DAT Quantitative Reasoning: Combined Work Rates

DAT High-Yield Concept

Work Equation: Work equals rate times time ( $\text{Work} = \text{Rate} \times \text{Time}$ ). A worker who completes 1 job in $t$ hours has a rate of $\frac{1}{t}$ jobs per hour.
Two Workers Together: Combined rate is $\frac{1}{t_A} + \frac{1}{t_B} = \frac{1}{T}$ . Use shortcut $T = \frac{t_A \times t_B}{t_A + t_B}$ ("Product over Sum").
$n$ Identical Workers: Combined rate scales to $\frac{n}{t}$ , so total time for 1 job is $T = \frac{t}{n}$ .
Opposing Rates (Pipes): Net rate is fill rate minus drain rate ( $\frac{1}{t_A} - \frac{1}{t_B} = \frac{1}{T}$ ). Use shortcut $T = \frac{t_A \times t_B}{t_B - t_A}$ when drain time $t_B > t_A$ .

1. Fundamental Work Formula & Individual Rates

Core Relationship:

\text{Work} = \text{Rate} \times \text{Time} \quad \implies \quad R = \frac{W}{T}

For a single job (

W = 1

), a worker requiring

t

hours works at a rate of

R = \frac{1}{t}

jobs per hour.

Quantitative Reasoning work problems require setting up relationships between individual productivity and total completed tasks. This structure is analogous to the motion formula ( $\text{Distance} = \text{Rate} \times \text{Time}$ , covered in depth in its own lesson), where completed work replaces distance. Crucially, while individual working times cannot be added directly, individual rates are strictly additive or subtractive.

2. Two Cooperative Workers (Additive Work Rates)

When Worker A (time $t_A$ ) and Worker B (time $t_B$ ) perform a job together, their combined rate $R_{\text{total}}$

References

Sources returned by Google Search grounding during lesson verification.

Geometry–Do - ResearchGate — vertexaisearch.cloud.google.com
Solve Any Work Word Problem EASILY - Table method vs Shortcut - YouTube — vertexaisearch.cloud.google.com

Combined Work Rates

Summary

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