2.2 Percentages, Discounts, Interest & Tax Calculations

2.2 Percentages, Discounts, Interest & Tax Calculations

Percentage word problems represent one of the highest-yield content areas on the AFQT Arithmetic Reasoning subtest. Mastery of percentage conversions, price changes, successive discounts, simple interest, and sales tax allows test-takers to solve complex multi-step financial problems quickly.

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The Core Percentage Formula: Part, Whole, and Rate

A percent literally means "per hundred" (/100). Every percentage calculation involves three fundamental components:

1. Base (Whole): The original total quantity ($100%$).
2. Rate (Percent): The ratio or percentage being applied ($r%$ or decimal $r$).
3. Part (Amount): The portion of the base resulting from the rate.

Fundamental Percentage Equations

$\text{Part} = \text{Percent (in decimal form)} \times \text{Whole}$ $\frac{\text{Is}}{\text{Of}} = \frac{\text{Percent}}{100}$ $\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\%$

Common Percent-to-Decimal-to-Fraction Conversions

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Percent Increase, Percent Decrease, and Markups

To calculate the percentage change between an old value and a new value, use the Percent Change Formula: $\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$

CRITICAL RULE: The denominator MUST always be the original (starting) value, never the new or updated value!

Shortcut Multiplier Method

Instead of calculating the change amount and adding/subtracting it in two separate steps, use a single decimal multiplier:

• Percent Increase / Markup: $\text{New Value} = \text{Original Value} \times (1 + r)$ Example: A $15%$ increase on $$80 \implies 80 \times 1.15 = $92$.
• Percent Decrease / Discount: $\text{New Value} = \text{Original Value} \times (1 - r)$ Example: A $20%$ discount on $$80 \implies 80 \times 0.80 = $64$.

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Successive Discounts and Combined Reductions

One of the most common AFQT trap questions involves successive (sequential) discounts (e.g., "Take an extra $20%$ off an item already marked down by $30%").

The Common Fallacy

Adding the percentages directly ($20% + 30% = 50%$) is WRONG. The second discount applies only to the already reduced price, not the original price.

Mathematical Formula for Successive Discounts

$\text{Final Price} = \text{Original Price} \times (1 - d_1) \times (1 - d_2)$ $\text{Effective Overall Discount} = 1 - [(1 - d_1) \times (1 - d_2)]$

Example: For an item costing $$100$ discounted by $30%$ and then an additional $20%$: $\text{First reduction } (30\%): \$100 \times (1 - 0.30) = \$70$ $\text{Second reduction } (20\%): \$70 \times (1 - 0.20) = \$56$ $\text{Effective Discount} = \frac{\$100 - \$56}{\$100} = 44\% \quad (\text{NOT } 50\%!)$

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Simple Interest Calculations ($I = Prt$)

Simple interest problems on the AFQT evaluate interest earned or charged over a specified timeframe without compounding.

The Simple Interest Formula

$I = P \cdot r \cdot t$

Where:

• $I$ = Interest amount ($)
• $P$ = Principal amount (original sum borrowed or invested)
• $r$ = Annual interest rate (expressed as a decimal, e.g., $6% = 0.06$)
• $t$ = Time (expressed in years)

Time Conversion Rules for $t$

• If time is given in months: $t = \frac{\text{months}}{12}$
• If time is given in days: $t = \frac{\text{days}}{365}$ (or $360$ if banking standard is specified)
• Total Future Balance ($A$): $A = P + I = P(1 + rt)$

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Sales Tax and Total Cost Calculations

When buying equipment or supplies, sales tax and shipping fees are calculated on the net selling price (after discounts have been subtracted).

$\text{Net Price} = \text{List Price} \times (1 - \text{Discount Rate})$ $\text{Sales Tax} = \text{Net Price} \times \text{Tax Rate}$ $\text{Total Cost} = \text{Net Price} + \text{Sales Tax} = \text{Net Price} \times (1 + \text{Tax Rate})$

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Step-by-Step Worked AFQT Examples

Worked Example 1: Percent Change with Base Value

Problem: A military surplus store purchased tactical backpacks for $$40$ each and sold them for $$66$ each. What was the percentage markup based on the cost price?

• Step 1: Identify original cost ($$40$) and new price ($$66$).
• Step 2: Calculate profit/markup amount. $\text{Markup Amount} = \$66 - \$40 = \$26$
• Step 3: Divide by the original cost price. $\text{Percent Markup} = \frac{\$26}{\$40} = 0.65$
• Step 4: Convert decimal to percentage: $0.65 \times 100% = 65%$.

Worked Example 2: Successive Discounts

Problem: A generator listed at $$800$ is on clearance at $25%$ off. A military veteran receives an additional $10%$ discount off the clearance price. How much does the veteran pay before tax?

• Step 1: Calculate price after first discount ($25% \implies 0.75$ multiplier). $\text{Clearance Price} = \$800 \times 0.75 = \$600$
• Step 2: Apply second discount ($10% \implies 0.90$ multiplier) to the clearance price. $\text{Final Price} = \$600 \times 0.90 = \$540$
• Alternative Single Multiplier Method: $\text{Final Price} = \$800 \times (1 - 0.25) \times (1 - 0.10) = \$800 \times 0.75 \times 0.90 = \$540$

Worked Example 3: Simple Interest over Months

Problem: A soldier borrows $$2,500$ from a credit union at a simple annual interest rate of $8%$. If the loan is paid off in full after $18$ months, what is the total amount repaid?

• Step 1: Identify $P = 2,500$, $r = 0.08$, and convert months to years: $t = \frac{18}{12} = 1.5 \text{ years}$.
• Step 2: Calculate interest amount $I$. $I = 2,500 \times 0.08 \times 1.5 = 200 \times 1.5 = \$300$
• Step 3: Add interest to principal to find total repayment. $\text{Total Repaid} = P + I = \$2,500 + \$300 = \$2,800$

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Common AFQT Traps & Shortcut Strategies

1. The Wrong Base Trap: Always divide by the starting value when computing percentage changes. If a price goes from $$50$ to $$40$, the decrease is $\frac{10}{50} = 20%$, NOT $\frac{10}{40} = 25%$.
2. The Month-to-Year Conversion Trap: In $I = Prt$, entering $t = 18$ instead of $t = 1.5$ for an 18-month loan will result in an answer $12$ times too large!