Speed, Distance, and Time
Apply the rate formula d = rt to solve for distance, speed, or time — including multi-leg journey problems.
How long will your road trip take? Can you make your flight if you drive 75 mph? If you jog at 6 mph, how far can you run in 45 minutes? The d = rt formula answers all of these in seconds. It's the rate formula you'll use more than any other — whether you're planning a commute, bidding on a delivery contract, or pacing yourself in a race.
- Use d = rt to solve for distance, rate (speed), or time given the other two
- Convert time units correctly before applying the formula
- Calculate average speed for multi-leg journeys using total distance and total time
The three-way rate formula connects distance, rate, and time:
\[ d = r \cdot t \qquad r = \frac{d}{t} \qquad t = \frac{d}{r} \]Know any two of the three variables and you can find the third. Critical rule: time must be in the same unit as the rate. If speed is in mph, time must be in hours — convert minutes before plugging in.
The Formula Triangle
d = rt Triangle — Cover the Variable You Want
d = r × t
r = d ÷ t
t = d ÷ r
A train travels at 65 mph for 2.5 hours. How far does it travel?
\[ d = r \cdot t = 65 \text{ mph} \times 2.5 \text{ hr} = 162.5 \text{ miles} \]The train travels 162.5 miles.
A runner finishes a half-marathon (13.1 miles) at an average pace of 6 mph. What is the finishing time in hours and minutes?
\[ t = \frac{d}{r} = \frac{13.1}{6} \approx 2.183 \text{ hours} \]Convert the decimal to minutes: \(0.183 \times 60 \approx 11\) minutes.
Finishing time: 2 hours 11 minutes.
A delivery driver travels 120 miles at 60 mph (highway), then 80 miles at 40 mph (city). What is the average speed for the entire trip?
- Find time for each leg: \(t_1 = 120/60 = 2\text{ hr}\), \(t_2 = 80/40 = 2\text{ hr}\).
- Total distance: \(120 + 80 = 200\text{ miles}\).
- Total time: \(2 + 2 = 4\text{ hours}\).
- Average speed: \(200 \div 4 = 50\text{ mph}\).
Note: In this case \(\frac{60+40}{2} = 50\) happens to match, but that's a coincidence because both legs took equal time. Always use total d ÷ total t.
Test yourself before moving on:
- A car drives at 55 mph for 3.5 hours. How far does it travel?
- A plane covers 1,440 miles in 3 hours. What is its speed in mph?
- You jog 4.5 miles at 5 mph. How long does the jog take? Give answer in minutes.
▶ Show Answers
- \(d = 55 \times 3.5 =\) 192.5 miles.
- \(r = 1440 \div 3 =\) 480 mph.
- \(t = 4.5 \div 5 = 0.9\text{ hr} \times 60 =\) 54 minutes.
- Averaging the speeds directly: For a trip of 120 miles at 60 mph then 60 miles at 30 mph, the average speed is NOT (60+30)/2 = 45 mph. It's (120+60)/(2+2) = 180/4 = 45 mph — coincidence again! Always use total d ÷ total t.
- Mixing unit systems: If speed is in mph, time must be in hours — not minutes. 45 minutes = 0.75 hours. Using 45 directly gives a distance 60× too large.
- Forgetting to find both legs' times separately: In multi-leg problems, calculate each segment's time independently before adding them to get total time.
- d = rt — rearrange to r = d/t or t = d/r depending on what you need.
- Units must match: mph needs time in hours; km/h needs time in hours. Convert before plugging in.
- Average speed = total distance ÷ total time — never average the individual speeds directly.
- Multi-leg: find each segment's time separately, then sum all distances and all times.
Logistics coordinators at companies like FedEx and Amazon build delivery routes using d = rt constantly. They estimate how many stops a driver can make in an 8-hour shift given average city speeds of 25 mph, or whether a 640-mile freight run can be completed within DOT's 11-hour driving limit. Miscalculating arrival times costs money in overtime, missed delivery windows, and penalty fees — making this formula foundational to every routing decision.
Calculator Connection
The Speed, Distance & Time Calculator solves for any of the three variables directly — choose what to calculate, enter the other two, and get step-by-step work. The Ratio Solver and Proportion Solver can also set up rate problems as proportions for more complex multi-step scenarios.
Try it with the Calculator
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Speed, Distance, and Time — Quiz
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