Study guide
2.4 Work Rates & Combined Labor 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
2.4 Work Rates & Combined Labor Problems
Key Takeaways Explain This Topic AI
Work rate is the fraction of a complete job accomplished per unit of time: R = 1 / T.
When two workers perform a job together, their individual rates add: R_total = R1 + R2 = 1/a + 1/b.
The two-person combined time shortcut formula is T = (a * b) / (a + b).
For partial shift problems, calculate the work done during the solo period first, then divide the remaining work by the combined rate.
Tank filling/draining problems treat inlet pipes as positive rates (+1/A) and drain pipes as negative rates (-1/B).
Last updated: July 2026
2.4 Work Rates & Combined Labor Problems
Work rate word problems on the AFQT evaluate your ability to compute how fast individuals, teams, or machines complete tasks working individually or collaboratively. Understanding the relationship between work completed ( W W W ), rate of work ( R R R ), and time ( t t t ) is essential for high scores.
The Core Work Rate Formula ( W = R t W = Rt W = Rt )
The fundamental work equation mirrors the distance formula ( d = r t d = rt d = r t ): W = R ⋅ t W = R \cdot t W = R ⋅ t
Where:
W W W = Total work completed (often expressed as 1 1 1 complete job).
R R R = Rate of work per unit of time (job/hour, job/day).
t t t = Time spent working.
Defining Individual Work Rates
If Worker A can complete 1 1 1 entire job alone in T T T hours, Worker A's hourly work rate ( R A R_A R A ) is: R A = 1 T of the job per hour R_A = \frac{1}{T} \text{ of the job per hour} R A = T 1 of the job per hour
Example : If a mechanic takes 6 6 6 hours to overhaul an engine, the mechanic overhauls 1 6 \frac{1}{6} 6 1 of the engine per hour.
Combined Labor: Two Workers Completing One Job
When two entities work simultaneously without interfering with each other, their individual rates add together to yield a combined rate ( R combined R_{\text{combined}} R combined ).
Deriving the Combined Work Equation R combined = R A + R B = 1 a + 1 b R_{\text{combined}} = R_A + R_B = \frac{1}{a} + \frac{1}{b} R combined = R A + R B = a 1 + b 1 1 T combined = b + a a ⋅ b \frac{1}{T_{\text{combined}}} = \frac{b + a}{a \cdot b} T combined 1 = a ⋅ b b + a
The Product-Over-Sum Shortcut Formula
Inverting the fraction yields the famous Product-over-Sum formula for two workers: T combined = a ⋅ b a + b T_{\text{combined}} = \frac{a \cdot b}{a + b} T combined = a + b a ⋅ b
Where:
a a a = Time required for Worker A alone.
b b b = Time required for Worker B alone.
Example : Tech A can configure a server in 4 4 4 hours ( a = 4 a = 4 a = 4 ), while Tech B can configure it in 6 6 6 hours ( b = 6 b = 6 b = 6 ). T combined = 4 × 6 4 + 6 = 24 10 = 2.4 hours (2 hours 24 minutes) T_{\text{combined}} = \frac{4 \times 6}{4 + 6} = \frac{24}{10} = 2.4 \text{ hours (2 hours 24 minutes)} T combined = 4 + 6 4 × 6 = 10 24 = 2.4 hours (2 hours 24 minutes)
Multiple Workers and Variable Efficiency
When three or more workers collaborate, or when a crew consists of multiple identical workers, add all individual rates: Combined Rate ( 3 workers ) = 1 a + 1 b + 1 c \text{Combined Rate } (3 \text{ workers}) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} Combined Rate ( 3 workers ) = a 1 + b 1 + c 1
Identical Worker Crew Formula
If n n n identical workers each take T T T hours to finish a job alone, the total time required for the crew of n n n workers is: T crew = T n T_{\text{crew}} = \frac{T}{n} T crew = n T
- Number of Workers ( n n n ) Individual Time ( T T T ) Combined Time ( T crew T_{\text{crew}} T crew )
- 1 1 1 worker 12 12 12 hours 12 12 12 hours
- 2 2 2 workers 12 12 12 hours 6 6 6 hours
- 3 3 3 workers 12 12 12 hours 4 4 4 hours
- 4 4 4 workers 12 12 12 hours 3 3 3 hours
- 6 6 6 workers 12 12 12 hours 2 2 2 hours
- Partial Shifts and Staggered Start Times
AFQT word problems often involve scenarios where one worker starts earlier, and a second worker joins later.
