Percentage: Concepts, Formulas, and Clear Examples

Percent means per hundred: $x\% = \frac{x}{100}$. MCQs chain successive percent changes, population/marked price, and reverse percent (find original from final).

Never add successive percents unless applied to the same base; use multipliers $(1\pm p)$.

Fraction ↔ percent table (e.g., $12.5\%=\frac{1}{8}$) saves time on campus tests.

Basics and reverse

Find part

$$\text{Part} = \frac{p}{100}\times \text{Whole}$$

Reverse whole

$$\text{Whole} = \frac{\text{Part}}{p/100}$$

Successive change

Net multiplier

$$\text{Final} = \text{Initial}\times\left(1+\frac{p}{100}\right)\left(1+\frac{q}{100}\right)$$

Net % (two changes)

$$p + q + \frac{pq}{100}$$

Example — +20% then −20%

Net = $1.2\times0.8=0.96$ → 4% decrease, not 0%.

Population / depreciation

Growth

$$P_n = P_0\left(1+\frac{r}{100}\right)^n$$

Depreciation

$$V_n = V_0\left(1-\frac{r}{100}\right)^n$$

Exam marks and failure

• If 32% failed and 272 passed, total = $\frac{272}{0.68}$.

Fraction shortcuts

• $25\%=\frac{1}{4}$, $33\frac{1}{3}\%=\frac{1}{3}$, $12.5\%=\frac{1}{8}$, $62.5\%=\frac{5}{8}$.

Comparison

• Compare by converting to same base or difference in percentage points vs relative percent.

Step-by-step method for exam questions

1. Identify whole (base) for each percent.
2. Convert % to multiplier.
3. Chain multipliers for successive changes.
4. For reverse, divide by multiplier.
5. Eliminate absurd bases (negative/zero).
6. Select option.

Formula cheat sheet

Frequently asked questions

Add 10% + 20%?

Only if both refer to same original; else multiply.

Percentage points vs percent?

30%→40% is +10 points, +33.3% relative.

Which base on discount?

Usually on marked price unless stated on CP.

Quick net zero?

+a% then −a% still net loss: $-(a/100)^2$ fraction.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.