3. Distance, Rate, and Time

Travel problems all rest on one formula:
\[ \text{distance} = \text{rate} \times \text{time}, \qquad d = rt. \]

Rearrange it for whatever is missing: \(t = \dfrac{d}{r}\) and \(r = \dfrac{d}{t}\).

Worked example: You drive 150 miles at 50 miles per hour. How long does it take?
\[ t = \frac{d}{r} = \frac{150}{50} = 3 \text{ hours.} \]

Worked example (find the speed): A train covers 280 miles in 4 hours. Its average speed is
\[ r = \frac{d}{t} = \frac{280}{4} = 70 \text{ mph.} \]

Keep your units consistent. If the rate is in miles per hour, time must be in hours — 30 minutes is \(0.5\) hours, not 30. A trip at 40 mph for 30 minutes covers \(d = 40 \times 0.5 = 20\) miles.