3.1 Word Problems: Rates, Ratios, Proportions & Percentages

Armed Forces Classification Test AFCT

Track your progress as you read

Finished sections get a check mark.

1 Chapter 1: Guide Scope, AFQT & Service-Specific Policy

1.1 Editorial Scope and Service-Specific AFCT Policy 1.2 AFQT Meaning and Study Priorities 1.3 Service Composites and Targeted Skill Planning 1.4 Format-Agnostic Test Preparation and Pacing

2 Chapter 2: General Science (GS)

2.1 Life Science: Cell Biology, Human Anatomy & Genetics 2.2 Physical Science: Matter, Energy, Forces & Motion 2.3 Earth & Space Science: Geology, Meteorology & Astronomy 2.4 Chemistry Fundamentals: Atomic Structure, Elements & Chemical Reactions

3 Chapter 3: Arithmetic Reasoning (AR)

3.1 Word Problems: Rates, Ratios, Proportions & Percentages 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time 3.3 Work & Mixture Problems: Joint Rates & Percentage Concentrations 3.4 Multi-Step Problem Solving & Algebraic Word Problems

4 Chapter 4: Word Knowledge (WK)

4.1 General Academic & Technical Vocabulary 4.2 Context Clues, Synonyms & Antonyms 4.3 Word Roots, Prefixes & Suffixes (Etymology) 4.4 Advanced Technical & Operational Terminology

5 Chapter 5: Paragraph Comprehension (PC)

5.1 Identifying Main Ideas, Central Themes & Topic Sentences 5.2 Extracting Supporting Details & Factual Information 5.3 Drawing Logical Inferences & Author's Purpose 5.4 Analyzing Text Structure & Drawing Conclusions

6 Chapter 6: Mathematics Knowledge (MK)

6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations 6.2 High School Algebra: Linear Equations, Inequalities & Polynomials 6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles 6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

7 Chapter 7: Electronics Information (EI)

7.1 Electrical Principles: Voltage, Current, Resistance & Ohm's Law (V=IR, P=VI) 7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis 7.3 Electrical Components: Resistors, Capacitors, Inductors & Transformers 7.4 Electronic Semiconductors, Transistors, Diodes & Logic Gates

8 Chapter 8: Auto & Shop Information (AS)

8.1 Automotive Systems: Internal Combustion Engines, Fuel & Ignition Systems 8.2 Automotive Mechanical Systems: Transmission, Suspension, Brakes & Cooling 8.3 Hand & Power Tools: Identification, Uses & Fasteners 8.4 Shop Practice, Woodworking, Metalworking & Safety Rules

9 Chapter 9: Mechanical Comprehension (MC)

9.1 Simple Machines: Levers, Pulleys, Inclined Planes & Mechanical Advantage 9.2 Force, Motion, Work & Power: Newton's Laws & Energy Transformations 9.3 Fluid Dynamics & Pressure: Pascal's Principle, Hydraulics & Pneumatics 9.4 Mechanical Components: Gears, Belts, Springs & Friction

10 Chapter 10: Spatial Assembly Practice (AO Skill)

10.1 Spatial Connections and Assembly Patterns

Free exam prep AFCT Armed Forces Classification Test Practice Questions 100 questions Study Guide 37 topics Cheat Sheet 58 facts Flashcards 50 cards

3.1 Word Problems: Rates, Ratios, Proportions & Percentages

Key Takeaways Explain This Topic AI

Ratios express a relative comparison between two quantities (a:b or a/b), while rates compare quantities measured in different units (e.g., miles per hour, dollars per gallon).

Proportions state that two ratios are equal (a/b = c/d); solve for missing variables using cross-multiplication (a * d = b * c).

In direct proportions, both quantities increase or decrease together (y = kx); in inverse proportions, as one quantity increases, the other decreases proportionally (x * y = k).

Percentage changes are calculated as |New - Old| / Old * 100%. Always divide by the original (starting) value, never the new value.

