
|---|---|---|
| 12 red tiles and 8 blue tiles | 12:8 or 3:2 | For every 3 red tiles, there are 2 blue tiles |
| $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 | 1:60 if both are inches | Scale drawing relationship |
Unit Rates
A unit rate makes comparison simple.
Worked example: Store A sells 3 notebooks for $8.25. Store B sells 5 notebooks for $13.50. Which is cheaper per notebook?
Store A: 8.25 / 3 = 2.75 dollars per notebook. Store B: 13.50 / 5 = 2.70 dollars per notebook. Store B is cheaper by $0.05 per notebook.
The GED trap is comparing total prices instead of equal units. Five notebooks cost more in total, but less per notebook. Always ask, per what? Per pound, per hour, per square mile, and per serving are different units, and the answer choice must match the unit requested.
Proportions
A proportion says two ratios are equal. You can solve by scaling or by cross multiplication.
Worked example: A recipe uses 2 cups of rice for 5 servings. How much rice is needed for 12 servings?
Set up cups/servings: 2/5 = x/12. Cross multiply: 5x = 24, so x = 4.8 cups. A reasonableness check helps: 12 servings is a little more than twice 5 servings, so the rice should be a little more than twice 2 cups. 4.8 cups fits.
Use proportions only when the relationship stays proportional. A taxi fare with a $4 starting fee plus $2 per mile is not proportional because the fixed fee changes the ratio.
Unit Conversion With Canceling
A conversion factor is a fraction equal to 1, such as 12 inches/1 foot or 1 hour/60 minutes. Place the factor so the unit you do not want cancels.
Worked example: A car travels 54 miles in 1.5 hours. What is the speed in feet per second?
First find miles per hour: 54 / 1.5 = 36 mph.
Now convert: 36 miles/hour x 5280 feet/1 mile x 1 hour/3600 seconds. Miles cancel, hours cancel, and the remaining units are feet/second. Compute 36 x 5280 / 3600 = 52.8 feet per second.
If a conversion makes the number move in the wrong direction, pause. Converting miles to feet should create a larger number because feet are smaller units. Converting seconds to hours should create a smaller number because hours are larger units.
Scale Factors
Scale factors compare a drawing or model to the actual object. If 0.75 inch on a map represents 6 feet, then 1 inch represents 6 / 0.75 = 8 feet. A 2.5-inch drawing length represents 2.5 x 8 = 20 feet.
For GED questions, write the units next to every number. If the units do not cancel to the unit asked for, the setup needs to be fixed before you calculate.
A store sells 3 cans of soup for $7.50 and another package sells 5 cans for $11.75. Which statement is correct?
A recipe uses 2.5 cups of flour for 4 servings. If the recipe is scaled proportionally, how many cups of flour are needed for 10 servings?