Everyday Percentage Math: Discounts, Markups, Percentage Changes, and Tips

The Four Fundamental Percentage Formulas#

Almost all percentage calculations revolve around four questions:

• 1. What is X% of Y?: $\text{Result} = (X / 100) \times Y$. (e.g. 15% of $80 = 0.15 \times 80 = $12).
• 2. X is what percent of Y?: $\text{Percentage} = (X / Y) \times 100$. (e.g. 25 of 200 = $(25/200) \times 100 = 12.5\%$).
• 3. What is the total if X is Y%?: $\text{Total} = X / (Y / 100)$. (e.g. If $45 is 30%, Total = $45 / 0.30 = $150).
• 4. What is the percentage change from A to B?: $\text{Change} = [(B - A) / A] \times 100$.

Percentage Increase and Decrease: $(B - A) / A \times 100$#

When web traffic grows from 50,000 to 75,000 visitors, the percentage increase is $[(75,000 - 50,000) / 50,000] \times 100 = +50\%$.

Notice the asymmetry of percentage drops: if your investment falls by 50% (from $100 to $50), it requires a **100% gain** (from $50 back to $100) just to break even.

The Stacked Discount Fallacy: Why 30% + 20% is NOT 50% Off#

Retailers often advertise "Take 30% off, plus an extra 20% off at checkout!". Shoppers often assume this equals 50% off.

In reality, the secondary 20% discount applies to the already-reduced price:

1. Original price: $100.

2. First discount (30% off): $100 - $30 = $70.

3. Second discount (20% off $70): $70 - $14 = **$56**.

The true total discount is 44% off, not 50%.

Markup vs. Gross Margin: A Critical Business Distinction#

• Markup: Percentage added on top of cost. If an item costs $80 to produce and you sell it for $100, the markup is $[(100 - 80) / 80] = 25\%$.

• Gross Margin: Profit as a percentage of selling price. The gross margin is $[(100 - 80) / 100] = 20\%$.

Everyday Mental Math Shortcuts for 10%, 15%, and 20%#

• To find 10%: Move the decimal point one place to the left ($64.00 \rightarrow $6.40).

• To find 20%: Find 10% and double it ($6.40 \times 2 = $12.80).

• To find 15%: Find 10% and add half of that amount ($6.40 + $3.20 = $9.60).