3.2 Percentages, Percent Change & Markup/Discount

Percentage Fundamentals: Conversions and the Percent Equation

A percent represents a fraction or ratio whose denominator is always $100$. The word percent originates from the Latin per centum, meaning "by the hundred." On the HiSET exam, percentages link foundational arithmetic to algebraic equations and practical financial scenarios.

The Three Equivalent Forms

Every percentage can be expressed interchangeably as a fraction, decimal, or percent:

$$P\% = \frac{P}{100} = P \times 0.01$$

• Percent to Decimal: Shift the decimal point two places to the left and drop the $\%$ sign ($42.5\% = 0.425$; $6\% = 0.06$).
• Decimal to Percent: Shift the decimal point two places to the right and append the $\%$ sign ($0.085 = 8.5\%$; $1.45 = 145\%$).
• Fraction to Percent: Convert the fraction to a decimal via division, then convert to a percent ($\frac{3}{8} = 0.375 = 37.5\%$).

The Fundamental Percent Equation

All standard percent problems involve three elements: the Base (Whole), the Rate (Percent), and the Amount (Part):

$$\text{Part} = \text{Percent (as a decimal)} \times \text{Whole} \quad \Longleftrightarrow \quad A = P \cdot W$$

Alternatively, this relationship can be structured as the Percent Proportion:

$$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \quad \Longleftrightarrow \quad \frac{\text{"is"}}{\text{"of"}} = \frac{\%}{100}$$

              The Percent Triangle Formula Helper
                           /   \ 
                          /  A  \          A = Part (Amount)
                         /───────\         P = Percent (as decimal)
                        / P  │  W \        W = Whole (Base)
                       /─────┴─────\ 
              
              • Cover A  ──►  A = P × W
              • Cover P  ──►  P = A / W
              • Cover W  ──►  W = A / P

Solving the Three Core Percent Cases

Percent Change: Measuring Increase and Decrease

Percent change quantifies the relative difference between an original value and a new value over time. Standardized tests frequently assess whether test-takers correctly identify the original baseline value.

$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% = \frac{\text{Amount of Change}}{\text{Original Base}} \times 100\%$$

• If $\text{New} > \text{Original}$, the result is positive, indicating a percent increase.
• If $\text{New} < \text{Original}$, the result is negative, indicating a percent decrease.

Universal Rule: The denominator of the percent change formula is always the original starting value, never the final or new value.

The Multiplier Shortcut for Percent Changes

Rather than computing the change in two separate steps (calculating the dollar change and adding/subtracting from the original), use single decimal multipliers:

• For an increase of $r\%$: Multiply the base by $(1 + r)$. For a $15\%$ increase, multiply by $1 + 0.15 = 1.15$.
• For a decrease of $r\%$: Multiply the base by $(1 - r)$. For a $20\%$ decrease, multiply by $1 - 0.20 = 0.80$.

The Asymmetry Trap of Percent Increase vs. Decrease

A common misconception is assuming that equal percentage increases and decreases cancel each other out. They do not, because each percentage applies to a different base value.

   The 25% Increase vs. 20% Decrease Asymmetry
   
   Start Value: $100
     │
     ▼ (+25% increase on $100)
   Value: $125
     │
     ▼ (-20% decrease on $125)
   Final Value: $125 × 0.80 = $100  (Back to start!)
   
   Notice: A 25% drop from $125 would be $125 × 0.75 = $93.75, NOT $100!

Worked Example: Price Increase and Reversal

An item priced at $\text{\textdollar}80$ is increased by $25\%$. What percent must the new price be decreased to return to the original $\text{\textdollar}80$ price?

1. Calculate the increased price: $$\text{New Price} = \$80 \times (1 + 0.25) = \$80 \times 1.25 = \$100$$
2. Calculate the required dollar reduction: $$\text{Reduction} = \$100 - \$80 = \$20$$
3. Compute the percent decrease relative to the NEW base of $\text{\textdollar}100$: $$\text{Percent Decrease} = \frac{\$20}{\$100} \times 100\% = 20\%$$

Thus, a $25\%$ increase requires a $20\%$ decrease to return to the initial value.

Retail Mathematics: Markups, Discounts, and Working Backwards

Retail arithmetic on the HiSET focuses on the relationships between wholesale cost, retail list price, discount rates, and final sale prices.

Key Retail Formulas

1. Markup Equation: $$\text{Selling Price} = \text{Wholesale Cost} + \text{Markup Amount} = \text{Cost} \times (1 + \text{Markup Rate})$$
2. Discount (Sale Price) Equation: $$\text{Sale Price} = \text{Original Tag Price} - \text{Discount Amount} = \text{Original Price} \times (1 - \text{Discount Rate})$$

Working Backwards to Find the Original Price

When a problem provides the discounted sale price and the discount percentage, students often make the fatal error of taking the discount percentage of the sale price and adding it back. That produces an incorrect result because the discount was originally calculated on the larger, unknown starting price.

