5.1 Percents, Percent Change, Markups, Discounts, Taxes & Profit/Loss
Key Takeaways
- A percent represents a ratio per hundred ($p\% = p/100$); the fundamental tripartite percentage relation is $\text{Percentage} = \text{Base} \times \text{Rate}$, where the Rate is always converted to decimal form before arithmetic operations.
- Percent change is strictly calculated relative to the original reference value: $\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$; a positive quotient indicates a percent increase, while a negative quotient indicates a percent decrease.
- Successive percentage changes cannot be combined via simple addition; a +25% price increase followed by a -20% discount restores the exact initial price (1.25 * 0.80 = 1.00), whereas a +20% increase followed by a -20% decrease results in a net -4% loss (1.20 * 0.80 = 0.96).
- Markup on Cost ($M_C = \frac{S - C}{C} \times 100\%$) is always strictly larger than Markup on Selling Price or Gross Margin ($M_S = \frac{S - C}{S} \times 100\%$) for any transaction where the Selling Price ($S$) exceeds Cost ($C$).
- Commercial retail transactions apply trade discounts to the list price before calculating sales tax: $\text{Total Customer Payment} = [\text{List Price} \times (1 - d)] \times (1 + t_{\text{tax}})$; Gross Profit equals $S - C$, while Net Profit deducts operating overhead from Gross Profit.
5.1 Percents, Percent Change, Markups, Discounts, Taxes & Profit/Loss
Commercial arithmetic and percentage computations represent a cornerstone of the Financial Mathematics domain, which accounts for 20% of the CLEP College Mathematics examination (approximately 12 out of 60 questions). Mastering these concepts requires moving beyond basic middle-school arithmetic into precise algebraic modeling of relative change, successive multi-step adjustments, retail pricing structures, and commercial profit-and-loss accounting.
1. Percent Fundamentals & Base-Rate-Percentage Relationships
The term percent originates from the Latin per centum, meaning "by the hundred." A percent is a dimensionless ratio expressing a fraction of 100:
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| THE BASE-RATE-PERCENTAGE TRIANGLE (P = B * R) |
| |
| / \ |
| / \ |
| / P \ <--- Percentage (Part / Portion)|
| /------- |
| / B | R \ <--- Base (Total) * Rate (%) |
| /____ | ____\ |
| |
| 1. Percentage (Part): P = B * R |
| 2. Base (Whole/Total): B = P / R |
| 3. Rate (Percentage Rate): R = P / B (Multiply by 100 for %) |
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Conversions Across Representations
Before executing algebraic operations, percentages must be converted into decimals or reduced fractions:
- Percent to Decimal: Shift the decimal point two places to the left and drop the
%symbol (e.g., 7.25% = 0.0725; 0.4% = 0.004; 245% = 2.45). - Decimal to Percent: Shift the decimal point two places to the right and append
%(e.g., 0.068 = 6.8%; 1.35 = 135%). - Fraction to Percent: Convert the fraction to a decimal via division, then multiply by 100% (e.g., $\frac{7}{8} = 0.875 = 87.5%$).
The Three Fundamental Operational Types
| Operational Goal | Algebraic Formula | Concrete Application Example |
|---|---|---|
| Find the Part ($P$) | $P = B \times R$ | What is 35% of $480? $\implies P = 480 \times 0.35 = \mathbf{168.00}$ (dollars) |
| Find the Base ($B$) | $B = \frac{P}{R}$ | $72 is 12% of what number? $\implies B = \frac{72}{0.12} = \mathbf{600}$ |
| Find the Rate ($R$) | $R = \frac{P}{B}$ | $45 is what percent of $180? $\implies R = \frac{45}{180} = 0.25 = \mathbf{25%}$ |
2. Percent Change & Successive Multi-Step Adjustments
Absolute Change vs. Relative (Percent) Change
When a numerical quantity shifts from an initial state ($V_{\text{initial}}$ or $V_{\text{old}}$) to a final state ($V_{\text{final}}$ or $V_{\text{new}}$):
- Absolute Change: $\Delta V = V_{\text{final}} - V_{\text{initial}}$ (measured in original units like dollars or units).