4-Step Algorithm for Staggered Work Problems
Find individual rates : R A = 1 a R_A = \frac{1}{a} R A = a 1 and R B = 1 b R_B = \frac{1}{b} R B = b 1 .
Calculate initial work completed : Multiply Worker A's rate by the solo time ( t solo t_{\text{solo}} t solo ). W initial = R A ⋅ t solo W_{\text{initial}} = R_A \cdot t_{\text{solo}} W initial = R A ⋅ t solo
Determine remaining work : Subtract initial work from 1 1 1 full job. W remaining = 1 − W initial W_{\text{remaining}} = 1 - W_{\text{initial}} W remaining = 1 − W initial
Divide remaining work by combined rate : Find remaining joint time ( t joint t_{\text{joint}} t joint ). t joint = W remaining R A + R B t_{\text{joint}} = \frac{W_{\text{remaining}}}{R_A + R_B} t joint = R A + R B W remaining
Tank Filling and Pipe Flow Problems
Pipe flow problems operate on the exact same principles as labor rates. However, when a drain or outlet pipe is open, its rate subtracts from the total filling rate. Net Rate = R inlet 1 + R inlet 2 − R outlet \text{Net Rate} = R_{\text{inlet 1}} + R_{\text{inlet 2}} - R_{\text{outlet}} Net Rate = R inlet 1 + R inlet 2 − R outlet 1 T net = 1 A + 1 B − 1 C \frac{1}{T_{\text{net}}} = \frac{1}{A} + \frac{1}{B} - \frac{1}{C} T net 1 = A 1 + B 1 − C 1
Example : Pipe A fills a water tank in 3 3 3 hours ( + 1 / 3 +1/3 + 1/3 ), Pipe B fills it in 6 6 6 hours ( + 1 / 6 +1/6 + 1/6 ), and Drain C empties it in 4 4 4 hours ( − 1 / 4 -1/4 − 1/4 ). Net Rate = 1 3 + 1 6 − 1 4 = 4 12 + 2 12 − 3 12 = 3 12 = 1 4 \text{Net Rate} = \frac{1}{3} + \frac{1}{6} - \frac{1}{4} = \frac{4}{12} + \frac{2}{12} - \frac{3}{12} = \frac{3}{12} = \frac{1}{4} Net Rate = 3 1 + 6 1 − 4 1 = 12 4 + 12 2 − 12 3 = 12 3 = 4 1 Time to fill tank = 4 hours \text{Time to fill tank} = 4 \text{ hours} Time to fill tank = 4 hours
- Step-by-Step Worked AFQT Examples
- Worked Example 1: Combined Labor Shortcut
Problem : Specialist Miller can dig a tactical trench in 5 5 5 hours alone. Specialist Davis can dig the same trench in 7.5 7.5 7.5 hours alone. If they work together at their respective constant rates, how many hours will it take them to dig the trench?
Step 1 : Identify individual times a = 5 a = 5 a = 5 and b = 7.5 b = 7.5 b = 7.5 .
Step 2 : Apply Product-over-Sum shortcut. T = 5 × 7.5 5 + 7.5 = 37.5 12.5 = 3 hours T = \frac{5 \times 7.5}{5 + 7.5} = \frac{37.5}{12.5} = 3 \text{ hours} T = 5 + 7.5 5 × 7.5 = 12.5 37.5 = 3 hours
Worked Example 2: Staggered Work Shifts
Problem : Carpenter A can build a command ramp in 10 10 10 hours, and Carpenter B can build it in 15 15 15 hours. Carpenter A begins working alone for 4 4 4 hours. Carpenter B then joins Carpenter A, and they work together until the ramp is completed. What is the total time spent building the ramp?
