Motion problems use one formula, $d = rt$: distance equals rate times time. You have used rates like miles per hour when converting units, and you can write equations from words. Here the two skills meet.
Why it works
A rate of $50$ miles per hour means $50$ miles for every hour. In $3$ hours that is $50 + 50 + 50 = 50 \cdot 3$ miles. So distance is rate times time, as long as the units match: if the rate is per hour, the time must be in hours.
Key idea
Make a table with a row for each traveler and columns for rate, time and distance, filling the distance as rate $\times$ time. Then write one equation from how the distances (or times) are related: they add, they are equal, or they differ.
Worked examples
Worked example: Opposite directions
Two cars leave the same town at the same time, one going east at $50$ mph and one going west at $30$ mph. When are they $160$ miles apart?
- Let $t$ be the time in hours. East: rate $50$, distance $50t$. West: rate $30$, distance $30t$.
- Moving apart, the distances add: $50t + 30t = 160$.
- $80t = 160$, so $t = 2$.
- Check: $100 + 60 = 160$ miles.
Answer: After $2$ hours.
Worked example: Catching up
A truck leaves at $45$ mph. One hour later a car follows on the same road at $60$ mph. How long does the car take to catch up?
- Let $t$ be the car's time in hours. The truck has driven one hour longer: $t + 1$.
- Truck: $45(t + 1)$. Car: $60t$. Same direction, so at the moment it catches up the distances are equal: $60t = 45(t + 1)$.
- Distribute: $60t = 45t + 45$. Subtract $45t$: $15t = 45$, so $t = 3$.
- Check: the car goes $180$ miles; the truck goes $45 \cdot 4 = 180$ miles.
Answer: $3$ hours.
Worked example: A round trip
Lena bikes to a lake at $12$ mph and back at $8$ mph. The trip takes $2.5$ hours in all. How far is the lake?
- Let $d$ be the one-way distance. Time is distance divided by rate: $\frac{d}{12}$ there and $\frac{d}{8}$ back.
- The times add: $\frac{d}{12} + \frac{d}{8} = 2.5$.
- Multiply every term by $24$: $2d + 3d = 60$, so $5d = 60$ and $d = 12$.
Answer: $12$ miles.
Common mistake
Adding the speeds when both travelers go the same way. Rates add only when the distances add (moving apart or toward each other). When one catches the other, the distances are equal. A second slip is mixing units: $20$ minutes at $36$ mph is $36 \cdot \frac{1}{3} = 12$ miles, not $36 \cdot 20$.