2.3 Percent Applications, Markup, Discount, & Simple Interest

Percent Fundamentals & Algebraic Translation

The word percent derives from the Latin per centum, meaning "out of one hundred" or "per hundredths". In all algebraic calculations, percentages must be converted to their decimal or fractional equivalents by dividing by $100$:

$p\% = \frac{p}{100} = 0.01 \cdot p$

Examples: $45\% = 0.45 = \frac{9}{20}$, $\; 7.5\% = 0.075 = \frac{3}{40}$, $\; 125\% = 1.25 = \frac{5}{4}$.

The Direct Translation Keyword Dictionary

Word problems on the ACCUPLACER test can be systematically translated into linear equations using standard verbal cues:

The Three Fundamental Percent Problem Types

Every single-step percent problem revolves around the core relationship:

$\text{Part} = \text{Percent} \times \text{Whole} \quad (\text{or } A = P \cdot W)$

1. Finding the Part ($A$ is unknown): Problem: What is $35\%$ of $160$? $\text{Equation: } x = 0.35 \times 160 = 56$
2. Finding the Percent ($P$ is unknown): Problem: What percent of $80$ is $28$? $\text{Equation: } 28 = \left(\frac{p}{100}\right) \times 80 \implies 28 = 0.80p \implies p = \frac{28}{0.80} = 35\%$
3. Finding the Base/Whole ($W$ is unknown): Problem: $42$ is $60\%$ of what number? $\text{Equation: } 42 = 0.60 \times W \implies W = \frac{42}{0.60} = 70$

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Percent Change & Percent Error

Percent change measures the relative increase or decrease of a quantity compared to its initial baseline value.

General Percent Change Formula

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

• 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.

(Critical Rule: The denominator MUST always be the Original (initial) value, never the new value!)

Worked Example 1: Calculating Percent Increase and Decrease

1. Percent Increase: A municipal transit authority raises the one-way bus fare from $\$2.50$ to $\$3.10$. Calculate the percent increase in fare. $\text{Percent Increase} = \frac{3.10 - 2.50}{2.50} \times 100\% = \frac{0.60}{2.50} \times 100\% = 0.24 \times 100\% = 24\%$
2. Percent Decrease: A digital camera originally retailing for $\$450$ is marked down on clearance to $\$324$. Calculate the percent discount. $\text{Percent Decrease} = \frac{324 - 450}{450} \times 100\% = \frac{-126}{450} \times 100\% = -0.28 \times 100\% = -28\% \; (28\%\text{ decrease})$

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Commercial Applications: Discounts, Tax, Tips, & Multipliers

In practical arithmetic, using decimal multipliers is significantly faster and less prone to calculation error than performing separate multi-step percentage additions or subtractions.

Decimal Multipliers for Common Transactions

• Discount of $d\%$: $\text{Sale Price} = \text{Original Price} \times (1 - d)$ (Example: $25\%$ off means you pay $1 - 0.25 = 0.75$ of original)
• Sales Tax of $t\%$: $\text{Total with Tax} = \text{Subtotal} \times (1 + t)$ (Example: $8.5\%$ tax means you pay $1 + 0.085 = 1.085$ of subtotal)
• Tip / Gratuity of $g\%$: $\text{Total with Tip} = \text{Meal Cost} \times (1 + g)$ (Example: $18\%$ tip means total paid is $1.18$ of meal cost)

Successive (Chained) Discounts

A pervasive trap on math exams involves successive percentage changes. When multiple discounts are applied in sequence, they are never additive because each subsequent percentage applies to the newly reduced intermediate price.

$\text{Final Multiplier} = (1 - d_1) \times (1 - d_2) \times \dots \times (1 - d_n)$ $\text{Effective Total Discount} = 1 - \text{Final Multiplier}$

Worked Example 2: Successive Discounts vs. Direct Sum

A designer winter jacket with an original retail price of $\$240$ is placed on a $30\%$ off clearance rack. During a weekend promotional sale, the store offers an additional $20\%$ discount off the clearance price. What is the final sale price of the jacket, and what is the effective total percentage discount?

1. Calculate intermediate clearance price (Multiplier $1 - 0.30 = 0.70$): $\text{Clearance Price} = \$240 \times 0.70 = \$168.00$
2. Apply the second $20\%$ discount (Multiplier $1 - 0.20 = 0.80$): $\text{Final Sale Price} = \$168.00 \times 0.80 = \$134.40$
3. Combine using overall chained multiplier: $\text{Net Multiplier} = 0.70 \times 0.80 = 0.56$ $\text{Final Price} = \$240 \times 0.56 = \$134.40$
4. Determine effective single discount rate: $\text{Effective Discount} = (1 - 0.56) \times 100\% = 44\%$

(Exam Trap Alert: The effective discount is $44\%$, NOT $30\% + 20\% = 50\%$. If a $50\%$ discount were applied, the price would be $\$120.00$, undercharging by $\$14.40$.)

