Percentages

• How percentages model increase, decrease, discounts, and successive changes.
• To compute percent of quantity, what percent, and base when two are known.
• To avoid successive percent traps (two 10% discounts ≠ 20% off).
• To apply percentage reasoning in profit-loss, population, and exam data questions.

Level 1 — Foundations

Verbal: Percent means "per hundred". x% of Q = (x/100) × Q.

Core formulas:

Quick fractions: 12.5%=1/8; 20%=1/5; 33⅓%=1/3 — speed mental math.

Base identification: "30% of what is 60?" → base = 60/0.30.

Level 2 — Exam depth

Successive discount: 20% then 10% off → 0.8×0.9=0.72 → 28% total discount, not 30%.

Percent point vs percent: Rate rises 4% to 5% → 1 percentage point increase, (5−4)/4=25% relative increase.

Population/compound style: +10% per year two years → 1.1² = 1.21 → 21% total growth.

Reverse profit: SP and loss% given → CP = SP/(1 − loss%).

Exam estimation: Round to nice numbers when MCQ options far apart.

Successive percentage change

Price increased 25%, then decreased 20%.
Net multiplier = 1.25 × 0.80 = 1.00 → **back to original** (not net +5% or −5%).
Always multiply factors, never add percents blindly.

Find whole from part percent

35% of a number is 140. Number = 140 × (100/35) = **400**.
Verify: 0.35 × 400 = 140 ✓.

• Express 3/8 as percent.
• 80 increased by 15% — result?
• Two 10% discounts equivalent to one discount of?
• Stretch: Population 50,000 grows 8% then falls 8% — final?

Revision tip: Revisit adjacent topics in Quantitative Reasoning before mixed practice on Percentages.

Open the Practice tab for graded questions on Percentages.

Replace successive percent changes with multipliers (increase 20% → ×1.2) before any addition temptation. For reverse problems (final given, find original), divide by the net multiplier, do not subtract percents. Memorise common fraction–percent swaps to save thirty seconds per item. Label what the base is in every calculation — circle it in the question stem.

Link percentages to ratio questions: a part-to-whole ratio converts directly to a percent of the whole. In profit and loss, CP is the base for markup; successive discount problems share the same multiplier logic as population growth. When a question mixes rupees and percent, convert to either form early — do not leave hybrid expressions until the final step. Board-style DI often embeds percent change inside table cells; treat each column as an independent series before comparing across products.

Keep a dedicated notebook spread for this topic: one page for methods, one for worked mistakes, and one for mixed drill from the Practice tab. Review weekly by explaining the core idea aloud in under sixty seconds without notes.