QUANTITATIVE REASONING · GMAT-Q-RT-01

Work and Motion Rates

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical model

\frac{1}{T} = \frac{1}{a} + \frac{1}{b}

Total distance over total elapsed time

v_{avg} = \frac{d _{1} + d _{2}}{t _{1} + t _{2}}

Approach rate

t = \frac{initial separation}{v _{1} + v _{2}}

A worker and a drain or opposing work rate

r_{net} = r_{in} - r_{out}

Think Like a GMAT Test-Taker

Translate completion time into jobs per hour, or distance into speed times time.

Faster Methods

Add rates, not hours; use consistent units.

Worked example 1

Workers take 3 and 6 hours alone. Find combined time.

\frac{1}{T} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2}

Together they take 2 hours.

Worked example 2

Opposing trains travel at 40 and 50 km/h for 2 hours. Find separation.

(40 + 50) \cdot 2 = 180

Their relative speeds add.

Worked example 3

A car travels 60 km at 30 km/h and returns 60 km at 60 km/h. Find overall average speed.

\frac{120}{60/30 + 60/60} = 40

Do not take the simple mean of unequal travel times.

Worked example 4

One pipe fills a tank in 4 h and a drain empties it in 12 h. Find the net fill time.

\frac{1}{T} = \frac{1}{4} - \frac{1}{12} = \frac{1}{6}

The drain subtracts from the net work rate, so T=6 h.

Worked example 5

A car travels 120 km at 60 km/h, rests for 30 minutes, then travels another 120 km at 80 km/h. What is its average speed for the entire trip, including the rest?

v_{avg} = \frac{240}{120/60 + 0.5 + 120/80} = 60 km/h

Average speed uses total distance divided by total elapsed time. Include the 0.5-hour rest: 240 km in 4 hours.

Worked example 6

Pump A fills a tank in 4 hours and Pump B in 6 hours. Both run for 1 hour, then A is turned off. How long after the start is the tank full?

\frac{1}{4} + \frac{1}{6} = \frac{5}{12}; 1 - \frac{5}{12} = \frac{7}{12}; \frac{7/12}{1/6} = \frac{7}{2}

Together the pumps fill 5/12 of the tank in the first hour. Pump B then needs 3.5 more hours to fill the remaining 7/12, so total time is 4.5 hours.

Common Traps

The combined completion time must be less than the fastest solo time.