Unit conversion in rate problems requires chain-link dimensional analysis to systematically cancel out unwanted units. Last updated: July 2026

3.1 Word Problems: Rates, Ratios, Proportions & Percentages The Arithmetic Reasoning (AR) subtest on the ASVAB-style classification practice evaluates your ability to analyze practical word problems, translate verbal statements into mathematical expressions, and solve them accurately. Rates, ratios, proportions, and percentages are useful arithmetic-reasoning practice topics. 1. Ratios and Ratio Scaling A ratio is a comparison of two quantities by division. Ratios can be written in three forms: as a fraction ( a b \frac{a}{b} b a ), with a colon ( a : b a:b a : b ), or using the word "to" ( a to b a\text{ to }b a to b ). Part-to-Part vs. Part-to-Whole Ratios Understanding whether a ratio represents a part-to-part relationship or a part-to-whole relationship is essential: Part-to-Part: Compares one subgroup to another subgroup. For example, if a unit has 15 officers and 45 enlisted personnel, the ratio of officers to enlisted is 15 : 45 15:45 15 : 45 , which simplifies to 1 : 3 1:3 1 : 3 . Part-to-Whole: Compares a subgroup to the total. In the same unit, the total personnel is 15 + 45 = 60 15 + 45 = 60 15 + 45 = 60 . The ratio of officers to total personnel is 15 : 60 = 1 : 4 15:60 = 1:4 15 : 60 = 1 : 4 . The Ratio Box / Total Parts Method When a total quantity must be divided according to a given ratio a : b : c a : b : c a : b : c : Add the ratio terms together to find the total number of equal parts : Total Parts = a + b + c \text{Total Parts} = a + b + c Total Parts = a + b + c . Divide the total quantity by the total number of parts to determine the value of one part : Value per Part = Total Quantity Total Parts \text{Value per Part} = \frac{\text{Total Quantity}}{\text{Total Parts}} Value per Part = Total Parts Total Quantity . Multiply each ratio term by the value of one part to find the specific quantity for each category. Step-by-Step Worked Example 1: A tactical supply unit receives a shipment of 450 cases of ammunition distributed among caliber types Alpha, Bravo, and Charlie in the ratio 2 : 4 : 3 2 : 4 : 3 2 : 4 : 3 . How many cases of Bravo ammunition were received? Step 1: Calculate total ratio parts: 2 + 4 + 3 = 9 parts 2 + 4 + 3 = 9\text{ parts} 2 + 4 + 3 = 9 parts . Step 2: Find value per part: 450 cases 9 parts = 50 cases per part \frac{450\text{ cases}}{9\text{ parts}} = 50\text{ cases per part} 9 parts 450 cases = 50 cases per part . Step 3: Calculate Bravo cases ( 4 parts 4\text{ parts} 4 parts ): 4 × 50 = 200 cases 4 \times 50 = 200\text{ cases} 4 × 50 = 200 cases . 2. Rates, Unit Rates & Dimensional Analysis A rate is a ratio that compares two quantities measured in different units (such as miles per hour, dollars per gallon, or fuel consumption per engine hour). A unit rate is a rate expressed with a denominator of 1. Dimensional Analysis (Unit Conversion) To convert rates across different measurement units, multiply by conversion factors structured as fractions equal to 1, ensuring unwanted units cancel out diagonally. Target Rate = Given Rate × ( Unit B Unit A ) × ( Unit C Unit B ) \text{Target Rate} = \text{Given Rate} \times \left( \frac{\text{Unit } B}{\text{Unit } A} \right) \times \left( \frac{\text{Unit } C}{\text{Unit } B} \right) Target Rate = Given Rate × ( Unit A Unit B ) × ( Unit B Unit C )