Step 1 : Find individual rates: R A = 1 10 R_A = \frac{1}{10} R A = 10 1 and R B = 1 15 R_B = \frac{1}{15} R B = 15 1 .
Step 2 : Calculate work done by Carpenter A in first 4 4 4 hours. W A solo = 4 × 1 10 = 4 10 = 2 5 of the job W_{\text{A solo}} = 4 \times \frac{1}{10} = \frac{4}{10} = \frac{2}{5} \text{ of the job} W A solo = 4 × 10 1 = 10 4 = 5 2 of the job
Step 3 : Calculate remaining work. W remaining = 1 − 2 5 = 3 5 of the job W_{\text{remaining}} = 1 - \frac{2}{5} = \frac{3}{5} \text{ of the job} W remaining = 1 − 5 2 = 5 3 of the job
Step 4 : Find combined rate R A + R B R_A + R_B R A + R B . R combined = 1 10 + 1 15 = 3 30 + 2 30 = 5 30 = 1 6 job/hour R_{\text{combined}} = \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ job/hour} R combined = 10 1 + 15 1 = 30 3 + 30 2 = 30 5 = 6 1 job/hour
Step 5 : Calculate joint time for remaining work. t joint = 3 / 5 1 / 6 = 3 5 × 6 = 18 5 = 3.6 hours t_{\text{joint}} = \frac{3/5}{1/6} = \frac{3}{5} \times 6 = \frac{18}{5} = 3.6 \text{ hours} t joint = 1/6 3/5 = 5 3 × 6 = 5 18 = 3.6 hours
Step 6 : Add initial solo time to joint time for total duration. Total Time = 4 hours + 3.6 hours = 7.6 hours (7 hours 36 minutes) \text{Total Time} = 4 \text{ hours} + 3.6 \text{ hours} = 7.6 \text{ hours (7 hours 36 minutes)} Total Time = 4 hours + 3.6 hours = 7.6 hours (7 hours 36 minutes)
Worked Example 3: Pipe Filling with Drain
Problem : An inlet pipe can fill a fuel bladder in 4 4 4 hours. A drainage valve can completely empty a full bladder in 12 12 12 hours. If the valve is accidentally left open while the inlet pipe is filling an empty bladder, how many hours will it take to fill the bladder?
Step 1 : Identify inlet rate ( + 1 / 4 +1/4 + 1/4 ) and outlet rate ( − 1 / 12 -1/12 − 1/12 ).
Step 2 : Subtract drain rate from inlet rate to find net rate. Net Rate = 1 4 − 1 12 = 3 12 − 1 12 = 2 12 = 1 6 bladder/hour \text{Net Rate} = \frac{1}{4} - \frac{1}{12} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \text{ bladder/hour} Net Rate = 4 1 − 12 1 = 12 3 − 12 1 = 12 2 = 6 1 bladder/hour
Step 3 : Invert net rate to find total filling time. Total Time = 6 hours \text{Total Time} = 6 \text{ hours} Total Time = 6 hours
Common AFQT Traps & Shortcut Strategies
The Direct Addition Fallacy : Never add completion times together! If Worker A takes 4 4 4 hours and Worker B takes 6 6 6 hours, working together takes LESS than 4 4 4 hours, never 4 + 6 = 10 4 + 6 = 10 4 + 6 = 10 hours!
The Product-over-Sum Restriction : Remember that T = a b a + b T = \frac{ab}{a+b} T = a + b ab ONLY works for TWO workers. For three workers, you must find a common denominator and sum their fractional rates: 1 a + 1 b + 1 c \frac{1}{a} + \frac{1}{b} + \frac{1}{c} a 1 + b 1 + c 1 .