Study guide
2.3 Rate, Distance, Time & Speed Word Problems
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
2.3 Rate, Distance, Time & Speed Word Problems
Key Takeaways Explain This Topic AI
The distance formula triangle links distance, rate, and time: d = r * t, r = d / t, and t = d / r.
Objects moving in opposite directions (towards or away from each other) close or open distance at a combined rate of r1 + r2.
Objects moving in the same direction close distance at a relative catch-up rate of r1 - r2.
Average speed over a multi-leg journey is TOTAL distance divided by TOTAL time (never average the speeds directly!).
Upstream/headwind speed equals still-water speed minus current speed (r - c); downstream/tailwind speed equals still-water speed plus current speed (r + c).
Last updated: July 2026
2.3 Rate, Distance, Time & Speed Word Problems
Motion and speed problems form a core component of the AFQT Arithmetic Reasoning subtest. These word problems require setting up equations based on the relationships between distance ( d d d ), rate or speed ( r r r ), and time ( t t t ).
- The Fundamental Distance Formula Triangle ( d = r t d = rt d = r t )
- All motion problems rely on the primary distance equation: d = r ⋅ t d = r \cdot t d = r ⋅ t
By rearranging the algebraic variables, you derive three working formulas: Distance: d = r ⋅ t \text{Distance: } d = r \cdot t Distance: d = r ⋅ t Rate (Speed): r = d t \text{Rate (Speed): } r = \frac{d}{t} Rate (Speed): r = t d Time: t = d r \text{Time: } t = \frac{d}{r} Time: t = r d
- Variable Common Units Metric Units
- Distance ( d d d ) Miles, Feet, Yards Kilometers, Meters
- Rate ( r r r ) Miles per hour (mph), Feet/sec km/h, Meters/sec
- Time ( t t t ) Hours, Minutes, Seconds Hours, Minutes, Seconds
CRITICAL RULE : Units MUST be consistent across all variables! If rate is given in miles per hour and time in minutes , you MUST convert time to hours ( t = minutes 60 t = \frac{\text{minutes}}{60} t = 60 minutes ) before multiplying!
Opposite-Direction Motion (Closing / Separating Distance)
When two vehicles or objects move in opposite directions (either traveling toward each other to meet, or departing from the same point in opposite directions), their speeds add together to determine the rate of distance change.
Combined Rate Formula for Opposite Directions r combined = r 1 + r 2 r_{\text{combined}} = r_1 + r_2 r combined = r 1 + r 2 Total Distance = ( r 1 + r 2 ) ⋅ t \text{Total Distance} = (r_1 + r_2) \cdot t Total Distance = ( r 1 + r 2 ) ⋅ t Time to Meet / Separate = Initial Distance r 1 + r 2 \text{Time to Meet / Separate} = \frac{\text{Initial Distance}}{r_1 + r_2} Time to Meet / Separate = r 1 + r 2 Initial Distance
Example : Two military convoys start 300 300 300 miles apart and drive directly toward each other. Convoy A travels at 45 mph 45 \text{ mph} 45 mph and Convoy B travels at 55 mph 55 \text{ mph} 55 mph . Combined Rate = 45 + 55 = 100 mph \text{Combined Rate} = 45 + 55 = 100 \text{ mph} Combined Rate = 45 + 55 = 100 mph Time to meet = 300 miles 100 mph = 3 hours \text{Time to meet} = \frac{300 \text{ miles}}{100 \text{ mph}} = 3 \text{ hours} Time to meet = 100 mph 300 miles = 3 hours
Same-Direction Catch-Up Problems
When two objects move in the same direction and the faster object attempts to catch up to the slower object (which had a head start), their speeds subtract to determine the relative rate of closure.
