Percentage

Module 07 — Lesson 0007 · In Depth · 35 min

Percent = per hundred — a fraction of 100. Every percentage calculation is part/whole × 100 with a multiplier: increase → ×(1+X/100), decrease → ×(1−X/100). Master conversions and the multiplier; profit, discount, and successive changes are just applications.

1. The Fundamental Conversions

Direction	Operation	Example
Fraction → %	Multiply by 100	`1/2×100=50%`, `3/4×100=75%`, `0.15×100=15%`
% → Fraction/Decimal	Divide by 100	`20%=20/100=1/5=0.2`, `5%=1/20=0.05`, `150%=3/2=1.5`

Key fractions to memorize: 1/2=50%, 1/3≈33.33%, 1/4=25%, 1/5=20%, 1/6≈16.67%, 1/8=12.5%, 1/9≈11.11%, 1/11≈9.09%, 1/20=5%. These turn "20% of 150" into "1/5 of 150 =30" instantly.

2. Basic Calculation Formulas

Type	Question	Formula	Example
A	X% of Y	`(X/100)×Y`	20% of 150 = 0.20×150=30
B	X is what % of Y?	`(X/Y)×100`	30 is what % of 50? 30/50×100=60%
C	X% of a number is Y → find number	`Number = Y/(X/100)`	10% is 30 → 30/0.10=300

All three are part = percent × whole rearranged. If you know any two, you find the third.

3. Percentage Increase and Decrease

Finding % Change

Initial → Final:

% Change = [(Final − Initial) / Initial] × 100

Final > Initial → % increase
Final < Initial → % decrease

Denominator is always initial — the starting point. 100→120 is 20% increase, but 120→100 is 16.67% decrease (same absolute change, different base).

Applying Change via Multipliers

Increase by X%: New = Old × (1 + X/100) — e.g., +20% → ×1.20 (100→120)
Decrease by X%: New = Old × (1 − X/100) — e.g., −20% → ×0.80 (100→80)

Why multipliers matter: They chain. +20% then −10% is ×1.20×0.90=×1.08 → net +8%. Adding percentages (20−10=10%) would be wrong because the second change acts on the new base.

4. Successive Percentage Changes

Change by X%, then by Y% (use signed values: + for increase, − for decrease):

Net Change % = X + Y + (XY / 100)

Worked: Increase 20%, then decrease 10% → +20 −10 + (20×−10/100)=10−2=8% increase. Check: 100→120→108 ✓.
Two increases 10%+10%: 10+10+1=21% (not 20%). Two decreases 10%+10%: −10−10+1=−19% (not −20%). The XY/100 corrects the overlap.

5. Product Constancy — Price × Consumption = Expenditure

If expenditure stays constant and price changes by X%:

Price ↑ X% → consumption must ↓ by [X/(100+X)]×100
Price ↓ X% → consumption must ↑ by [X/(100−X)]×100

Derivation: P×C = k (constant). If P becomes P(1+X/100), C must become C/(1+X/100) = C×100/(100+X) → reduction = 1 − 100/(100+X) = X/(100+X).
Worked: Price ↑ 25% → consumption must fall by 25/125×100=20%. Check: 100×10=1000, new price 125, new consumption 8 → 125×8=1000 ✓. If price ↓ 20% → consumption must rise 20/80×100=25%.

6. Profit, Loss, and Discount Basics

Cost Price (CP): 100% base
Selling Price: SP = CP × (1 ± P/100) — profit → +, loss → −. 20% profit → SP=120% of CP.
Marked Price (MP): tag value
Discount: always on MP: SP = MP × (1 − D/100)
Profit/Loss always on CP (unless specified "on SP")

Full treatment in Module 14.

7. Mixture and Population Problems

Mixtures: weighted average — Total solute = V₁×%₁ + V₂×%₂, final % = total solute / total volume.
Population / depreciation: compound growth — Value after N years = Initial × (1 ± R/100)ⁿ (same as CI — Module 16). + for growth, − for depreciation.

8. Summary of Variables & Multipliers

Symbol	Meaning
`I, F`	Initial and final values
`CP, SP, MP`	Cost, selling, marked price

Common multipliers: 1.10 (10% ↑), 0.90 (10% ↓), 1.25 (25% ↑), 0.80 (20% ↓), 1.50 (50% ↑).

20% as fraction?

20% of 150 = ?

30 is what % of 50?

10% of a number is 30 → number?

Increase 100 by 20% → ?

Increase 20% then decrease 10% → net?

Price ↑20%, expenditure constant → consumption ↓?

Discount is always on?

Notes

Percentage :- Basics :-

Per 100 ; fraction→% ×100 ; %→fraction ÷100

Key ⇒ 1/2=50%, 1/4=25%, 1/5=20%, 1/8=12.5%, 1/20=5%

Formulas :-

X% of Y ⇒ (X/100)Y ; X is what % of Y ⇒ (X/Y)×100 ; whole: Y/(X/100)

% change ⇒ (F−I)/I×100 ; initial is denominator

Apply ⇒ +X%: ×(1+X/100) ; −X%: ×(1−X/100)

Successive & Constancy :-

Net ⇒ X+Y+XY/100 (signed) (eg +20−10−2=+8%)

Product const ⇒ Price↑X→consumption↓ X/(100+X) ; Price↓X→↑ X/(100−X)

Commercial :-

CP 100% base ; SP=CP(1±P/100) ; Discount on MP: SP=MP(1−D/100)

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