Conversion Metric Equivalent Relationship Length 1 mile = 5 , 280 feet = 1 , 760 yards 1\text{ mile} = 5,280\text{ feet} = 1,760\text{ yards} 1 mile = 5 , 280 feet = 1 , 760 yards Time 1 hour = 60 minutes = 3 , 600 seconds 1\text{ hour} = 60\text{ minutes} = 3,600\text{ seconds} 1 hour = 60 minutes = 3 , 600 seconds Volume 1 gallon = 4 quarts = 8 pints 1\text{ gallon} = 4\text{ quarts} = 8\text{ pints} 1 gallon = 4 quarts = 8 pints Mass / Weight 1 ton = 2 , 000 pounds = 32 , 000 ounces 1\text{ ton} = 2,000\text{ pounds} = 32,000\text{ ounces} 1 ton = 2 , 000 pounds = 32 , 000 ounces Step-by-Step Worked Example 2: A generator consumes diesel fuel at a rate of 12 quarts per hour. If fuel costs $4.00 per gallon, what is the operational fuel cost of running the generator for an 8-hour shift? Step 1: Find total quarts consumed in 8 hours: 12 quarts/hr × 8 hrs = 96 quarts 12\text{ quarts/hr} \times 8\text{ hrs} = 96\text{ quarts} 12 quarts/hr × 8 hrs = 96 quarts . Step 2: Convert quarts to gallons ( 4 quarts = 1 gallon 4\text{ quarts} = 1\text{ gallon} 4 quarts = 1 gallon ): 96 quarts 4 quarts/gal = 24 gallons \frac{96\text{ quarts}}{4\text{ quarts/gal}} = 24\text{ gallons} 4 quarts/gal 96 quarts = 24 gallons . Step 3: Multiply by cost per gallon: 24 gallons × $ 4.00 = $ 96.00 24\text{ gallons} \times \text{\textdollar}4.00 = \text{\textdollar}96.00 24 gallons × $ 4.00 = $ 96.00 . 3. Direct vs. Inverse 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 . Proportions fall into two distinct categories: direct and inverse. Direct Proportions ( y = k x y = kx y = k x ) In a direct proportion , as one variable increases, the other variable increases at the same multiplier rate. The ratio y x = k \frac{y}{x} = k x y = k remains constant. a 1 b 1 = a 2 b 2 ⟹ a 1 ⋅ b 2 = b 1 ⋅ a 2 \frac{a_1}{b_1} = \frac{a_2}{b_2} \implies a_1 \cdot b_2 = b_1 \cdot a_2 b 1 a 1 = b 2 a 2 ⟹ a 1 ⋅ b 2 = b 1 ⋅ a 2 Inverse Proportions ( x ⋅ y = k x \cdot y = k x ⋅ y = k ) In an inverse proportion , as one variable increases, the other variable decreases proportionally. The product of the two variables x ⋅ y = k x \cdot y = k x ⋅ y = k remains constant. Common classification practice inverse proportion scenarios include workers vs. completion time or travel speed vs. travel time . a 1 ⋅ b 1 = a 2 ⋅ b 2 a_1 \cdot b_1 = a_2 \cdot b_2 a 1 ⋅ b 1 = a 2 ⋅ b 2 Step-by-Step Worked Example 3 (Inverse Proportion): If 8 soldiers can dig a trench system in 15 hours, how many hours will it take 12 soldiers working at the exact same rate to dig the identical trench system? Step 1: Identify relationship: More soldiers means fewer hours (inverse proportion). Step 2: Set up product equality: Soldiers 1 × Hours 1 = Soldiers 2 × Hours 2 \text{Soldiers}_1 \times \text{Hours}_1 = \text{Soldiers}_2 \times \text{Hours}_2 Soldiers 1 × Hours 1 = Soldiers 2 × Hours 2 . Step 3: Substitute known values: 8 × 15 = 12 × H 2 8 \times 15 = 12 \times H_2 8 × 15 = 12 × H 2 . Step 4: Solve for H 2 H_2 H 2 : 120 = 12 ⋅ H 2 ⟹ H 2 = 10 hours 120 = 12 \cdot H_2 \implies H_2 = 10\text{ hours} 120 = 12 ⋅ H 2 ⟹ H 2 = 10 hours . 4. Percentages, Percent Change & Multi-Step Discounts A percentage is a fraction expressed out of 100. The foundational percentage relationship is: Part = Percent (in decimal form) × Whole \text{Part} = \text{Percent (in decimal form)} \times \text{Whole} Part = Percent (in decimal form) × Whole Percent Increase and Decrease When calculating percentage increase or decrease, the change is always evaluated relative to the original starting amount : 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% classification practice Common Pitfall: Never divide the change by the new value! Always divide by the starting (original) value. Successive (Multi-Step) Percentages When multiple percentage changes occur sequentially (e.g., a 20 % 20\% 20% discount followed by an additional 10 % 10\% 10% discount), do not add the percentages together . Step-by-Step Worked Example 4: A piece of tactical equipment originally priced at $500 is placed on sale for 20 % 20\% 20% off. A military personnel discount reduces the sale price by an additional 10 % 10\% 10% . What is the final price? Step 1: Calculate price after first discount ( 20 % 20\% 20% off means paying 80 % 80\% 80% ): $ 500 × 0.80 = $ 400 \text{\textdollar}500 \times 0.80 = \text{\textdollar}400 $ 500 × 0.80 = $ 400 . Step 2: Apply second discount to the new price ( 10 % 10\% 10% off of $400 means paying 90 % 90\% 90% ): $ 400 × 0.90 = $ 360 \text{\textdollar}400 \times 0.90 = \text{\textdollar}360 $ 400 × 0.90 = $ 360 . Step 3: Note total discount is $ 500 − $ 360 = $ 140 \text{\textdollar}500 - \text{\textdollar}360 = \text{\textdollar}140 $ 500 − $ 360 = $ 140 , which is 140 500 = 28 % \frac{140}{500} = 28\% 500 140 = 28% , not 30 % 30\% 30% . 5. classification practice Word Problem Translation Strategy To translate word problems quickly into solvable mathematical equations, use this translation matrix:

