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Percentage Changes

The Story Behind the Math

Percentages emerged from commerce. Italian merchants in the 15th century used "per cento" (by the hundred) to express interest rates and profits. Before percentages, they used fractions like "one part in twenty"—hard to compare!

The breakthrough: Expressing everything "out of 100" created a universal language. A 5% profit in Venice meant the same as a 5% profit in Florence.

But percentages hide a subtle trap that confuses even professionals today: successive changes don't add up. A 10% increase followed by a 10% decrease doesn't return you to where you started. This fact has cost investors, shoppers, and policymakers billions.

Understanding percentage changes means understanding multiplicative thinking—a fundamental shift from additive thinking.

Why It Matters

Percentage changes appear everywhere:

  • Finance: Investment returns, inflation, interest rates, stock market moves
  • Science: Error margins, population growth, radioactive decay
  • Medicine: Treatment efficacy, survival rates, dosage adjustments
  • Economics: GDP growth, unemployment rates, price indices
  • Everyday life: Sales discounts, tips, tax calculations

Misunderstanding percentage changes leads to poor decisions—from choosing the wrong credit card to misinterpreting medical studies.

Prerequisites

The Core Insight

What Percentages Really Are

"Percent" means "per hundred."

\[ 50\% = \frac{50}{100} = 0.5 = \frac{1}{2} \]

A percentage is just a fraction with denominator 100. This makes comparisons easy.

The Fundamental Formula

To find what percentage \(a\) is of \(b\):

\[ \text{Percentage} = \frac{a}{b} \times 100\% \]

To apply a percentage \(p\) to a value \(b\):

\[ \text{Result} = b \times \frac{p}{100} \]

Calculating Percentage Changes

Percentage Increase

Example: A shirt costs $40. The price increases by 25%. What's the new price?

Method 1: Calculate increase, then add - Increase: \(40 \times 0.25 = 10\) - New price: \(40 + 10 = 50\)

Method 2 (Faster): Multiply by \((1 + \text{percentage})\)

\[ \text{New Price} = 40 \times 1.25 = 50 \]

The multiplier \(1.25\) means "100% of original + 25% increase = 125% of original."

Percentage Decrease

Example: A $80 item is on sale for 30% off. What's the sale price?

Method 1: Calculate discount, then subtract - Discount: \(80 \times 0.30 = 24\) - Sale price: \(80 - 24 = 56\)

Method 2 (Faster): Multiply by \((1 - \text{percentage})\)

\[ \text{Sale Price} = 80 \times 0.70 = 56 \]

The multiplier \(0.70\) means "100% - 30% = 70% of original."

The Multiplier Method

For any percentage change: - Increase: Multiply by \((1 + p)\) where \(p\) is the decimal percentage - Decrease: Multiply by \((1 - p)\) where \(p\) is the decimal percentage

This is the key to understanding successive changes.

The Trap: Successive Changes Don't Add

Example: 10% Up, Then 10% Down

You have $100.

Year 1: 10% increase

\[ 100 \times 1.10 = 110 \]

Year 2: 10% decrease

\[ 110 \times 0.90 = 99 \]

Result: You have $99, not $100!

Why? The second 10% is calculated on $110, not the original $100.

The General Rule

Two successive changes of \(+p\%\) and \(-p\%\) result in:

\[ (1 + p)(1 - p) = 1 - p^2 < 1 \]

You always lose! For 10%: \(1 - 0.01 = 0.99\) (1% loss).

Real-World Consequence

A stock falls 50%, then rises 50%. Is it back to the original price?

\[ \$100 \xrightarrow{-50\%} \$50 \xrightarrow{+50\%} \$75 \]

No! It's down 25% from the original!

Compound Percentage Changes

Successive Multiplication

Multiple percentage changes multiply, not add.

Example: A $200 item gets 20% off, then an additional 15% off. Is that 35% off total?

Calculate:

\[ 200 \times 0.80 \times 0.85 = 200 \times 0.68 = 136 \]

The final price is $136, which is 32% off (not 35%).

Why?: The second discount applies to the already-discounted price.

Compound Interest

If you invest $1000 at 5% annual interest for 3 years:

\[ 1000 \times 1.05 \times 1.05 \times 1.05 = 1000 \times (1.05)^3 = 1157.63 \]

Not \(1000 + 15\% = 1150\)! The extra $7.63 is "interest on interest."

