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Distance, rate and time
Trains, cars, runners and boats all follow one formula. The test makes it hard in three ways: minutes instead of hours, two travelers at once, and average speed. Here is how to handle each one.
Distance = rate × time
d = r × t
The rate (speed) is how far you go in one unit of time. Go at that rate for some number of hours and the distance is the rate added up hour after hour: rate × time.
Units have to match
Miles per hour needs time in hours. If the problem gives minutes, divide by 60 first. Then check that your answer is in the unit the question asks for: miles, hours or minutes.
How to solve them
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1
One formula, three forms
You only need to remember d = r × t. Cover the letter you want and the other two tell you what to do: multiply for distance, divide for rate or time.
Next step: Minutes to hoursFind the distance46 mph for 3.5 hours
46 × 3.5 = 161 miles
Find the rate171 miles in 3 hours
171 ÷ 3 = 57 mph
Find the time280 miles at 56 mph
280 ÷ 56 = 5 hours
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2
Convert minutes before you multiply
Divide minutes by 60 to get hours. Never write 30 minutes as 0.30: it is 0.5 of an hour.
15 minutes15 ÷ 600.25 hour20 minutes20 ÷ 601/3 hour40 minutes40 ÷ 602/3 hour90 minutes90 ÷ 601.5 hoursNext step: Toward or apartExampleA cyclist rides at 18 mph for 40 minutes. How far does she go?
- 40 minutes = 40 ÷ 60 = 2/3 hour.
- 18 × 2/3 = 12.
Answer 12 miles
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3
Toward each other or apart: add the speeds
When two travelers move toward each other, or away from each other, the gap between them changes by both speeds at once. Add the speeds and treat the sum as one rate.
Next step: Catching upExampleTwo trains start 330 miles apart and head toward each other at 50 mph and 60 mph. When do they meet?
- The gap closes at 50 + 60 = 110 mph.
- 330 ÷ 110 = 3.
Answer In 3 hours
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4
One chasing the other: subtract the speeds
When both travel in the same direction, the faster one gains only the difference in speeds. Divide the head start by that difference.
Next step: Average speedExampleA car is 30 miles behind a bus on the same road. The bus goes 45 mph, the car 65 mph. How long until the car catches the bus?
- The car gains 65 − 45 = 20 mph.
- 30 ÷ 20 = 1.5.
Answer 1.5 hours (90 minutes)
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5
Average speed is total distance ÷ total time
It is not the average of the two speeds. You spend more time at the slower speed, so that speed counts more. Find the time of each part, add up distances and times, then divide.
average speed = total distance ÷ total time
Next step: Make a tableExampleYou drive 90 miles at 45 mph and come back the same 90 miles at 30 mph. What is your average speed?
- Out: 90 ÷ 45 = 2 hours. Back: 90 ÷ 30 = 3 hours.
- In total: 180 miles in 5 hours.
- 180 ÷ 5 = 36 mph, not 37.5.
Answer 36 mph
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6
Harder problems: one row per traveler
When the two travelers start at different times, give each one a row with rate, time and distance. Write the unknown time as a letter, and look for what is the same for both (often the distance).
ExampleJo leaves at noon at 40 mph. Sam leaves from the same place at 1 p.m. at 60 mph on the same road. When does Sam catch up?
- Jo drives t hours; Sam drives t − 1 hours.
- Same distance: 40t = 60(t − 1), so 40t = 60t − 60 and 20t = 60.
- t = 3: Sam catches up 3 hours after noon, at 3 p.m. Check: 40 × 3 = 120 and 60 × 2 = 120.
Answer 3 p.m.
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Adding speeds in a chase
A car at 70 mph chases a truck at 55 mph that is 6 miles ahead:
Wrong6 ÷ (70 + 55) hours
Right6 ÷ (70 − 55) = 0.4 hour = 24 minutes
Minutes written as decimals
50 mph for 30 minutes:
Wrong50 × 0.30 = 15 miles
Right50 × 0.5 = 25 miles
Averaging the two speeds
60 miles at 30 mph, then 60 miles at 60 mph:
Wrong(30 + 60) ÷ 2 = 45 mph
Right120 ÷ (2 + 1) = 40 mph
The answer in the wrong unit
The question asks for minutes and you found 0.75 hour:
Wrong0.75 minutes
Right0.75 × 60 = 45 minutes
5 questions at test level
Pick an answer, then read why. No timer, no account.
A truck travels at 58 mph for 3.5 hours. How far does it go?
Show the answer
D. d = r × t = 58 × 3.5. 58 × 3 = 174 and half of 58 is 29, so 174 + 29 = 203. 174 forgets the half hour. It goes 203 miles.
A runner covers 9 miles in 1 hour 15 minutes. What is her average speed?
Show the answer
B. 1 hour 15 minutes is 1.25 hours, not 1.15. r = d ÷ t = 9 ÷ 1.25 = 7.2. 7.8 reads the time as 1.15 hours, and 11.25 multiplies instead of dividing. Her speed is 7.2 mph.
Two cars leave the same town at the same time and drive in opposite directions at 48 mph and 62 mph. How far apart are they after 2.5 hours?
Show the answer
C. Moving apart, the gap grows by both speeds: 48 + 62 = 110 mph, and 110 × 2.5 = 275. 35 subtracts the speeds, which is only right when they go the same way, and 155 counts one car. They are 275 miles apart.
A boat is 12 miles behind a ferry. The ferry moves at 18 mph and the boat at 26 mph in the same direction. How long until the boat catches the ferry?
Show the answer
C. Same direction, so the boat gains only 26 − 18 = 8 mph, and 12 ÷ 8 = 1.5. 0.27 hour adds the speeds and 0.46 hour ignores the ferry’s speed. It takes 1.5 hours.
Lee drives 150 miles to a lake at 50 mph and drives back at 75 mph. What is the average speed for the whole trip?
Show the answer
A. Out: 150 ÷ 50 = 3 hours. Back: 150 ÷ 75 = 2 hours. In total, 300 miles in 5 hours: 300 ÷ 5 = 60. 62.5 averages the two speeds and ignores that more time was spent at the slower one. The average speed is 60 mph.
9 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 9 cards known.
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