Rate, Time and Distance

Rate, time and distance is one formula and three rearrangements, and almost all the difficulty is in deciding which distance and which time the question means.

The formula and its three faces

Distance = rate x time.

From that, by ordinary algebra:

The units tell you which one you need without thinking about it. If the answer choices are in hours, you are dividing something by a speed.

A convoy travels at 48 miles per hour for 2 hours 45 minutes. How far does it go?

Two hours 45 minutes is 2.75 hours, not 2.45. This conversion is the most common error in the whole topic.

Distance: 48 times 2.75 is 48 times 2 plus 48 times 0.75, which is 96 plus 36, or 132 miles.

Minutes to a decimal hour

Memorize these four and interpolate the rest.

For anything else, divide the minutes by 60. Forty minutes is 40/60, which is two thirds, or 0.667.

Average speed is not the average of the speeds

This is the most reliable trap on the topic, and it catches strong test takers.

A driver covers 120 miles at 40 miles per hour, then the same 120 miles back at 60 miles per hour. What is the average speed for the round trip?

The tempting answer is 50, and 50 will be a choice. It is wrong, because the driver spends more time at the slower speed, so the slow leg gets more weight.

Do it properly:

• Out: 120 divided by 40 is 3 hours.
• Back: 120 divided by 60 is 2 hours.
• Total distance: 240 miles. Total time: 5 hours.
• Average: 240 divided by 5 is 48 miles per hour.

Average speed is always total distance over total time. Never average the rates. The only case where averaging the rates happens to work is when the two legs take equal time, which is not what these questions describe.

A useful sanity check: the true average always lands closer to the slower speed. If your answer sits exactly in the middle, you averaged the rates.

Two objects

Three arrangements, and the only thing that changes is whether you add or subtract the speeds.

Approaching each other

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Zoom in

Two vehicles start 330 miles apart and drive toward each other, one at 55 miles per hour and one at 65. How long until they meet?

They close the gap at 55 plus 65, which is 120 miles per hour. Time is 330 divided by 120, or 2.75 hours, which is 2 hours 45 minutes.

The combined speed is the whole trick. You are not solving two problems; you are solving one problem about a shrinking gap.

Moving apart

Same arithmetic. Two vehicles leaving the same point in opposite directions at 55 and 65 separate at 120 miles per hour, so after 3 hours they are 360 miles apart.

One chasing another

A truck leaves at 40 miles per hour. Two hours later a car leaves from the same point at 60 miles per hour. How long does the car take to catch up?

Two things to handle: the head start and the closing speed.

• Head start: the truck has been going 2 hours at 40, so it is 80 miles ahead.
• Closing speed: 60 minus 40 is 20 miles per hour.
• Catch-up time: 80 divided by 20 is 4 hours after the car leaves.

Read the question's clock carefully. Four hours after the car leaves is six hours after the truck leaves, and both numbers will be choices.

With and against a current

A current adds to a boat's speed going downstream and subtracts going upstream. (Wind does the same to a plane.) So the two trip speeds straddle the boat's own speed, and the current is half the difference between them.

A boat goes 24 miles downstream in 2 hours and back upstream in 3 hours.

Downstream speed: 24 / 2 = 12 mph. Upstream: 24 / 3 = 8 mph. The current is half of 12 - 8: 2 mph, and the boat's own speed is halfway between: 10 mph.

What you can skip

Across the 96 questions on this topic:

• Acceleration never appears. Every speed is constant for the part of the trip being asked about.
• Wind problems come up once; the current method above handles them.

Where people lose points

Reading 2 hours 45 minutes as 2.45. Minutes are sixtieths, not hundredths.

Averaging two speeds. Covered above. If a round trip is described, expect this trap and expect the naive average to be a choice.

Mismatched units. A speed in miles per hour multiplied by a time in minutes gives a number sixty times too large. Convert the time first, every time.

Adding speeds when the objects move in the same direction, or subtracting when they move toward each other. Draw the arrows if you need to.

Answering the wrong clock on a catch-up problem - time since the chaser started against time since the leader started.

Work one in under a minute

A trainee marches 9 miles in 3 hours, rests 1 hour, then marches 5 more miles in 2 hours. What was the average speed for the whole period, including the rest?

Total distance: 9 plus 5 is 14 miles.

Total time: 3 plus 1 plus 2 is 6 hours. The rest counts, because the question said so.

Average: 14 divided by 6 is about 2.33 miles per hour.

Dropping the rest hour gives 2.8, which is a choice, and the question told you not to.

Where this leads

Work rate questions use the same idea with jobs in place of miles, and the combined-speed move is exactly the combined-rate move.