Distance, Rate, and Time Problems: Six Worked Examples
To solve a distance, rate, and time problem, use distance = rate × time with matching units. Define the unknown, organize each mover or trip leg in a chart, and write an equation from the stated relationship. The six worked situations below assume constant speed on each leg; when speed changes, split the trip into suitable legs or use a known average speed.
This guide covers catch-up, separation, meeting, round trips, average speed, and wind or current examples. A chart helps you track units and different starting times.
The DRT Formula
\[d = r \cdot t\]
Distance equals rate times time. The two rearrangements:
r = d / t (rate equals distance divided by time).
t = d / r (time equals distance divided by rate).
The units must match. If rate is in miles per hour and time is in minutes, convert one of them before multiplying.
The Chart Method
For these constant-speed examples, start with one row per mover or trip leg:
Mover
Rate
Time
Distance
A
B
Fill in any two columns from the problem, then use d = rt to fill the third. The equation almost always comes from a relationship between the two distances or two times.
Type 1: Same Direction (Catching Up)
Maria leaves home jogging at 6 mph. Two hours later, her brother bikes after her at 14 mph. How long until her brother catches up?
Let t = time the brother bikes. Maria has been moving for t + 2 hours.
Mover
Rate
Time
Distance
Maria
6
t + 2
6(t + 2)
Brother
14
t
14t
They have traveled the same distance when he catches up:
Two trains leave the same station at the same time, heading in opposite directions. One travels at 50 mph, the other at 70 mph. After how many hours are they 360 miles apart?
Train
Rate
Time
Distance
A
50
t
50t
B
70
t
70t
Distances add to 360:
50t + 70t = 360 → 120t = 360 → t = 3 hours.
Type 3: Toward Each Other
Two cars are 240 miles apart and drive toward each other. One goes 35 mph, the other 45 mph. When do they meet?
Same setup as opposite directions:
35t + 45t = 240 → 80t = 240 → t = 3 hours.
Type 4: Round Trip
A boat travels 24 miles downstream and then back. The current is 2 mph. The boat’s still-water speed is 10 mph. How long is the round trip?
A car drives 60 miles at 30 mph, then 60 miles at 60 mph. What is the average rate?
Wrong answer: (30 + 60) / 2 = 45 mph. That averages rates, not distance-time.
Correct method: total distance over total time.
Time 1: 60/30 = 2 hours.
Time 2: 60/60 = 1 hour.
Total distance: 120. Total time: 3.
Average rate: 120 / 3 = 40 mph.
When the distances are equal, the average rate equals the harmonic mean of the two rates: 2 · 30 · 60 / (30 + 60) = 40. Memorize this trick.
Type 6: Wind or Current
A plane with a tailwind goes faster; against the wind, slower. A boat downstream goes faster; upstream, slower.
If the still speed is s and the wind or current is w:
With (downstream / tailwind): rate is s + w.
Against (upstream / headwind): rate is s − w.
A plane flies 600 miles with the wind in 2 hours and the same distance against the wind in 3 hours. Find the plane’s still speed and the wind speed.
Two equations:
– With wind: (s + w)(2) = 600 → s + w = 300.
– Against wind: (s − w)(3) = 600 → s − w = 200.
Add the equations: 2s = 500 → s = 250 mph. Then w = 50 mph.
Common Mistakes
Mixing units. Convert minutes to hours, or miles to feet, before multiplying.
Averaging rates without weighting by time. Always go total distance over total time.
Forgetting the head start. When one mover leaves early, that mover’s time variable is t + (head start), not t.
Adding distances when they should be set equal. For catch-up from the same starting point, traveled distances are equal. With different starting points, include the initial gap. For the meeting and separation examples here, add the traveled distances.
Skipping the chart. A chart can help you track each mover’s rate, time, and distance; it is a tool, not a required method.
A Quick Cheat Sheet
Situation
Equation
Catch up (same direction)
d_A = d_B
Meet or apart (opposite directions)
d_A + d_B = total
Round trip
t_down + t_up = total time
Average rate
total distance / total time
Wind / current
rate = still ± wind
Tape this inside the front cover of your binder.
Frequently Asked Questions
Do I have to use the chart?
No. Use it when it helps you organize the quantities. Label the units and define what your time variable measures, especially when a mover has a head start.
What units should I use?
Whatever the problem gives you, as long as they match. If the answer needs different units (minutes instead of hours), convert at the end.
Are these problems on the SAT?
Distance, rate, and time provide useful practice with algebraic modeling. Do not assume that a particular motion subtype or number of questions will appear on your test; use the current official test specifications for planning.
What is the most-missed DRT problem?
A common mistake is taking the arithmetic mean of two speeds when the travel times differ. Average speed is total distance divided by total elapsed time, including any stops counted in the trip.
How fast should I be at these by test day?
First practice setting up correct equations and checking units without a timer. Then use the actual timing and question formats of your test to guide timed practice; these examples do not establish a universal target time.
Closing Thought
These six situations share d = rt, but the relationship between the distances and times determines the equation. Check your answer against the original story, account for head starts, and calculate average speed from total distance and total time.
For more practice, browse our Algebra 1 worksheets and our full Math Topics library. For additional practice, check a workbook’s current preview and contents before choosing it; this article does not verify coverage of every motion subtype in a product.
by: Effortless Math Team about
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