- Relative / Percent Change: The ratio of the absolute change to the original base value, expressed as a percentage:
[!IMPORTANT] The Base Value Rule: The denominator in the percent change formula is always the starting (old) value, never the new or average value. A change from $80 to $100 is a +25% increase ($\frac{100 - 80}{80} = +0.25$), whereas a change from $100 to $80 is a -20% decrease ($\frac{80 - 100}{100} = -0.20$).
The Multiplier (Growth / Decay Factor) Method
Rather than computing the change in two separate steps (multiplying and then adding/subtracting), use the single-step growth multiplier $M = (1 \pm r)$:
- Percent Increase of $r$: $V_{\text{new}} = V_{\text{old}} \times (1 + r)$. For example, a 15% increase implies $V_{\text{new}} = V_{\text{old}} \times 1.15$.
- Percent Decrease of $d$: $V_{\text{new}} = V_{\text{old}} \times (1 - d)$. For example, a 30% discount implies $V_{\text{new}} = V_{\text{old}} \times 0.70$.
Successive (Compound) Percentage Changes
When a quantity undergoes multiple consecutive percentage changes, the individual percentages cannot be added linearly. Each successive percentage acts upon the updated intermediate balance, not the original starting base.
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| SUCCESSIVE PERCENTAGE CHANGE ANALYSIS |
| |
| Scenario A: Price increases by 25%, then decreases by 20% |
| - Multiplier: M = (1 + 0.25) * (1 - 0.20) = 1.25 * 0.80 = 1.00 |
| - Net Result: 1.00 - 1.00 = 0.00 ===> 0% Net Change (Price Restored) |
| |
| Scenario B: Price increases by 20%, then decreases by 20% |
| - Multiplier: M = (1 + 0.20) * (1 - 0.20) = 1.20 * 0.80 = 0.96 |
| - Net Result: 0.96 - 1.00 = -0.04 ===> 4% Net Loss |
| |
| Scenario C: Price decreases by 30%, then increases by 30% |
| - Multiplier: M = (1 - 0.30) * (1 + 0.30) = 0.70 * 1.30 = 0.91 |
| - Net Result: 0.91 - 1.00 = -0.09 ===> 9% Net Loss |
| |
| Scenario D: Price increases by 10%, then increases by 10% |
| - Multiplier: M = (1 + 0.10) * (1 + 0.10) = 1.10 * 1.10 = 1.21 |
| - Net Result: 1.21 - 1.00 = +0.21 ===> +21% Net Gain (Not 20%) |
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General Mathematical Principle for Symmetric Changes: An increase of $x%$ followed by a decrease of $x%$ (or vice versa) always results in a net loss of $\left(\frac{x}{100}\right)^2 \times 100% = \frac{x^2}{100}%$. For $x = 20%$, net loss $= \frac{20^2}{100}% = \frac{400}{100}% = 4%$.
3. Retail Mathematics: Markup on Cost vs. Markup on Selling Price
In commercial trade, the difference between the wholesale Cost ($C$) paid by a merchant and the retail Selling Price ($S$) is the Dollar Markup ($M$):
However, businesses compute percentage markups using two distinct accounting bases: Cost versus Selling Price (Gross Margin).
