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2.2 Percentages, Discounts, Interest & Tax Calculations

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Armed Forces Qualification Test AFQT

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1 Chapter 1: AFQT Structure, Scoring & Enlistment Requirements

1.1 AFQT Overview, Subtest Breakdown & CAT-ASVAB vs. P&P Format 1.2 The AFQT Scoring Formula (2×VE + AR + MK) & Percentile Norming 1.3 Branch Minimum Scores, AFQT Categories & Retake Policies

2 Chapter 2: Arithmetic Reasoning (AR) — Ratios, Rates & Percentages

2.1 Ratios, Proportions & Scale Conversions 2.2 Percentages, Discounts, Interest & Tax Calculations 2.3 Rate, Distance, Time & Speed Word Problems 2.4 Work Rates & Combined Labor Problems

3 Chapter 3: Arithmetic Reasoning (AR) — Mixtures, Sequences & Applied Math

3.1 Mixture, Concentration & Price-Blending Problems 3.2 Sequences, Patterns & Series Calculations 3.3 Averages, Weighted Averages & Statistical Word Problems

4 Chapter 4: Word Knowledge (WK) — Vocabulary & Word Structure

4.1 High-Frequency AFQT Vocabulary & Synonym Recognition 4.2 Latin & Greek Roots, Prefixes & Suffixes for Word Decoding 4.3 Context Clues & Words with Multiple Meanings

5 Chapter 5: Paragraph Comprehension (PC) — Reading Analysis & Inference

5.1 Identifying Main Ideas & Primary Purpose 5.2 Locating Supporting Details & Direct Facts 5.3 Drawing Logical Inferences & Implied Conclusions 5.4 Analyzing Tone, Author Perspective & Contextual Vocabulary

6 Chapter 6: Mathematics Knowledge (MK) — Fundamentals & Linear Algebra

6.1 Arithmetic Operations, Order of Operations & Number Properties 6.2 Exponents, Scientific Notation & Square Roots / Radicals 6.3 Linear Equations, Inequalities & Absolute Value 6.4 Systems of Linear Equations & Word-to-Algebra Translation

7 Chapter 7: Mathematics Knowledge (MK) — Advanced Algebra & Functions

7.1 Polynomial Operations & Factoring Techniques 7.2 Quadratic Equations, Factoring & Quadratic Formula 7.3 Coordinate Geometry: Slope, Distance, Midpoint & Linear Graphs

8 Chapter 8: Mathematics Knowledge (MK) — Geometry & Spatial Math

8.1 Angles, Triangles & Pythagorean Theorem 8.2 Quadrilaterals, Polygons & Circle Geometry 8.3 3D Geometry: Surface Area & Volume of Solids 8.4 Basic Geometric Transformations & Trigonometric Ratios

2.2 Percentages, Discounts, Interest & Tax Calculations

Key Takeaways Explain This Topic AI

The core percentage equation is Part = Percent * Whole (or Is/Of = %/100).

Percent change equals the absolute difference divided by the original base value: |New - Old| / Old * 100%.

Successive discounts cannot be added directly; they must be applied sequentially using multiplier factors: Final Price = Original Price * (1 - d1) * (1 - d2).

Simple interest uses the formula I = P * r * t, where t MUST be expressed in years (e.g., 6 months = 0.5 years).

Total cost including tax and markup is calculated as Original Price * (1 + markup rate) * (1 + tax rate).

Last updated: July 2026

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.

The Core Percentage Formula: Part, Whole, and Rate

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

Base (Whole) : The original total quantity ( 100 % 100\% 100% ).

Rate (Percent) : The ratio or percentage being applied ( r % r\% r % or decimal r r r ).

Part (Amount) : The portion of the base resulting from the rate.

