5.1 Percents, Percent Change, Markups, Discounts, Taxes & Profit/Loss

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.

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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:

$$p\% = \frac{p}{100} = p \times 10^{-2}$$

+-----------------------------------------------------------------------------+
|                   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 %)           |
+-----------------------------------------------------------------------------+

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

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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:

$$\text{Percent Change} = \frac{V_{\text{new}} - V_{\text{old}}}{V_{\text{old}}} \times 100\% = \left(\frac{\Delta V}{V_{\text{old}}}\right) \times 100\%$$

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.

+-----------------------------------------------------------------------------+
|                   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%)           |
+-----------------------------------------------------------------------------+

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 $= \frac{20^2}{100}\% = \frac{400}{100}\% = 4\%$.

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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 ($M$):

$$S = C + M \iff M = S - C$$

However, businesses compute percentage markups using two distinct accounting bases: Cost versus Selling Price (Gross Margin).

+-----------------------------------------------------------------------------+
|                     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)        |
+-----------------------------------------------------------------------------+

Formulas and Direct Conversions

To convert directly between the two markup rates without finding dollar amounts:

$$M_S = \frac{M_C}{1 + M_C}, \qquad M_C = \frac{M_S}{1 - M_S}$$

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?

1. Identify the given parameters: $C = 1400$ and $M_S = 0.30$.
2. Apply the selling price formula: $$S = \frac{C}{1 - M_S} = \frac{1400}{1 - 0.30} = \frac{1400}{0.70} = \mathbf{2000.00}$$
3. 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.

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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 ($N$):

$$N = L(1 - d)$$

When a manufacturer offers a chain discount (e.g., "20/10/5"), the net price factor is the product of the complement factors:

$$N = L \times (1 - d_1) \times (1 - d_2) \times (1 - d_3) = L \times 0.80 \times 0.90 \times 0.95 = L \times 0.684$$

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:

$$\text{Sales Tax Amount} = N \times t_{\text{tax}}$$ $$\text{Total Invoice Amount} = N(1 + t_{\text{tax}}) = [L(1 - d)](1 + t_{\text{tax}})$$

Commercial Profit and Loss Structure

+-----------------------------------------------------------------------------+
|                        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)     |
+-----------------------------------------------------------------------------+

Common Test Traps Summary