SAT Ratios, rates, proportions, units
Scale quantities and convert between units.
How to score it
- Set up the proportion so units cancel diagonally — track the labels, not just the numbers.
- Chain conversion factors until you reach the target unit.
- For density, mass = density × volume; keep units consistent.
Common traps
- Flips the conversion factor (multiplies when you should divide).
- Drops a unit step in a multi-step conversion.
- Mismatched units (feet vs. inches) inside one calculation.
The 5 question types, with real examples
Rate conversion across units
“[Rate in unit A]. What is the rate in [unit B]?”
A cyclist rides 72 miles in 4 hours. At this rate, how many miles does the cyclist ride in 11 hours?
- A
- B
- C✓
- D
The cyclist rides 72 miles in 4 hours, so the rate is 72/4 = 18 miles per hour. In 11 hours, the cyclist rides 18 × 11 = 198 miles.
Direct proportion / similar figures
“How [tall / long] is the other [object]?”
A car travels 240 miles in 4 hours. At this rate, how many miles will the car travel in 7 hours?
- A✓
- B
- C
- D
The car travels 240 miles in 4 hours, so its rate is 240/4 = 60 miles per hour. In 7 hours, the car will travel 60 × 7 = 420 miles.
Density / mass / volume
“To the nearest [unit], what is the [mass / volume]?”
Steel has a density of 7.85 grams per cubic centimeter. A cube of steel has a mass of 3925 grams. To the nearest centimeter, what is the edge length of the cube?
- A✓
- B
- C
- D
Using mass = density × volume, the volume is 3925 ÷ 7.85 = 500 cubic centimeters. For a cube with edge length s, volume = , so = 500 and s = ∛500 ≈ 7.94, which rounds to 8 centimeters.
Unit conversion via given equivalence
“A distance of [n] furlongs is equivalent to how many feet? (1 furlong = 660 ft)”
A distance of 8 miles is equivalent to how many feet? 1 mile = 5280 feet
- A
- B✓
- C
- D
Since 1 mile equals 5280 feet, 8 miles equals 8 × 5280 = 42240 feet.
Reading a complex table with rate / growth-factor
“[TABLE: species -> growth factor] [context: age = diameter x growth factor]. [question]”
The table below shows the growth factor for three tree species in a forest. The age of a tree, in years, can be estimated by multiplying the diameter of the tree, in inches, by the growth factor for that species. | Species | Growth Factor | |---------|---------------| | Oak | 4.2 | | Maple | 5.8 | | Pine | 6.5 |
A maple tree has a diameter of 18 inches. What is the estimated age, in years, of this tree?
- A
- B✓
- C
- D
The age is found by multiplying the diameter by the growth factor for maple trees. Age = 18 × 5.8 = 104.4 years.
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