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Retail Markup vs. Gross Profit Margin

In retail business arithmetic, students must distinguish between markup on cost and gross profit margin.

Worked Example 3: Markup vs. Margin

A boutique electronics retailer purchases wireless headphones from a wholesale distributor for $\$60.00$ per unit and applies a $65\%$ markup on cost. Calculate the retail selling price and the retailer's gross profit margin.

1. Calculate selling price with $65\%$ markup: $\text{Selling Price} = \text{Cost} \times (1 + 0.65) = \$60.00 \times 1.65 = \$99.00$
2. Determine gross dollar profit: $\text{Profit} = \$99.00 - \$60.00 = \$39.00$
3. Compute gross profit margin: $\text{Gross Margin} = \frac{\$39.00}{\$99.00} \times 100\% = 0.3939\dots \times 100\% \approx 39.39\%$

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Simple Interest ($I = Prt$)

Simple interest is interest calculated solely on the original principal amount deposited or borrowed, ignoring any interest compounding over time.

The Simple Interest Formula

$I = P \cdot r \cdot t$

• $I$ (Interest): The dollar amount of interest earned by an investor or paid by a borrower.
• $P$ (Principal): The original starting capital deposited or borrowed.
• $r$ (Annual Interest Rate): The stated annual interest rate expressed in decimal form (e.g., $5.25\% = 0.0525$).
• $t$ (Time in Years): The duration of the loan or investment expressed in years.

Converting Time Intervals to Years

• If time is given in months ($m$): $t = \frac{m}{12}$ (Example: $18\text{ months} = \frac{18}{12} = 1.5\text{ years}$; $\; 9\text{ months} = \frac{9}{12} = 0.75\text{ years}$)
• If time is given in days ($d$): $t = \frac{d}{365}$ (or $\frac{d}{360}$ if using ordinary interest conventions)

Total Accumulated Balance ($A$)

The total future value or repayment balance $A$ combines the original principal with the total interest earned:

$A = P + I = P + Prt = P(1 + rt)$

Worked Example 4: Calculating Simple Interest Loan Repayment

A small business owner secures a $\$12,000$ equipment loan with a simple annual interest rate of $6.5\%$ for a term of $42\text{ months}$. Calculate the total interest accrued over the life of the loan and the total amount required to settle the balance.

1. Convert duration $42\text{ months}$ to years: $t = \frac{42}{12} = 3.5\text{ years}$
2. Calculate total interest accrued ($I = Prt$): $I = \$12,000 \times 0.065 \times 3.5 = \$2,730.00$
3. Calculate total future repayment balance ($A = P + I$): $A = \$12,000 + \$2,730 = \$14,730.00$

Worked Example 5: Solving for the Unknown Interest Rate

An investor deposits $\$5,000$ into a guaranteed fixed-rate certificate of deposit. After $4\text{ years}$, the total balance in the account grows to $\$5,900$. What was the annual simple interest rate paid by the certificate?

1. Find total interest earned ($I = A - P$): $I = \$5,900 - \$5,000 = \$900$
2. Set up the simple interest formula to isolate $r$: $I = P \cdot r \cdot t$ $\$900 = \$5,000 \cdot r \cdot 4$ $900 = 20,000 \cdot r$
3. Solve for $r$ and convert to percent: $r = \frac{900}{20,000} = 0.045 = 4.5\%\text{ per year}$

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Common Pitfalls & ACCUPLACER Exam Traps

1. Base Confusion in Percent Increase/Decrease: Always divide the difference $\Delta$ by the original starting value. If an item drops from $\$100$ to $\$80$, the decrease is $\frac{20}{100} = 20\%$. If it rises back from $\$80$ to $\$100$, the increase is $\frac{20}{80} = 25\%$, NOT $20\%$.
2. Linear Addition of Successive Percentages: Remember that a $20\%$ increase followed by a $20\%$ decrease does NOT return you to the original value: $(1 + 0.20)(1 - 0.20) = 1.20 \times 0.80 = 0.96$ (a net $4\%$ loss).
3. Using Months Directly for $t$ in $I = Prt$: If a loan lasts $18\text{ months}$, plugging in $t = 18$ calculates interest for $18\text{ years}$! Always convert months to years ($t = \frac{18}{12} = 1.5$).
4. Applying Sales Tax to Discounted vs. Original Price: In retail problems, sales tax is almost universally assessed on the discounted subtotal, not the pre-sale original sticker price.