Percentages – Quick‑Start Study Notes
(MCAT – Quantitative Aptitude)
1. Simple Explanation
Percent is a way to express a part of a whole as a fraction of 100.
- “x % of y” = (x/100) × y
- “y is x % of z” → x = (y/z) × 100
Percentages let you compare quantities on a common scale, calculate changes over time, or convert between fractions, decimals, and percentages.
2. Key Points & Formulas
| Problem type | What you’re asked to find | Formula | Quick Check |
|---|---|---|---|
| Percent of a number | x % of N | result = (x/100) * N |
Multiply by x/100 (e.g., 25 % → 0.25) |
| Number from a percent | Find N when percent and part are known | N = (part / (percent/100)) |
Divide part by fraction of percent |
| Percent change | Change from old to new |
percent_change = ((new-old)/old)*100 |
Positive = increase, negative = decrease |
| Rate of change (compound) | e.g., growth/decay over time | final = initial * (1 + r)^t (where r is the rate per period) |
For simple change use linear formula above |
| Percent composition | Percentage of a component in a mixture | percent = (component / total) * 100 |
Often used in chemistry, finance |
| Inverse problems | Find the original value given a percent after change | original = new / (1 + r) |
Use reciprocal of growth factor |
Common “cancelling” tricks
- Percent of a percent:
(a% of b%) = (a/100) * (b/100) * 100 = (a*b)/100 - Percent of a percent of a number:
(a% of b% of N) = (a*b/10000) * N
3. Worked Example (Step‑by‑Step)
Problem
A student’s score on a practice test increased from 72 % to 84 %.
- What is the percentage increase?
- If the final score represents 84 % of a 100‑point test, how many points did the student earn?
- If the original score represented 72 % of the same 100‑point test, how many points did the student earn before the increase?
1. Percentage Increase
[ \text{percent_increase} = \frac{84-72}{72}\times 100 = \frac{12}{72}\times 100 = 0.1667 \times 100 = 16.67% ]
Answer: 16.67 % increase.
2. Final Score in Points
[ \text{points_final} = \frac{84}{100}\times 100 = 84 \text{ points} ]
3. Original Score in Points
[ \text{points_original} = \frac{72}{100}\times 100 = 72 \text{ points} ]
Summary of the Example
| Quantity | Value |
|---|---|
| Initial percent | 72 % |
| Final percent | 84 % |
| Percent increase | 16.67 % |
| Final score | 84 pts |
| Original score | 72 pts |
4. Quick Revision Summary
- Percent = Part / Whole × 100
- Part = Percent × Whole / 100
- Percent change = ((new – old) / old) × 100
- For compound growth/decay:
final = initial * (1 + r)^t - Percent of a percent: multiply fractions then convert back to percent.
- Check units: Percent is dimensionless; multiply or divide by 100 as needed.
Mini‑Practice Questions (Answer in the margin)
- What is 30 % of 250?
- A medicine dose is 5 % of body weight. What is the dose for a 68‑kg patient?
- A population grew from 1 200 to 1 350 in one year. What was the percent growth?
- If a 500‑point test was scored at 76 %, how many points were earned?
(Answers: 75, 3.4 kg, 12.5 %, 380 points)
Use these notes to quickly refresh the core concepts and practice calculations before the MCAT. Happy studying!