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| MARKUP ON COST VS. MARKUP ON SELLING PRICE |
| |
| Wholesale Cost: C = $60 Retail Selling Price: S = $100 |
| ----------------------------------------------------------------------- |
| Dollar Markup: M = S - C = $100 - $60 = $40 |
| |
| MARKUP ON COST (M_C): MARKUP ON SELLING PRICE (M_S): |
| M_C = (S - C) / C * 100% M_S = (S - C) / S * 100% |
| M_C = 40 / 60 = 2/3 = 66.67% M_S = 40 / 100 = 40.00% |
| Base = $60 (Cost) Base = $100 (Selling Price) |
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Formulas and Direct Conversions
| Metric | Definition & Formula | Algebraic Inversion |
|---|---|---|
| Markup on Cost ($M_C$) | $M_C = \frac{S - C}{C} = \frac{S}{C} - 1$ | $S = C(1 + M_C)$ |
| Markup on Selling Price ($M_S$) | $M_S = \frac{S - C}{S} = 1 - \frac{C}{S}$ | $S = \frac{C}{1 - M_S}$ |
To convert directly between the two markup rates without finding dollar amounts:
Worked Example: Pricing Under Desired Margin
Problem: A distributor buys a commercial espresso machine for $C = 1400$ dollars. If management requires a 30% markup on selling price (gross margin), what must the retail selling price be?
- Identify the given parameters: $C = 1400$ and $M_S = 0.30$.
- Apply the selling price formula:
- Verification: Dollar markup is $2000 - 1400 = 600$, and $\frac{600}{2000} = 30.0%$.
Trap Check: If the distributor incorrectly applied 30% to the cost ($1400 \times 1.30 = 1820$), the resulting margin on selling price would only be $\frac{1820 - 1400}{1820} = \frac{420}{1820} \approx 23.08%$, falling short of corporate targets.
4. Discounts, Sales Taxes & Profit / Loss Accounting
Trade Discounts and Chain Discounts
Retailers frequently discount the original List Price ($L$) to arrive at a discounted Net Sale Price ($N$):
When a manufacturer offers a chain discount (e.g., "20/10/5"), the net price factor is the product of the complement factors:
The single equivalent discount rate is $1 - 0.684 = 0.316 = 31.6%$ (not $20 + 10 + 5 = 35%$).
Sales Tax Calculations
Sales tax is a statutory percentage applied to the final negotiated or discounted sale price, not the initial list price:
Commercial Profit and Loss Structure
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| INCOME STATEMENT ACCOUNTING FLOW |
| |
| Gross Revenue / Net Sales (S) |
| - Cost of Goods Sold (COGS / C) |
| ======================================================================= |
| = GROSS PROFIT (S - C) Gross Margin = (Gross Profit / S) |
| - Operating Expenses (Overhead, Rent, Utilities, Wages) |
| ======================================================================= |
| = NET PROFIT (or Net Loss if negative) Net Margin = (Net Profit / S) |
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Common Test Traps Summary
| Error Type | Incorrect Assumption | Correct Mathematical Rule |
|---|---|---|
| Additive Successive Changes | +20% then -20% = 0% change. | Multiply factors: $1.20 \times 0.80 = 0.96 \implies -4%$ net change. |
| Markup Confusion | $M_C$ and $M_S$ are interchangeable. | $M_C = \frac{S-C}{C}$ uses Cost base; $M_S = \frac{S-C}{S}$ uses Selling Price base. |
| Sales Tax on List Price | Applying tax before trade discount. | Trade discounts reduce the taxable transaction base first. |
| Percent Point vs. Percent | Increasing from 4% to 5% is a 1% gain. | It is a 1 percentage point increase, but a $\frac{5-4}{4} \times 100% = \mathbf{25%}$ relative increase. |
A specialty electronics retailer increases the price of a computer monitor by 25%. During a holiday sale, the new price is discounted by 20%. If the original price of the monitor was $320, what is the final sale price, and what is the net percentage change from the original price?
A furniture retailer purchases a dining table for a wholesale cost of $450 and sets the retail selling price at $750. What are the retailer's markup percentage on cost and markup percentage on selling price (gross margin), rounded to the nearest tenth of a percent?
A designer winter coat has an original list price of $280. The store offers a storewide trade discount of 30%, and the customer must pay a state and local sales tax of 8.5% on the final discounted purchase price. What is the total amount the customer pays at checkout?