The compound interest formula:

\[ A = P(1 + r)^n \]

Where: - \(P\) = principal (initial amount) - \(r\) = interest rate (decimal) - \(n\) = number of periods

Working Backwards: Finding Original Values

After a Percentage Increase

Example: After a 20% price increase, a product costs $120. What was the original price?

Common error: \(120 - 20\% = 120 - 24 = 96\) ❌

Correct approach: The $120 is 120% of original (100% + 20%).

Let \(x\) be the original price:

\[ x \times 1.20 = 120 \]
\[ x = \frac{120}{1.20} = 100 \]

The original price was $100.

After a Percentage Decrease

Example: After a 30% discount, you paid $70. What was the original price?

The $70 is 70% of original (100% - 30%).

Let \(x\) be the original price:

\[ x \times 0.70 = 70 \]
\[ x = \frac{70}{0.70} = 100 \]

The original price was $100.

The Division Rule

To find the original amount after a percentage change:

\[ \text{Original} = \frac{\text{New Amount}}{1 \pm \text{percentage}} \]

Use \(+\) for increase, \(-\) for decrease.

Percentage Points vs. Percent

Critical Distinction

Example: Unemployment rises from 5% to 6%.

  • Percentage point increase: \(6\% - 5\% = 1\) percentage point
  • Percent increase: \(\frac{6 - 5}{5} \times 100\% = 20\%\)

The unemployment rate increased by 1 percentage point (or 20%).

When to use which: - Percentage points: Comparing absolute changes in rates - Percent: Comparing relative changes

Real-World Importance

Headlines saying "Taxes increase 5%" are ambiguous: - Tax rate goes from 20% to 25% (5 percentage points, 25% relative increase) - Tax rate goes from 20% to 21% (1 percentage point, 5% relative increase)

Always check which is meant!

Common Calculations

Finding What Percentage One Number Is of Another

Example: 15 is what percentage of 60?

\[ \frac{15}{60} \times 100\% = 0.25 \times 100\% = 25\% \]

Finding a Number Given a Percentage

Example: 30% of what number is 45?

Let \(x\) be the number:

\[ 0.30x = 45 \]
\[ x = \frac{45}{0.30} = 150 \]

Percentage of a Percentage

Example: 20% of 50%

\[ 0.20 \times 0.50 = 0.10 = 10\% \]

This is how tax brackets work—you pay X% on the amount in that bracket.

Visual Understanding

The Area Model

Imagine a rectangle where the area represents 100%:

+------------------+
|                  |
|       100%       |
|                  |
+------------------+

A 20% increase makes it 120% of the original area:

+------------------+------+
|                  |      |
|       100%       | 20%  |
|                  |      |
+------------------+------+

The Number Line

Starting at 100:

0-----50-----100-----150-----200
            ↑
          Start

+20%:      100 ----→ 120
-20%:      100 ←---- 80
+20% then -20%: 100 → 120 → 96 (not back to 100!)

Common Misconceptions

  1. "10% + 10% = 20% increase": Only if applied to the same base. Successive changes multiply.

  2. "Double is 100% more": Yes! But "200% of" is triple (200% more = 300% of).

  3. "Percentages above 100% don't make sense": 150% means 1.5 times, or 50% more than original.

  4. "I saved 50% + 30% = 80%": Two successive discounts compound to less than the sum.

  5. "If I lose 50% then gain 50%, I'm even": No, you're down 25%. See proof above.

Real-World Applications

Shopping: Comparing Discounts

"Buy one get one 50% off" vs "25% off everything"

BOGO 50%: Two items at prices \(A\) and \(B\):

\[ \text{Total} = A + 0.5B \]

If \(A = B\): Total = \(1.5A\), so average price per item is \(0.75A\) (25% off).

But if \(A \neq B\), the savings vary!

Investing: Average Returns

Investment returns: +20%, -20%, +20%, -20% over 4 years.

Average return = 0%, but actual result:

\[ 1.20 \times 0.80 \times 1.20 \times 0.80 = 0.9216 \]

You're down 7.84%! Volatility hurts returns.

Medicine: Risk Reduction

"Treatment reduces risk by 50%" - Absolute risk from 2% to 1% (1 percentage point difference) - Or from 40% to 20% (20 percentage points difference)

Same relative reduction, vastly different practical significance!

References

  • Bennett, J. (2021). Randomness and Coincidence: How to Think Clearly About Chance. Cambridge University Press.
  • Gigerenzer, G. (2002). Calculated Risks: How to Know When Numbers Deceive You. Simon & Schuster.
  • Paulos, J. A. (1988). Innumeracy: Mathematical Illiteracy and Its Consequences. Hill and Wang.