Relative Speed Formula for Same Direction r relative = r faster − r slower r_{\text{relative}} = r_{\text{faster}} - r_{\text{slower}} r relative = r faster − r slower Head Start Distance = r slower ⋅ t head start \text{Head Start Distance} = r_{\text{slower}} \cdot t_{\text{head start}} Head Start Distance = r slower ⋅ t head start Time to Catch Up = Head Start Distance r faster − r slower \text{Time to Catch Up} = \frac{\text{Head Start Distance}}{r_{\text{faster}} - r_{\text{slower}}} Time to Catch Up = r faster − r slower Head Start Distance
Alternatively, set distance equations equal ( d 1 = d 2 d_1 = d_2 d 1 = d 2 ): r slower ⋅ ( t + t head start ) = r faster ⋅ t r_{\text{slower}} \cdot (t + t_{\text{head start}}) = r_{\text{faster}} \cdot t r slower ⋅ ( t + t head start ) = r faster ⋅ t
Round-Trip and Average Speed Calculations
One of the most frequently missed question types on the AFQT is finding the average speed for a round trip or multi-leg journey.
The Arithmetic Mean Trap
If a vehicle travels to a destination at 30 mph 30 \text{ mph} 30 mph and returns along the exact same route at 60 mph 60 \text{ mph} 60 mph , the average speed is NOT 30 + 60 2 = 45 mph \frac{30 + 60}{2} = 45 \text{ mph} 2 30 + 60 = 45 mph ! Because the vehicle spends more time traveling at the slower speed, 45 mph 45 \text{ mph} 45 mph is mathematically incorrect.
The Harmonic Average Speed Formula Average Speed = Total Distance Total Time = d 1 + d 2 t 1 + t 2 \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d_1 + d_2}{t_1 + t_2} Average Speed = Total Time Total Distance = t 1 + t 2 d 1 + d 2
For a round trip of distance d d d each way at speeds r 1 r_1 r 1 and r 2 r_2 r 2 : R avg = 2 ⋅ r 1 ⋅ r 2 r 1 + r 2 R_{\text{avg}} = \frac{2 \cdot r_1 \cdot r_2}{r_1 + r_2} R avg = r 1 + r 2 2 ⋅ r 1 ⋅ r 2
Validation using example ( 30 mph 30 \text{ mph} 30 mph and 60 mph 60 \text{ mph} 60 mph ) : R avg = 2 ( 30 ) ( 60 ) 30 + 60 = 3600 90 = 40 mph R_{\text{avg}} = \frac{2(30)(60)}{30 + 60} = \frac{3600}{90} = 40 \text{ mph} R avg = 30 + 60 2 ( 30 ) ( 60 ) = 90 3600 = 40 mph
Current and Wind Resistance Problems
Naval vessels traveling in water currents or aircraft flying through wind experience rate adjustments based on direction.
Downstream / Tailwind (With the current/wind) : Current speed adds to still-water/air speed. r effective = r vehicle + c r_{\text{effective}} = r_{\text{vehicle}} + c r effective = r vehicle + c
Upstream / Headwind (Against the current/wind) : Current speed subtracts from still-water/air speed. r effective = r vehicle − c r_{\text{effective}} = r_{\text{vehicle}} - c r effective = r vehicle − c
- Scenario Effective Rate Formula Distance Equation
- With Current / Tailwind r + c r + c r + c d = ( r + c ) ⋅ t 1 d = (r + c) \cdot t_1 d = ( r + c ) ⋅ t 1
- Against Current / Headwind r − c r - c r − c d = ( r − c ) ⋅ t 2 d = (r - c) \cdot t_2 d = ( r − c ) ⋅ t 2
- Step-by-Step Worked AFQT Examples
- Worked Example 1: Opposite Direction Motion
Problem : Two naval cutters depart from the same port at the same time, one heading east at 22 knots 22 \text{ knots} 22 knots and the other heading west at 18 knots 18 \text{ knots} 18 knots . How many hours will it take for them to be 200 200 200 nautical miles apart?