English Phrase Mathematical Equivalent Example Translation "Is", "was", "equals", "results in" = = = "The total cost is $80" ⟹ C = 80 \implies C = 80 ⟹ C = 80 "Of", "times", "product of" × \times × " 30 % 30\% 30% of 150 150 150 " ⟹ 0.30 × 150 \implies 0.30 \times 150 ⟹ 0.30 × 150 "Per", "out of", "ratio of" ÷ \div ÷ or fraction bar "Miles per hour" ⟹ miles hours \implies \frac{\text{miles}}{\text{hours}} ⟹ hours miles "What number", "how many" Variable ( x , n x, n x , n ) "How many items..." ⟹ x \implies x ⟹ x "More than", "increased by" + + + " 5 5 5 more than twice x x x " ⟹ 2 x + 5 \implies 2x + 5 ⟹ 2 x + 5 "Less than", "decreased by" − - − " 4 4 4 less than y y y " ⟹ y − 4 \implies y - 4 ⟹ y − 4

Test Your Knowledge

A military supply depot stocks 420 total field rations consisting of Meal Ready-to-Eat (MRE) Type A, Type B, and Type C in the ratio 4 : 5 : 3, respectively. How many Type B rations are in the inventory?

175

140

105

210 Check Answer Need a hint? AI

Test Your Knowledge

An armored convoy travels 180 miles in 4 hours. At this same constant rate of speed, how many miles will the convoy cover in 7 hours?

280 miles

315 miles

360 miles

245 miles Check Answer Need a hint? AI

Test Your Knowledge

A tactical radio unit was originally priced at $850. During a procurement discount, its price was reduced to $680. What was the percentage discount applied to the tactical radio unit?

15%

25%

20%

30% Check Answer Need a hint? AI

Test Your Knowledge

If 6 military technicians can inspect a fleet of aircraft in 10 hours, how many hours would it take 15 technicians working at the exact same rate to inspect the same fleet?

2 hours

3 hours

5 hours

4 hours Check Answer Need a hint? AI

Sections you finish are checked off in the contents.

Up Next 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time Continue learning

Independent exam preparation notice OpenExamPrep is an independent education provider, not affiliated with or endorsed by the exam sponsors named on this site. Exam and certification names identify what our resources help you study for; they do not imply sponsorship, approval, or review of our materials. Unless a resource expressly identifies authorized public material, our practice questions are independently developed simulations, not actual, recalled, or sponsor-provided exam items. We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.