Ratio & Proportion – MCAT Quantitative Aptitude Study Notes
Audience: Students preparing for the MCAT (or any competitive exam) who need a solid, derivation‑based understanding of ratios and proportions.
Goal:
- Master the formal definition of a ratio and a proportion.
- Derive the fundamental formulae.
- Learn to spot common pitfalls and “trick” questions.
- Apply the concepts to real‑world problems (chemistry, biology, physics, economics).
- Solve a hard, multi‑step practice problem.
1. Basic Definitions
| Term | Symbol | Explanation |
|---|---|---|
| Ratio | ( \frac{a}{b} ) or ( a:b ) | A comparison of two quantities of the same type. |
| Proportion | ( \frac{a}{b} = \frac{c}{d} ) | An equation stating that two ratios are equal. |
| Direct Proportion | ( y = kx ) | When one variable grows linearly with another. |
| Inverse (Reciprocal) Proportion | ( y = \frac{k}{x} ) | When the product of two variables is constant. |
Important:
- Ratios are ordered (2:3 ≠ 3:2).
- Proportions are unordered; swapping the two ratios keeps the equality true.
2. Cross‑Multiplication (Derivation)
Given a proportion
[ \frac{a}{b} = \frac{c}{d}, ]
multiplying both sides by (bd) (the product of the denominators) gives
[ ad = bc . ]
This is the cross‑product rule and is the cornerstone of solving proportion problems.
Derivation in one line
[ \frac{a}{b} = \frac{c}{d} ;\Longrightarrow; a\cdot d = b\cdot c ]because multiplying by (bd) eliminates the fractions.
3. Key Properties & Formulas
| Property | Formula | Notes |
|---|---|---|
| Scale factor in a proportion | ( k = \frac{a}{b} = \frac{c}{d} ) | Used in similar triangles, resizing, and in many MCAT quantitative stems. |
| Compound Ratio | ( r_1 \times r_2 \times \dotsb \times r_n ) | For converting units or successive scaling. |
| Direct proportion | ( y = kx ) | If (k) is known, (x) is found by dividing. |
| Inverse proportion | ( y = \frac{k}{x} ) | If (k) is known, (x) is found by dividing. |
| Mixed proportion | ( \frac{a}{b} = \frac{c}{d} = \frac{e}{f} ) | All pairs share the same ratio. |
| Proportional change | ( \frac{\Delta y}{y} = \frac{\Delta x}{x} ) | Useful for percent changes (percent change in a quantity equals percent change in its proportional partner). |
4. Applications in MCAT‑Style Problems
| Context | Example Problem | Quick Strategy |
|---|---|---|
| Similar Triangles / Geometry | If (\triangle ABC \sim \triangle DEF) and (AB = 3), (DE = 6), find (EF). | Use ( \frac{AB}{DE} = \frac{BC}{EF} ). |
| Scale & Resizing | A map shows a city as (1\text{ in} : 50{,}000\text{ in}). A street on the map is (2.5\text{ in}). What is the real length? | Multiply (2.5 \times 50{,}000). |
| Time & Speed | A runner covers 10 km in 45 min, how long for 25 km? | Direct proportion: ( \frac{t_1}{d_1} = \frac{t_2}{d_2} ). |
| Percent & Mixtures | 40 % saline solution + 30 % saline solution → ? | Convert to proportions, then find weighted average. |
| Biological Ratios | Protein A to Protein B ratio in a cell is 5:3. If total protein = 400 mg, find each. | Set up ( \frac{A}{B} = \frac{5}{3} ) and ( A + B = 400 ). |
| Pharmacology | A drug’s concentration halves every 8 h. Find concentration after 24 h. | Exponential decay: (C_t = C_0 (1/2)^{t/8}). |
5. Common Pitfalls & Tricky Points
| Pitfall | Example & Fix |
|---|---|
| Confusing ratio order | 2:3 ≠ 3:2. Always keep the order when substituting. |
| Assuming “direct proportion” means “equal” | Direct proportion means linearly related, not necessarily equal numerically. |
| **Mix |