Step 1 : Identify opposite directions ⟹ \implies ⟹ add rates. r combined = 22 + 18 = 40 knots (nautical miles/hour) r_{\text{combined}} = 22 + 18 = 40 \text{ knots (nautical miles/hour)} r combined = 22 + 18 = 40 knots (nautical miles/hour)
Step 2 : Use t = d r t = \frac{d}{r} t = r d to find time. t = 200 nautical miles 40 knots = 5 hours t = \frac{200 \text{ nautical miles}}{40 \text{ knots}} = 5 \text{ hours} t = 40 knots 200 nautical miles = 5 hours
Worked Example 2: Same Direction Catch-Up
Problem : A supply truck leaves an airbase traveling at 40 mph 40 \text{ mph} 40 mph . Two hours later, a fast courier jeep leaves the same airbase along the same route traveling at 60 mph 60 \text{ mph} 60 mph . How long will it take the courier jeep to catch up to the supply truck?
Step 1 : Calculate head start distance of the supply truck. Head Start Distance = 40 mph × 2 hours = 80 miles \text{Head Start Distance} = 40 \text{ mph} \times 2 \text{ hours} = 80 \text{ miles} Head Start Distance = 40 mph × 2 hours = 80 miles
Step 2 : Determine relative closing speed. r relative = 60 mph − 40 mph = 20 mph r_{\text{relative}} = 60 \text{ mph} - 40 \text{ mph} = 20 \text{ mph} r relative = 60 mph − 40 mph = 20 mph
Step 3 : Divide head start distance by relative closing speed. Catch-up Time = 80 miles 20 mph = 4 hours \text{Catch-up Time} = \frac{80 \text{ miles}}{20 \text{ mph}} = 4 \text{ hours} Catch-up Time = 20 mph 80 miles = 4 hours
Worked Example 3: Upstream vs Downstream River Motion
Problem : A patrol boat travels at 15 mph 15 \text{ mph} 15 mph in still water. It travels 36 36 36 miles downstream with a 3 mph 3 \text{ mph} 3 mph current and then returns 36 36 36 miles upstream against the same current. What is the total time for the round trip?
Step 1 : Calculate downstream rate and time. r downstream = 15 + 3 = 18 mph r_{\text{downstream}} = 15 + 3 = 18 \text{ mph} r downstream = 15 + 3 = 18 mph t downstream = 36 miles 18 mph = 2 hours t_{\text{downstream}} = \frac{36 \text{ miles}}{18 \text{ mph}} = 2 \text{ hours} t downstream = 18 mph 36 miles = 2 hours
Step 2 : Calculate upstream rate and time. r upstream = 15 − 3 = 12 mph r_{\text{upstream}} = 15 - 3 = 12 \text{ mph} r upstream = 15 − 3 = 12 mph t upstream = 36 miles 12 mph = 3 hours t_{\text{upstream}} = \frac{36 \text{ miles}}{12 \text{ mph}} = 3 \text{ hours} t upstream = 12 mph 36 miles = 3 hours
Step 3 : Sum downstream and upstream times. Total Trip Time = 2 + 3 = 5 hours \text{Total Trip Time} = 2 + 3 = 5 \text{ hours} Total Trip Time = 2 + 3 = 5 hours
Common AFQT Traps & Shortcut Strategies
The Minute-Hour Conversion Trap : If a problem states a speed of 60 mph 60 \text{ mph} 60 mph and a duration of 45 minutes 45 \text{ minutes} 45 minutes , do NOT multiply 60 × 45 = 2 , 700 60 \times 45 = 2,700 60 × 45 = 2 , 700 ! Convert 45 minutes 45 \text{ minutes} 45 minutes to 45 60 = 0.75 hours \frac{45}{60} = 0.75 \text{ hours} 60 45 = 0.75 hours , yielding 60 × 0.75 = 45 miles 60 \times 0.75 = 45 \text{ miles} 60 × 0.75 = 45 miles .
The Average Speed Distractor : On round trip average speed problems, test creators will almost always include the simple average as Option A or B. Always use Total Distance Total Time \frac{\text{Total Distance}}{\text{Total Time}} Total Time Total Distance !