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2.1 Ratios, Proportions & Scale Conversions

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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

2.1 Ratios, Proportions & Scale Conversions

Key Takeaways Explain This Topic AI

A ratio compares two quantities, written as a:b, a/b, or 'a to b', and must be expressed in reduced terms.

Part-to-part ratios compare subsets (e.g., 3 recruits to 5 veterans), whereas part-to-whole ratios compare a subset to the entire group (e.g., 3 recruits to 8 total personnel).

The total-shares method solves ratio distribution problems by adding the ratio parts to find total shares, dividing the total quantity by total shares, and multiplying by individual parts.

A proportion equates two ratios (a/b = c/d) and is solved using cross-multiplication: a * d = b * c.

Map and blueprint scale conversions require setting up equivalent ratios while maintaining consistent measurement units (e.g., converting feet to inches before solving).

Last updated: July 2026

2.1 Ratios, Proportions & Scale Conversions

Arithmetic Reasoning (AR) on the Armed Services Vocational Aptitude Battery (ASVAB) / Armed Forces Qualification Test (AFQT) heavily emphasizes word problems involving ratios , proportions , and scale conversions . These concepts test your ability to convert written real-world relationships into mathematical equations and solve them under strict time constraints.

Understanding Ratios: Part-to-Part vs. Part-to-Whole

A ratio is a mathematical comparison of two numbers or quantities showing how many times one value contains another. Ratios can be written in three distinct formats:

  • Fraction form: a b \frac{a}{b} b a ​
  • Colon notation: a : b a : b a : b
  • Verbal statement: " a a a to b b b "
  • Distinguishing Part-to-Part and Part-to-Whole Ratios

The most common error on the AFQT is confusing a part-to-part ratio with a part-to-whole ratio.

Part-to-Part Ratio : Compares one subgroup to another subgroup within a set. Example : In a motor pool with 12 12 12 Humvees and 8 8 8 trucks, the ratio of Humvees to trucks is 12 : 8 12:8 12 : 8 , which simplifies to 3 : 2 3:2 3 : 2 .

Part-to-Whole Ratio : Compares one subgroup to the total set. Example : In the same motor pool of 20 20 20 total vehicles ( 12 + 8 = 20 12 + 8 = 20 12 + 8 = 20 ), the ratio of Humvees to total vehicles is 12 : 20 12:20 12 : 20 , which simplifies to 3 : 5 3:5 3 : 5 .

  • Ratio Description Unsimplified Form Simplified Form Fraction Equivalent
  • Humvees to Trucks (Part-to-Part) 12 : 8 12 : 8 12 : 8 3 : 2 3 : 2 3 : 2 3 2 \frac{3}{2} 2 3 ​
  • Trucks to Humvees (Part-to-Part) 8 : 12 8 : 12 8 : 12 2 : 3 2 : 3 2 : 3 2 3 \frac{2}{3} 3 2 ​
  • Humvees to Total (Part-to-Whole) 12 : 20 12 : 20 12 : 20 3 : 5 3 : 5 3 : 5 3 5 \frac{3}{5} 5 3 ​
  • Trucks to Total (Part-to-Whole) 8 : 20 8 : 20 8 : 20 2 : 5 2 : 5 2 : 5 2 5 \frac{2}{5} 5 2 ​

Key Exam Tip : Always check whether the question asks for a ratio comparing two components against each other or a component against the total quantity!

The Total-Shares Method for Multi-Part Ratios

When a word problem distributes a total quantity among multiple entities based on a given ratio (such as 2 : 3 : 5 2:3:5 2 : 3 : 5 ), use the Total-Shares Method .

