Percentage Change in the Real World: Sales, Stocks, and Salaries

Why Percentage Change Is Everywhere

Every time a price, salary, stock price, or economic statistic changes, it is reported as a percentage change. Understanding how that number is calculated — and where it can mislead — is one of the most practical mathematical skills available.

Percentage

The Core Formula

Percentage change measures how a quantity moved relative to its starting value:

Percentage change = ((New Value − Old Value) / Old Value) × 100

• If the result is positive: the quantity increased
• If the result is negative: the quantity decreased
• If the result is 0: no change

Examples: - Stock price: $40 → $52. Change = ((52 − 40) / 40) × 100 = +30% - House price: £350,000 → £315,000. Change = ((315k − 350k) / 350k) × 100 = −10% - Salary: ₩3,200,000 → ₩3,600,000. Change = ((3.6M − 3.2M) / 3.2M) × 100 = +12.5%

Percentage Change Formula

The Asymmetry Trap: Gains vs. Losses

The most counterintuitive property of percentage change is its asymmetry. A percentage loss and an equal percentage gain do not cancel out.

Scenario	Starting	Loss	Result after loss	Gain needed to recover	Net after recovery
A	$100	−20%	$80	+25%	$100
B	$100	−50%	$50	+100%	$100
C	$100	−75%	$25	+300%	$100

The rule: If an asset falls by X%, it needs a gain of X/(1−X/100)% to recover. - A 20% loss requires: 20/(1−0.20) = 20/0.80 = 25% to recover - A 50% loss requires: 50/0.50 = 100% to recover - A 10% loss requires: 10/0.90 ≈ 11.1% to recover

This asymmetry explains why experienced investors obsess over avoiding large losses — it takes disproportionately large gains to undo them.

Sales and Retail: Price vs. Discount Math

Markups vs. markdowns use the same formula but different base values:

• Markup percentage = ((Selling Price − Cost) / Cost) × 100
• A retailer buys a jacket for $60 and sells it for $90. Markup = (30/60) × 100 = 50%
• Markdown percentage = ((Original Price − Sale Price) / Original Price) × 100
• The same $90 jacket is discounted to $63. Markdown = (27/90) × 100 = 30%

Note: a 50% markup followed by a 30% markdown does NOT restore the original cost price. The new selling price is $63, giving the retailer $3 profit — exactly 5% on cost.

Stock Markets and Investing

Investors encounter percentage change constantly: - Daily returns: How much a stock or index moved in a single session - Year-to-date (YTD) return: How much an investment has changed since January 1 - Compound annual growth rate (CAGR): The smoothed annual rate of return over multiple years

One critical mistake: adding daily returns instead of compounding them. A stock that rises 10% on day 1 and falls 10% on day 2 does NOT end flat. It ends at 0.10 × (1 − 0.10) = 0.99, meaning a −1% net result on a $100 investment.

Day	Price	Daily return
Start	$100	—
Day 1	$110	+10%
Day 2	$99	−10%

Daily returns compound multiplicatively, not additively. This is why volatile assets underperform their average return (the "variance drain" effect).

Salary Negotiations: Playing the Percentage Game

Understanding percentage change makes you a better salary negotiator.

Scenario: Your salary is $72,000 and you want $80,000. - Required increase: ((80,000 − 72,000) / 72,000) × 100 = 11.1%

Counter-framing: The employer offers $75,000 instead. - That offer from their framing: +4.2% over your current salary - Difference from your target: $5,000 — which you can express as $5,000 / $75,000 = 6.7% from the offer

Knowing both framings lets you choose the one that better supports your position.

The annual raise trap: A 3% raise for two consecutive years is NOT the same as a 6% total raise. - Year 1: $72,000 × 1.03 = $74,160 - Year 2: $74,160 × 1.03 = $76,385 - Actual total increase: 6.09%, not 6.00%

Over many years, compounded raises significantly exceed simple additions — which is why compound interest and compound salary growth are more powerful than they appear.

Economic Statistics: GDP, Inflation, Trade

Macroeconomic reporting uses percentage change almost exclusively: - GDP growth rate: How much an economy's total output changed year-over-year - Inflation rate: How much consumer prices changed (usually compared to same month, prior year) - Trade deficit change: How the gap between imports and exports moved

A common misuse: comparing percentage changes across different base years. If GDP was $20 trillion and grew 3%, that is $600 billion in new output. If it then grew 2% the following year from $20.6 trillion, that is $412 billion. The declining percentage understates that the absolute amounts of growth may be large and growing.

Checking Percentage Change Calculations

A quick sanity-check process:

1. Identify which is the old value (base) and which is the new value
2. Compute the absolute difference (new − old)
3. Divide by the old value (not the new one — this is the most common error)
4. Multiply by 100

The denominator is always the reference point — what the change is measured relative to. Using the new value as the denominator gives a different metric (sometimes called "percentage share") and is not the standard definition of percentage change.

Percentage