Worked Example: Reconstructing Original Tag Price

A winter coat is sold during an end-of-season clearance for $\text{\textdollar}105$ after a $30\%$ discount. What was the original tag price of the coat?

1. Identify the percent relationship: A $30\%$ discount means the customer paid $100\% - 30\% = 70\%$ of the original tag price $P$.
2. Write the algebraic equation: $$0.70 \cdot P = 105$$
3. Solve for $P$ by dividing by $0.70$: $$P = \frac{105}{0.70} = \$150$$
4. Check the result: $$30\% \text{ of } \$150 = 0.30 \times 150 = \$45 \implies \$150 - \$45 = \$105 \quad (\text{Correct})$$

(Error Check: Taking $30\%$ of $\text{\textdollar}105$ gives $\text{\textdollar}31.50$, and $\text{\textdollar}105 + \text{\textdollar}31.50 = \text{\textdollar}136.50$, which is completely incorrect).

Successive (Compound) Discounts and Markups

In real-world retail, multiple promotions often apply sequentially (e.g., a clearance markdown plus a store coupon). Successive discounts cannot be added directly together. A $20\%$ discount followed by an additional $10\%$ coupon does not equal a $30\%$ discount.

The Successive Multiplier Rule

To find the final price after consecutive percentage changes, multiply the original price by the product of individual decimal multipliers:

$$P_{\text{final}} = P_0 \times (1 - d_1) \times (1 - d_2) \times \cdots \times (1 - d_n)$$ $$\text{Effective Single Discount Rate} = 1 - [(1 - d_1)(1 - d_2)]$$

Mathematical Demonstration

Suppose an item has an initial retail price of $\text{\textdollar}200$, is discounted by $20\%$, and then an extra $10\%$ coupon is applied at the register:

1. Step-by-Step Method:
   • First discount: $\text{\textdollar}200 \times 0.20 = \text{\textdollar}40$ off $\implies \text{Intermediate Price} = \text{\textdollar}200 - \text{\textdollar}40 = \text{\textdollar}160$
   • Second discount ($10\%$ on $\text{\textdollar}160$): $\text{\textdollar}160 \times 0.10 = \text{\textdollar}16$ off $\implies \text{Final Price} = \text{\textdollar}160 - \text{\textdollar}16 = \text{\textdollar}144$
2. Multiplier Method: $$P_{\text{final}} = \$200 \times (1 - 0.20) \times (1 - 0.10) = \$200 \times 0.80 \times 0.90 = \$200 \times 0.72 = \$144$$
3. Effective Overall Discount: $$\text{Overall Savings} = \$200 - \$144 = \$56 \implies \frac{\$56}{\$200} = 0.28 = 28\%$$

The true combined savings is $28\%$, not $30\%$ ($20\% + 10\%$), because the second discount applied only to the already-reduced $\text{\textdollar}160$ subtotal.

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Sales Tax, Gratuities, and Commission Calculations

Sales Tax and Gratuities (Tips)

Sales tax and tips represent percentage additions to a transaction's subtotal:

• Tax or Tip Amount: $\text{Add-on} = \text{Subtotal} \times \text{Rate}$
• Total Bill (Single Multiplier): $\text{Total} = \text{Subtotal} \times (1 + \text{Tax Rate} + \text{Tip Rate})$ (when tip is based on pre-tax subtotal)

Worked Example: Dining Receipt

A restaurant bill subtotal is $\text{\textdollar}68.00$. A sales tax of $8.5\%$ is added, and the customer leaves an $18\%$ tip calculated on the pre-tax food subtotal. What is the total expenditure?

1. Calculate Tax Amount: $\text{\textdollar}68.00 \times 0.085 = \text{\textdollar}5.78$
2. Calculate Tip Amount: $\text{\textdollar}68.00 \times 0.18 = \text{\textdollar}12.24$
3. Sum the Total: $\text{Total} = \text{\textdollar}68.00 + \text{\textdollar}5.78 + \text{\textdollar}12.24 = \text{\textdollar}86.02$

Commission Compensation Structures

Sales representatives are frequently compensated using percentage commissions. Standard compensation structures include:

1. Straight Commission: $\text{Earnings} = \text{Sales Volume} \times \text{Commission Rate}$
2. Salary Plus Commission: $\text{Earnings} = \text{Base Salary} + (\text{Sales Volume} \times \text{Commission Rate})$
3. Quota-Tiered Commission: $\text{Earnings} = \text{Base Salary} + [(\text{Sales Volume} - \text{Quota Threshold}) \times \text{Commission Rate}]$