Distance, rate, and time: complete GMAT guide

Distance, rate, and time: complete GMAT guide | GMAT Sprint

Distance, rate, and time questions are among the most reliable scoring opportunities in GMAT Quant. The core equation is simple, but the exam varies how the motion is described: two travelers, a round trip, a pause, a current, or a circular track. The method stays the same if you organize the information in a table and keep the units consistent.

What the GMAT tests

Most questions use one of these structures:

A single trip with a constant rate
Two trips that share a distance or a time
Two moving objects, either in opposite directions or in the same direction
A round trip with different rates
A trip with a stop or a change in rate
Motion in a current or on a circular track

The formula is $d=rt$ , where $d$ is distance, $r$ is rate, and $t$ is time. Rearrange it as $r=d/t$ or $t=d/r$ depending on what the question asks. The units must match. If a rate is in miles per hour, the time must be in hours, or the rate must be converted.

Build a table with columns for distance, rate, and time. Fill in the values you know, use the same variable for the unknown that links the rows, and write an equation from the relationship the question describes. The table prevents the most common error: mixing up which quantity is equal for the two trips.

Single trips and unit conversion

Example. A car travels 150 miles in 2.5 hours. What is its average speed?

$r=d/t=150/2.5=60$ miles per hour.

If a rate is given in minutes and the distance in miles, convert one of them before dividing. For example, 30 miles in 40 minutes is $30/(40/60)=45$ miles per hour, not $30/40=0.75$ miles per hour.

Using proportions

When the rate is constant, distance and time are proportional. If a car travels 90 miles in 1.5 hours, the time for 150 miles at the same rate is $1.5 \times \frac{150}{90}=2.5$ hours. Set up the proportion with matching units and label each ratio. Proportions are faster than computing the rate when the question gives one complete trip and asks about a second trip at the same speed.

Proportions also work with maps and scale drawings. If 2 inches on a map represent 30 miles, then 5 inches represent $30 \times \frac{5}{2}=75$ miles. Keep the same order in both ratios: map over actual, or actual over map, but not one of each.

Average speed

Average speed is total distance divided by total time, not the average of the speeds. If a car travels 60 miles per hour for one hour and 40 miles per hour for one hour, the average speed is 50 miles per hour because the times are equal. But if the car travels the same distance at each speed, the average speed is the harmonic mean:

\text{average speed}=\frac{2r_1r_2}{r_1+r_2}

For a round trip at 60 miles per hour and 40 miles per hour, the average speed is $\frac{2(60)(40)}{60+40}=48$ miles per hour, not 50.

Example. A cyclist rides 12 miles to a town at 12 miles per hour and returns along the same route at 18 miles per hour. What is the average speed for the round trip?

Time out: $12/12=1$ hour. Time back: $12/18=2/3$ hour. Total distance: 24 miles. Total time: $5/3$ hours.

Average speed: $24/(5/3)=72/5=14.4$ miles per hour.

Two objects in opposite directions

When two objects move toward each other, their rates add. The time until they meet is the initial distance divided by the sum of the rates.

Example. Two towns are 210 miles apart. A train leaves one town at 50 miles per hour and another leaves the other town at 60 miles per hour at the same time. How long until they meet?

Combined rate: $50+60=110$ miles per hour. Time: $210/110=21/11$ hours, about 1 hour 55 minutes.

When two objects move in the same direction, subtract the rates. The faster object closes the gap at the difference of the rates.

Example. A police car traveling at 70 miles per hour chases a car traveling at 50 miles per hour. The cars start 40 miles apart. How long does the chase last?

Relative rate: $70-50=20$ miles per hour. Time: $40/20=2$ hours.

Round trips and shared distances

For a round trip with the same distance each way, let the distance be $d$ . The total time is $d/r_1+d/r_2$ . The average speed is $2d$ divided by that time. Do not average the rates unless the times are equal.

For two trips that share a distance but have different rates, set the distances equal. For two trips that share a time, set the times equal. The phrase "same distance" or "same time" tells you which column to equate.

Stops and changing rates

A stop adds time but no distance. If a trip includes a 30-minute stop, add $0.5$ hours to the total time. If a trip has two legs at different rates, calculate each leg separately and add the distances and times.

Example. A delivery truck travels 120 miles at 40 miles per hour, stops for 30 minutes, then travels 90 miles at 45 miles per hour. What is the average speed for the whole trip?

First leg: $120/40=3$ hours. Stop: 0.5 hours. Second leg: $90/45=2$ hours. Total distance: 210 miles. Total time: 5.5 hours. Average speed: $210/5.5=420/11 \approx 38.2$ miles per hour.

Currents and opposing forces

In a river, the boat's speed relative to the water is $b$ and the current is $c$ . Downstream speed is $b+c$ ; upstream speed is $b-c$ . If a boat takes the same time each way over equal distances, the current must be zero. Otherwise, use the two equations to solve for $b$ and $c$ .

Example. A boat travels 30 miles downstream in 2 hours and returns the same distance in 3 hours. Find the boat's speed in still water and the current speed.

Downstream: $b+c=15$ . Upstream: $b-c=10$ . Add: $2b=25$ , so $b=12.5$ . Then $c=2.5$ .

Circular tracks

On a circular track, two runners starting together meet again when the faster runner has gained one full lap on the slower runner. If the track length is $L$ , the time until the next meeting is $L/(r_1-r_2)$ when they run in the same direction. When they run in opposite directions, the time is $L/(r_1+r_2)$ .

A repeatable approach

Draw a table with distance, rate, and time.
Convert all rates and times to compatible units.
Write the relationship: equal distances, equal times, combined distance, or relative rate.
Solve for the target, not necessarily for every variable.
Check whether the question asks for total time, average speed, or a single leg.

Common traps

Averaging speeds when the times are not equal.
Forgetting to add a stop to total time.
Adding rates when objects move in the same direction.
Mixing minutes and hours in the same equation.
Using the one-way distance as total distance for a round trip.
Assuming the average speed is the arithmetic mean of two rates.
Ignoring the current when a boat returns upstream.

Quick reference

Situation	Equation
Basic motion	$d=rt$
Average speed	$\text{total distance}/\text{total time}$
Equal distances at two rates	$\frac{2r_1r_2}{r_1+r_2}$
Opposite directions	$r_1+r_2$
Same direction	$r_1-r_2$
Downstream	$b+c$
Upstream	$b-c$
Circular track, same direction	$L/(r_1-r_2)$

Most distance questions become simple once the table is complete. Fill in the known values, mark the equal quantity, and write one equation. If you find yourself juggling three equations without a table, stop and organize the information first.

Distance, rate, and time: complete GMAT guide

What the GMAT tests

Single trips and unit conversion

Using proportions

Average speed

Two objects in opposite directions

Round trips and shared distances

Stops and changing rates

Currents and opposing forces

Circular tracks

A repeatable approach

Common traps

Quick reference

Practice Problem Solving: Distance Rate Problems

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