Fundamental Percentage Equations Part = Percent (in decimal form) × Whole \text{Part} = \text{Percent (in decimal form)} \times \text{Whole} Part = Percent (in decimal form) × Whole Is Of = Percent 100 \frac{\text{Is}}{\text{Of}} = \frac{\text{Percent}}{100} Of Is ​ = 100 Percent ​ Percent = Part Whole × 100 % \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\% Percent = Whole Part ​ × 100%

Common Percent-to-Decimal-to-Fraction Conversions

  • Percentage Decimal Equivalent Simplified Fraction
  • 10 % 10\% 10% 0.10 0.10 0.10 1 10 \frac{1}{10} 10 1 ​
  • 12.5 % 12.5\% 12.5% 0.125 0.125 0.125 1 8 \frac{1}{8} 8 1 ​
  • 20 % 20\% 20% 0.20 0.20 0.20 1 5 \frac{1}{5} 5 1 ​
  • 25 % 25\% 25% 0.25 0.25 0.25 1 4 \frac{1}{4} 4 1 ​

33.33 % 33.33\% 33.33% ( 33 1 3 % 33 \frac{1}{3}\% 33 3 1 ​ % ) 0.333... 0.333... 0.333... 1 3 \frac{1}{3} 3 1 ​

  • 37.5 % 37.5\% 37.5% 0.375 0.375 0.375 3 8 \frac{3}{8} 8 3 ​
  • 50 % 50\% 50% 0.50 0.50 0.50 1 2 \frac{1}{2} 2 1 ​

66.67 % 66.67\% 66.67% ( 66 2 3 % 66 \frac{2}{3}\% 66 3 2 ​ % ) 0.666... 0.666... 0.666... 2 3 \frac{2}{3} 3 2 ​

  • 75 % 75\% 75% 0.75 0.75 0.75 3 4 \frac{3}{4} 4 3 ​
  • Percent Increase, Percent Decrease, and Markups

To calculate the percentage change between an old value and a new value, use the Percent Change Formula : Percent Change = ∣ New Value − Original Value ∣ Original Value × 100 % \text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\% Percent Change = Original Value ∣ New Value − Original Value ∣ ​ × 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 : New Value = Original Value × ( 1 + r ) \text{New Value} = \text{Original Value} \times (1 + r) New Value = Original Value × ( 1 + r ) Example : A 15 % 15\% 15% increase on $ 80 ⟹ 80 × 1.15 = $ 92 \text{\textdollar}80 \implies 80 \times 1.15 = \text{\textdollar}92 $ 80 ⟹ 80 × 1.15 = $ 92 .

Percent Decrease / Discount : New Value = Original Value × ( 1 − r ) \text{New Value} = \text{Original Value} \times (1 - r) New Value = Original Value × ( 1 − r ) Example : A 20 % 20\% 20% discount on $ 80 ⟹ 80 × 0.80 = $ 64 \text{\textdollar}80 \implies 80 \times 0.80 = \text{\textdollar}64 $ 80 ⟹ 80 × 0.80 = $ 64 .

Successive Discounts and Combined Reductions

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

The Common Fallacy

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

Mathematical Formula for Successive Discounts Final Price = Original Price × ( 1 − d 1 ) × ( 1 − d 2 ) \text{Final Price} = \text{Original Price} \times (1 - d_1) \times (1 - d_2) Final Price = Original Price × ( 1 − d 1 ​ ) × ( 1 − d 2 ​ ) Effective Overall Discount = 1 − [ ( 1 − d 1 ) × ( 1 − d 2 ) ] \text{Effective Overall Discount} = 1 - [(1 - d_1) \times (1 - d_2)] Effective Overall Discount = 1 − [( 1 − d 1 ​ ) × ( 1 − d 2 ​ )]

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

Simple Interest Calculations ( I = P r t I = Prt I = P r t )

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

The Simple Interest Formula I = P ⋅ r ⋅ t I = P \cdot r \cdot t I = P ⋅ r ⋅ t

Where:

  • I I I = Interest amount ($)
  • P P P = Principal amount (original sum borrowed or invested)
  • r r r = Annual interest rate (expressed as a decimal, e.g., 6 % = 0.06 6\% = 0.06 6% = 0.06 )
  • t t t = Time (expressed in years )
  • Time Conversion Rules for t t t
  • If time is given in months : t = months 12 t = \frac{\text{months}}{12} t = 12 months ​

If time is given in days : t = days 365 t = \frac{\text{days}}{365} t = 365 days ​ (or 360 360 360 if banking standard is specified)

