Rate of travel: word problems

key notes :

🔍Understanding Rate of Travel

Rate refers to how fast something moves, typically measured as distance per unit of time (e.g., km/h, m/s).

The basic formula is:

Distance = Rate × Time

Rate = Distance ÷ Time

Time = Distance ÷ Rate

🔍Types of Word Problems

  • Direct travel: One vehicle travels a certain distance at a constant speed.
  • Round trips: Going to a place and returning (different speeds possible).
  • Two objects moving toward/away from each other: Boats in a river, cars on a road, etc.
  • Catch-up problems: One object starts later and tries to catch up to the first.

🛠️Solving Word Problems Step-by-Step

  • Read the problem carefully — Identify what’s given (distance, rate, time) and what you need to find.
  • Define variables — Let r, t, or d represent unknowns.
  • Set up an equation — Use d=rtd and plug in the known values.
  • Solve the equation — Apply algebraic methods.
  • Check your answer — Make sure it makes sense in context (e.g., a negative speed doesn’t work).

🧠Common Example Problems

Single journey: A car travels 180 km at 60 km/h. How long does it take?

t = 180/60 = 3hours

Round trip: A boat travels 30 km downstream (with the current) at 10 km/h and returns against the current at 6 km/h. How long is the total trip?

Downstream: t1 = 30/10 = 3 hours

Upstream: t2 = 30/6 = 5 hours

Total time = 3 + 5 = 8 hours

Catch-up problem: A cyclist leaves a town at 20 km/h. Two hours later, a car leaves the same place at 50 km/h. How long until the car catches up?

Distance covered by the cyclist: 20t

Distance covered by the car: 50 (t−2)

Set up equation:

20t = 50 (t−2)

Solve:

20t = 50t − 100  ⟹  30t = 100  ⟹  t =100/30 ≈ 3.33 hours

Learn with an example

🎯 After spending the winter in Clay County, two flocks of birds are now migrating home. The first flock flies due east at 45 kilometres per hour, and the second flies west at 65 kilometres per hour. In how much time will they be 750 kilometres apart?

🎯 If necessary, round your answer to the nearest minute.

______ hours and__________  minutes

  • The distance to be covered is 750 kilometres.
  • The first flock’s rate is 45 kilometres per hour. The second flock’s rate is 65 kilometres per hour. Together, they are increasing the gap between them at a rate of 45 + 65 = 110 kilometres per hour.
  • Now write an equation and solve for t.
  • d = rt
  • 750 = 110t Plug in d = 750 and r = 110
  • 6.818181… = t Divide both sides by 110
  • The units for t are hours. Convert the units to be in hours and minutes, rounded to the nearest minute.
  • The whole-number portion is 6. That is 6 full hours.
  • Convert the decimal portion.
  • 0.818181… hours × 60 minutes/1 hour ≈ 49 minutes
  • So, t is approximately 6 hours and 49 minutes.
  • The flocks of birds will be 750 kilometres apart in about 6 hours and 49 minutes.

🎯 A Coast Guard rescue craft just picked up the captain of a disabled sailboat. The rescue craft is now heading back to shore at a rate of 57 kilometres per hour. Meanwhile, the sailboat is drifting in the other direction at 6 kilometres per hour. How long will it be before the rescue craft is 29 kilometres from the sailboat?

🎯 If necessary, round your answer to the nearest minute.

 ___hours and ____ minutes

  • The distance to be covered is 29 kilometres.
  • The sailboat’s rate is 6 kilometres per hour. The rescue craft’s rate is 57 kilometres per hour. Together, they are increasing the gap between them at a rate of 6 + 57 = 63 kilometres per hour.
  • Now write an equation and solve for t.
  • d = rt
  • 29 = 63tPlug in d = 29 and r = 63
  • 0.460317… = tDivide both sides by 63
  • The units for t are hours. Convert the units to be in hours and minutes, rounded to the nearest minute.
0.460317… hours × 60 minutes/ 1 hour ≈ 28 minutes
  • So, t is approximately 28 minutes.
  • The rescue craft will be 29 kilometres from the sailboat in about 28 minutes.

🎯 A ranch hand herded his cattle to some grazing land. Then he began riding back to the ranch, which is due west, at 5 kilometres per hour. Simultaneously, the cows began moving away from him, too, headed due east at 3 kilometres per hour. How long did it take for the herd and the ranch hand to be 2 kilometres apart?

🎯 If necessary, round your answer to the nearest minute.

 _____hours and _____  minutes

  • The distance to be covered is 2 kilometres.
  • The herd’s rate is 3 kilometres per hour. The ranch hand’s rate is 5 kilometres per hour. Together, they are increasing the gap between them at a rate of 3 + 5 = 8 kilometres per hour.
  • Now write an equation and solve for t.
  • d = rt
    2 = 8tPlug in d = 2 and r = 8
28
  •  = tDivide both sides by 8
14
  •  = tSimplify
  • The units for t are hours. Convert the units to be in hours and minutes.
14of an hour × 60 minutes/ 1 hour = 15 minutes
  • So, t is 15 minutes.
  • The herd and the ranch hand were 2 kilometres apart in 15 minutes.

Let’s practice!🖊️