Ratios, rates, proportions and units
Problem-Solving and Data Analysis is about 15% of SAT Math (5–7 of 44 questions), and ratios, rates and units are one of its most common topics. Questions are usually short real-life stories with a rate, a scale or a unit change.
Ratios and proportions
A ratio compares two amounts. If the ratio of boys to girls is 4 : 3, then for every 4 boys there are 3 girls. This is a part-to-part ratio; the whole group is 4 + 3 = 7 parts. A proportion says that two ratios are equal, for example 4/3 = 36/x. Cross-multiply to solve: 4x = 108, so x = 27. Always write the same quantity on top in both fractions (boys on top, girls on the bottom) so the ratio is not turned upside down.
| Boys | Girls | Whole group |
|---|---|---|
| 4 | 3 | 7 |
| 8 | 6 | 14 |
| 36 | 27 | 63 |
Rates and unit rates
A rate compares two different kinds of things: miles and gallons, dollars and kilograms, pages and minutes. To find a unit rate, divide so that the bottom number becomes 1. If a car uses 8 gallons for 280 miles, the unit rate is 280 ÷ 8 = 35 miles per gallon. Once you have the unit rate, multiplication does the rest: in 12 gallons the car goes 12 × 35 = 420 miles.
Unit conversion: multiply by fractions equal to 1
To change units, multiply by a conversion factor written so the old unit cancels. 1 hour = 60 minutes gives two factors: 60 min / 1 h and 1 h / 60 min. Choose the one with the unit you want to remove on the bottom. For a chain of conversions, write all factors in one line and cancel units as you go. The SAT usually gives the conversion facts you need (such as 1 inch = 2.54 cm), so you do not have to memorize them.
Squared and cubic units
When units are squared or cubed, the conversion factor must be squared or cubed as well. Since 1 m = 100 cm, one square meter is 100 × 100 = 10,000 square centimeters, and one cubic meter is 100 × 100 × 100 = 1,000,000 cubic centimeters. Forgetting this is the most common unit mistake on the test.
| Length | Area | Volume |
|---|---|---|
| 1 m = 100 cm | 1 m² = 10,000 cm² | 1 m³ = 1,000,000 cm³ |
| 1 km = 1,000 m | 1 km² = 1,000,000 m² | 1 km³ = 1,000,000,000 m³ |
| 1 ft = 12 in | 1 ft² = 144 in² | 1 ft³ = 1,728 in³ |
Proportional relationships and their graphs
A relationship is proportional when y = kx: the graph is a straight line through the origin (0, 0), and k is the unit rate. If a graph or table is not proportional, the ratio y/x changes, or the line does not pass through the origin. To read a rate from a graph, pick one clear point and divide y by x.
- Unit rate: 18 ÷ 45 = 0.4 kilogram of flour per loaf.
- For 70 loaves: 70 × 0.4 = 28 kilograms.
- Check with a proportion: 18/45 = 28/70, and both fractions equal 0.4 ✓
- Write the speed with conversion factors: 90 km/h × (1,000 m / 1 km) × (1 h / 3,600 s).
- The kilometers and the hours cancel.
- 90 × 1,000 ÷ 3,600 = 25.
- 15
- 150
- 225
- 22,500
- Side length = √2.25 = 1.5 meters.
- 1.5 m × 100 = 150 cm.
- Another way: 2.25 m² = 22,500 cm², and √22,500 = 150.
- The line passes through the origin, so the relationship is proportional.
- Take a clear point: (4, 30).
- Rate = 30 ÷ 4 = 7.5 pages per minute.
- Turning the ratio upside downLabel each number (boys : girls = 4 : 3) and keep the same order in both sides of the proportion.
- Confusing a part with the wholeA ratio of 4 : 3 means 7 equal parts in all. Boys are 4/7 of the group, not 4/3.
- Converting squared or cubed units with a single factorSquare or cube the conversion factor: 1 m² = 100² cm², 1 m³ = 100³ cm³.
- Units that do not cancelWrite the units in every fraction. If the unit you want to remove is not on the opposite side, flip the conversion factor.
- Mixing minutes and hoursChange all times to the same unit before dividing. A rate “per minute” and a time given in hours need one conversion.
- Rounding too earlyKeep the exact fraction or full calculator value until the last step; round only the final answer.
Desmos can solve a proportion for you. Type the proportion exactly as it is written, using x for the unknown, and Desmos draws a vertical line at the solution. Click where the line meets the x-axis to read the value. For a unit conversion chain, just type the whole product of fractions and Desmos shows the result on the right.
- Type 18/45 = x/70
- A vertical line appears at x = 28
- Type 90 × 1000 ÷ 3600 on a new line to see 25
Use Desmos to check long decimal calculations. Setting up the proportion or the conversion factors is the real skill, and Desmos cannot do that for you.
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