  • Total Future Balance ( A A A ): A = P + I = P ( 1 + r t ) A = P + I = P(1 + rt) A = P + I = P ( 1 + r t )
  • 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). Net Price = List Price × ( 1 − Discount Rate ) \text{Net Price} = \text{List Price} \times (1 - \text{Discount Rate}) Net Price = List Price × ( 1 − Discount Rate ) Sales Tax = Net Price × Tax Rate \text{Sales Tax} = \text{Net Price} \times \text{Tax Rate} Sales Tax = Net Price × Tax Rate Total Cost = Net Price + Sales Tax = Net Price × ( 1 + Tax Rate ) \text{Total Cost} = \text{Net Price} + \text{Sales Tax} = \text{Net Price} \times (1 + \text{Tax Rate}) Total Cost = Net Price + Sales Tax = Net Price × ( 1 + Tax Rate )

  • Step-by-Step Worked AFQT Examples
  • Worked Example 1: Percent Change with Base Value

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

Step 1 : Identify original cost ( $ 40 \text{\textdollar}40 $ 40 ) and new price ( $ 66 \text{\textdollar}66 $ 66 ).

Step 2 : Calculate profit/markup amount. Markup Amount = $ 66 − $ 40 = $ 26 \text{Markup Amount} = \$66 - \$40 = \$26 Markup Amount = $66 − $40 = $26

Step 3 : Divide by the original cost price. Percent Markup = $ 26 $ 40 = 0.65 \text{Percent Markup} = \frac{\$26}{\$40} = 0.65 Percent Markup = $40 $26 ​ = 0.65

Step 4 : Convert decimal to percentage: 0.65 × 100 % = 65 % 0.65 \times 100\% = 65\% 0.65 × 100% = 65% .

Worked Example 2: Successive Discounts

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

Step 1 : Calculate price after first discount ( 25 % ⟹ 0.75 25\% \implies 0.75 25% ⟹ 0.75 multiplier). Clearance Price = $ 800 × 0.75 = $ 600 \text{Clearance Price} = \$800 \times 0.75 = \$600 Clearance Price = $800 × 0.75 = $600

Step 2 : Apply second discount ( 10 % ⟹ 0.90 10\% \implies 0.90 10% ⟹ 0.90 multiplier) to the clearance price. Final Price = $ 600 × 0.90 = $ 540 \text{Final Price} = \$600 \times 0.90 = \$540 Final Price = $600 × 0.90 = $540

Alternative Single Multiplier Method : Final Price = $ 800 × ( 1 − 0.25 ) × ( 1 − 0.10 ) = $ 800 × 0.75 × 0.90 = $ 540 \text{Final Price} = \$800 \times (1 - 0.25) \times (1 - 0.10) = \$800 \times 0.75 \times 0.90 = \$540 Final Price = $800 × ( 1 − 0.25 ) × ( 1 − 0.10 ) = $800 × 0.75 × 0.90 = $540

Worked Example 3: Simple Interest over Months

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

Step 1 : Identify P = 2 , 500 P = 2,500 P = 2 , 500 , r = 0.08 r = 0.08 r = 0.08 , and convert months to years: t = 18 12 = 1.5 years t = \frac{18}{12} = 1.5 \text{ years} t = 12 18 ​ = 1.5 years .

Step 2 : Calculate interest amount I I I . I = 2 , 500 × 0.08 × 1.5 = 200 × 1.5 = $ 300 I = 2,500 \times 0.08 \times 1.5 = 200 \times 1.5 = \$300 I = 2 , 500 × 0.08 × 1.5 = 200 × 1.5 = $300

Step 3 : Add interest to principal to find total repayment. Total Repaid = P + I = $ 2 , 500 + $ 300 = $ 2 , 800 \text{Total Repaid} = P + I = \$2,500 + \$300 = \$2,800 Total Repaid = P + I = $2 , 500 + $300 = $2 , 800

Common AFQT Traps & Shortcut Strategies

The Wrong Base Trap : Always divide by the starting value when computing percentage changes. If a price goes from $ 50 \text{\textdollar}50 $ 50 to $ 40 \text{\textdollar}40 $ 40 , the decrease is 10 50 = 20 % \frac{10}{50} = 20\% 50 10 ​ = 20% , NOT 10 40 = 25 % \frac{10}{40} = 25\% 40 10 ​ = 25% .

The Month-to-Year Conversion Trap : In I = P r t I = Prt I = P r t , entering t = 18 t = 18 t = 18 instead of t = 1.5 t = 1.5 t = 1.5 for an 18-month loan will result in an answer 12 12 12 times too large!