4-Step Total-Shares Algorithm

Add the ratio numbers to find the total number of equal shares. Total Shares = a + b + c \text{Total Shares} = a + b + c Total Shares = a + b + c

Divide the total quantity by the total number of shares to calculate the value of one single share ( x x x ). Value per Share ( x ) = Total Quantity Total Shares \text{Value per Share } (x) = \frac{\text{Total Quantity}}{\text{Total Shares}} Value per Share ( x ) = Total Shares Total Quantity ​

Multiply the value of one share by each individual ratio part to find the specific amounts. Part A = a ⋅ x , Part B = b ⋅ x , Part C = c ⋅ x \text{Part } A = a \cdot x, \quad \text{Part } B = b \cdot x, \quad \text{Part } C = c \cdot x Part A = a ⋅ x , Part B = b ⋅ x , Part C = c ⋅ x

Verify your results by adding the calculated parts together to ensure they sum to the original total.

Setting Up and Solving Proportions

A proportion is an equation stating that two ratios are equal: a b = c d \frac{a}{b} = \frac{c}{d} b a ​ = d c ​

The Cross-Multiplication Rule

For any valid proportion, the product of the means equals the product of the extremes: a ⋅ d = b ⋅ c a \cdot d = b \cdot c a ⋅ d = b ⋅ c

To solve for an unknown quantity x x x when three values are known: a b = x d ⟹ x = a ⋅ d b \frac{a}{b} = \frac{x}{d} \implies x = \frac{a \cdot d}{b} b a ​ = d x ​ ⟹ x = b a ⋅ d ​

Direct Proportions in Word Problems

In direct proportions, as one quantity increases, the other increases at a constant rate. Example : If 4 4 4 tactical radios require 14 14 14 batteries, how many batteries are required for 10 10 10 tactical radios? 4 radios 14 batteries = 10 radios x batteries \frac{4 \text{ radios}}{14 \text{ batteries}} = \frac{10 \text{ radios}}{x \text{ batteries}} 14 batteries 4 radios ​ = x batteries 10 radios ​ 4 x = 140 ⟹ x = 35 batteries 4x = 140 \implies x = 35 \text{ batteries} 4 x = 140 ⟹ x = 35 batteries

Scale Conversions for Maps and Blueprints

Scale problems present a ratio between map/blueprint measurements and real-world physical distances. These problems test your ability to convert units before setting up proportions.

Unit Conversion Reference Table

  • Unit Relationship Equivalent Factor
  • Feet to Inches 1 foot = 12 inches 1 \text{ foot} = 12 \text{ inches} 1 foot = 12 inches
  • Yards to Feet 1 yard = 3 feet 1 \text{ yard} = 3 \text{ feet} 1 yard = 3 feet
  • Miles to Feet 1 mile = 5 , 280 feet 1 \text{ mile} = 5,280 \text{ feet} 1 mile = 5 , 280 feet

Meters to Centimeters 1 meter = 100 centimeters 1 \text{ meter} = 100 \text{ centimeters} 1 meter = 100 centimeters

Kilometers to Meters 1 kilometer = 1 , 000 meters 1 \text{ kilometer} = 1,000 \text{ meters} 1 kilometer = 1 , 000 meters

Solving Scale Problems Step-by-Step

When given a scale such as 1 4 inch = 5 miles \frac{1}{4} \text{ inch} = 5 \text{ miles} 4 1 ​ inch = 5 miles :

Express the scale as a ratio: Scale Inches Actual Miles = 0.25 5 \frac{\text{Scale Inches}}{\text{Actual Miles}} = \frac{0.25}{5} Actual Miles Scale Inches ​ = 5 0.25 ​ .

Set up a proportion matching the scale ratio to the problem parameters.

Solve for the missing variable using cross-multiplication.

  • Step-by-Step Worked AFQT Examples
  • Worked Example 1: Multi-Part Ratio Distribution

Problem : A supply sergeant needs to distribute 1 , 400 1,400 1 , 400 rounds of ammunition among three squads in the ratio 2 : 3 : 5 2 : 3 : 5 2 : 3 : 5 . How many rounds does the squad with the largest share receive?

