Overview
Ratios and Proportions is a foundational topic within the Problem-Solving and Data Analysis domain, appearing in both the easier and harder modules of the digital SAT Math section. Questions test the ability to write and simplify ratios, set up and solve proportions, apply unit rates, perform dimensional analysis, and work with direct and inverse variation in real-world contexts. All Problem-Solving and Data Analysis questions are calculator-active (no calculator-off section), and these questions typically appear at the beginning of each module due to their lower difficulty.
Key Points
1. Ratios: Part-to-Part vs. Part-to-Whole
A ratio compares two quantities. When the ratio of A to B is written as A : B, always determine whether you need a part-to-part or part-to-whole comparison before computing.
| Ratio | Total Parts | A as fraction of whole | B as fraction of whole |
|---|---|---|---|
| 3 : 5 | 8 | 3/8 | 5/8 |
| 2 : 7 | 9 | 2/9 | 7/9 |
Key rule: If A : B = 3 : 5, then A = 3/8 of the total, NOT 3/5.
2. Setting Up and Solving Proportions
A proportion sets two ratios equal: a/b = c/d
Cross-multiply to solve: ad = bc
Always place the same type of quantity in the same position on both sides.
- Correct: miles/hour = miles/hour
- Wrong: miles/hour = hour/miles
Example: If 4 workers complete a job in 6 days, how many days for 3 workers?
- Set up: 4/6 = 3/x → x = 18/4 = 4.5 days (for direct proportion, not inverse — check context)
3. Unit Rates
A unit rate expresses a ratio with a denominator of 1.
- 150 miles in 3 hours → 50 miles per hour
- 2.50 per pound**
Strategy: Always convert to the unit rate first, then multiply to find totals.
4. Dimensional Analysis (Unit Conversion)
Multiply by conversion fractions equal to 1, arranging each so that unwanted units cancel.
Example: Convert 72 km/h to m/s
Write out every unit — if they do not cancel correctly, the setup is wrong.
5. Scale Factors
In scale problems (maps, models, blueprints), the same scale factor k applies to all linear measurements.
- If scale is 1 cm : 50 km and a map distance is 3.5 cm → actual = 3.5 × 50 = 175 km
- When quantities are proportional, if one doubles, the other doubles — apply the scale factor directly rather than cross-multiplying to save time.
6. Direct and Inverse Variation
| Type | Formula | Behavior |
|---|---|---|
| Direct variation | y = kx | As x increases, y increases proportionally |
| Inverse variation | y = k/x | As x increases, y decreases |
For direct variation, the ratio y/x = k is constant. For inverse variation, the product xy = k is constant.
SAT clue words: “varies directly” → y = kx; “varies inversely” → y = k/x; “is proportional to” → y = kx.
Pitfalls and Common Mistakes
Mistake 1: Confusing part-to-part with part-to-whole ratios. A problem states the ratio of boys to girls is 3:5 and asks what fraction of the class are boys. Students write 3/5 instead of 3/8. Fix: Always add ratio parts to find the total (3+5=8) before computing the fraction of the whole.
Mistake 2: Setting up the proportion in mismatched order. Students write miles/hours = hours/miles on opposite sides, then cross-multiply incorrectly. Fix: Label each fraction with units before cross-multiplying. If units do not match position, flip one fraction.
Mistake 3: Applying direct-variation thinking to an inverse-variation scenario. If more workers are hired, students assume the job takes proportionally more time (direct), when it actually takes less time (inverse). Fix: Identify whether the relationship is direct (y = kx) or inverse (y = k/x) from context before setting up the equation.
Mistake 4: Forgetting to convert units before or after solving. Students solve a proportion in different units and report the result without converting. Fix: Check that all quantities are in the same units before setting up the proportion; convert first if needed.
Related Entries
- Percentages_Percent_Change
- Unit_Conversion
- Scatterplots_Regression
- Statistical_Measures
- Data_Distribution_Graphs
Quick Reference Card
| Concept | Formula / Rule |
|---|---|
| Proportion | a/b = c/d → ad = bc |
| Part-to-whole from ratio A:B | A/(A+B) |
| Direct variation | y = kx (ratio y/x = k is constant) |
| Inverse variation | y = k/x (product xy = k is constant) |
| Unit rate | Ratio with denominator = 1 |
| Scale factor | Multiply all linear measures by the same k |
| Dimensional analysis | Multiply by (wanted unit / unwanted unit) until target unit remains |