Chapter 2

Percentage Change & Sequential Changes

% increase/decrease, chained changes, percentage points vs relative %, reverse percentages

2.1 Percentage change formula

% change = (End − Start) ÷ Start × 100

Example: exports were €3.2 billion last year and are €3.6 billion this year. % change = (3.6 − 3.2) ÷ 3.2 × 100 = 12.5%.

2.2 Applying a percentage change: the multiplier

Increase by r%  →  multiply by (1 + r/100)
Decrease by r%  →  multiply by (1 − r/100)

Converting changes to multipliers: +20% → 1.20, −15% → 0.85, +3.5% → 1.035. Example: a salary of €38 000 rises by 6%. New salary = 38 000 × 1.06 = €40 280.

💡 Common trap: +6% then −6% is NOT zero net change. 1.06 × 0.94 = 0.9964, a net loss of 0.36%. Each change applies to a different base.

2.3 Sequential (chained) percentage changes

When a value changes multiple times, each change applies to the result of the previous step, not to the original value. Multiply all multipliers together — never add raw percentages.

Net multiplier = first × second × third × ...
Net % change   = (Net multiplier − 1) × 100

Example: €240 changes by +12%, −7%, +4%, −9%. Multipliers: 1.12 × 0.93 × 1.04 × 0.91 ≈ 0.9733. Final value ≈ €233.60. Net change ≈ −2.67%, even though 12−7+4−9 = 0.

2.4 Percentage points vs relative percentage change

Percentage point (pp) change = End rate − Start rate
Relative % change            = (End − Start) ÷ Start × 100

Example: employment rate rises from 64% to 67%. pp change = +3 pp. Relative change = 3 ÷ 64 × 100 = +4.7%. Both are correct but describe different things.

2.5 Reverse percentage (finding the original value)

Original = Final ÷ (1 + r/100)

For multiple successive changes:
Original = Final ÷ (multiplier_1 × multiplier_2 × ...)

Example: a product costs €94.60 after a 15% price rise. Original = 94.60 ÷ 1.15 = €82.26.

💡 Common trap: 94.60 × 0.85 = €80.41. Subtracting 15% from the final gives the wrong result. Always divide by the multiplier.

2.6 Revenue change: price × quantity

Net revenue factor = price multiplier × quantity multiplier

Example: prices rise 7% but volumes fall 4%. Net = 1.07 × 0.96 = 1.0272. Revenue increases by 2.72%, not 3% (7−4).

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