Step 1 : Find total shares. Total Shares = 2 + 3 + 5 = 10 \text{Total Shares} = 2 + 3 + 5 = 10 Total Shares = 2 + 3 + 5 = 10

Step 2 : Calculate value per share. Value per Share = 1 , 400 rounds 10 shares = 140 rounds/share \text{Value per Share} = \frac{1,400 \text{ rounds}}{10 \text{ shares}} = 140 \text{ rounds/share} Value per Share = 10 shares 1 , 400 rounds ​ = 140 rounds/share

Step 3 : Identify largest share ratio part ( 5 5 5 ) and multiply. Largest Share = 5 × 140 = 700 rounds \text{Largest Share} = 5 \times 140 = 700 \text{ rounds} Largest Share = 5 × 140 = 700 rounds

Step 4 : Verify all shares sum to 1 , 400 1,400 1 , 400 . 2 ( 140 ) + 3 ( 140 ) + 5 ( 140 ) = 280 + 420 + 700 = 1 , 400 rounds. (Correct) 2(140) + 3(140) + 5(140) = 280 + 420 + 700 = 1,400 \text{ rounds. (Correct)} 2 ( 140 ) + 3 ( 140 ) + 5 ( 140 ) = 280 + 420 + 700 = 1 , 400 rounds. (Correct)

Worked Example 2: Part-to-Part to Part-to-Whole Conversion

Problem : In an infantry platoon, the ratio of officers to enlisted personnel is 2 : 15 2 : 15 2 : 15 . If the total platoon strength is 68 68 68 personnel, how many enlisted personnel are in the platoon?

Step 1 : Identify the ratio components. Ratio of officers to enlisted is 2 : 15 2:15 2 : 15 .

Step 2 : Determine the part-to-whole relationship for enlisted personnel. Total Ratio Parts = 2 + 15 = 17 \text{Total Ratio Parts} = 2 + 15 = 17 Total Ratio Parts = 2 + 15 = 17 Enlisted Fraction = 15 17 \text{Enlisted Fraction} = \frac{15}{17} Enlisted Fraction = 17 15 ​

Step 3 : Multiply total personnel by the enlisted fraction. Enlisted Personnel = 15 17 × 68 = 15 × 4 = 60 \text{Enlisted Personnel} = \frac{15}{17} \times 68 = 15 \times 4 = 60 Enlisted Personnel = 17 15 ​ × 68 = 15 × 4 = 60

Worked Example 3: Map Scale Conversion

Problem : On a military tactical map, a scale of 1 2 inch = 15 miles \frac{1}{2} \text{ inch} = 15 \text{ miles} 2 1 ​ inch = 15 miles is used. If two outposts are 3.5 inches 3.5 \text{ inches} 3.5 inches apart on the map, what is the actual physical distance between them in miles?

Step 1 : Set up a proportion comparing map inches to actual miles. 0.5 in. 15 miles = 3.5 in. x miles \frac{0.5 \text{ in.}}{15 \text{ miles}} = \frac{3.5 \text{ in.}}{x \text{ miles}} 15 miles 0.5 in. ​ = x miles 3.5 in. ​

Step 2 : Cross-multiply to solve for x x x . 0.5 ⋅ x = 15 ⋅ 3.5 0.5 \cdot x = 15 \cdot 3.5 0.5 ⋅ x = 15 ⋅ 3.5 0.5 x = 52.5 0.5x = 52.5 0.5 x = 52.5

Step 3 : Divide by 0.5 0.5 0.5 (or multiply by 2 2 2 ). x = 52.5 0.5 = 105 miles x = \frac{52.5}{0.5} = 105 \text{ miles} x = 0.5 52.5 ​ = 105 miles

Common AFQT Traps & Shortcut Strategies

The Inverted Ratio Trap : Always double-check which order the ratio asks for! If a question asks for the ratio of rejections to acceptances , writing acceptances to rejections will lead to a distractor choice.

The Fractional Scale Shortcut : When multiplying by scale fractions like 1 4 in = 10 miles \frac{1}{4} \text{ in} = 10 \text{ miles} 4 1 ​ in = 10 miles , remember that 1 in = 40 miles 1 \text{ in} = 40 \text{ miles} 1 in = 40 miles . Multiplying the scale factor by 4 4 4 immediately simplifies your arithmetic.