Speed, distance and time questions

The speed distance time formula and the triangle

Every speed, distance and time question uses the same relationship, written three ways:

• Distance = speed x time
• Speed = distance / time
• Time = distance / speed

The formula triangle helps you pick the right version under pressure. Draw a triangle with D at the top and S and T side by side underneath. Cover the letter you want. What is left tells you the calculation: cover D and you see S next to T, so multiply. Cover S and you see D over T, so divide. Cover T and you see D over S, so divide.

A quick check: units tell you whether you have the right version. Miles per hour literally means miles divided by hours. If your answer for a speed is not "distance per time", you have used the wrong arrangement.

Three short examples:

• A van covers 150 miles in 2.5 hours. Speed = 150 / 2.5 = 60 mph.
• A courier rides at 16 km/h for 3 hours. Distance = 16 x 3 = 48 km.
• A train covers 210 km at 84 km/h. Time = 210 / 84 = 2.5 hours, which is 2 hours 30 minutes.

Unit conversions: where most marks are lost

The arithmetic in these questions is rarely hard. The units are where candidates slip. Convert first, then calculate.

Minutes to hours: divide by 60. 45 minutes is 0.75 hours, 20 minutes is 1/3 of an hour, and 1 hour 12 minutes is 1.2 hours.

Hours (decimal) back to minutes: multiply the decimal part by 60. 2.4 hours is 2 hours and 0.4 x 60 = 24 minutes. The trap: reading 2.4 hours as 2 hours 40 minutes. Answer options often include both.

Kilometers per hour to meters per second: divide by 3.6. This works because 1 km/h = 1,000 meters in 3,600 seconds. So 72 km/h = 20 m/s and 90 km/h = 25 m/s. To go back, multiply by 3.6.

Miles and kilometers: if a question mixes them, it will give you the conversion (for example, 5 miles is about 8 km). Use the figure it gives, not one you remember.

Worked example: a cyclist rides at 18 km/h for 40 minutes. How far? 40 minutes is 2/3 of an hour, so the distance is 18 x 2/3 = 12 km. The wrong option will be 18 x 0.4 = 7.2 km, from treating 40 minutes as 0.4 hours.

Average speed: the classic trap

Average speed is total distance divided by total time. It is not the average of the speeds, because you spend longer at the slower speed.

Worked example: a driver goes 60 miles to a client site at 30 mph and returns the same 60 miles at 60 mph. What is the average speed for the round trip?

1. Time out: 60 / 30 = 2 hours.
2. Time back: 60 / 60 = 1 hour.
3. Total: 120 miles in 3 hours, so the average speed is 40 mph.

The option 45 mph, the simple average of 30 and 60, is the trap. It would only be right if the driver spent the same time, not the same distance, at each speed.

Stops count when the question says "overall". If the same driver also stopped for 1 hour at the client site, the overall average speed would be 120 / 4 = 30 mph. Read whether the question wants the average while moving or for the whole journey.

Shortcut for equal distances: when the two legs are the same length, the average speed is 2 x a x b / (a + b). For 30 and 60 mph: 2 x 30 x 60 / 90 = 40 mph. You can use it to check your answer, but the total distance over total time method works for every case.

Meeting problems: moving toward each other

When two vehicles move toward each other, the gap closes at the sum of their speeds. This is called the closing speed (or relative speed).

Worked example: two trucks are 300 km apart and drive toward each other on the same road, one at 70 km/h and one at 80 km/h. When do they meet?

• Closing speed: 70 + 80 = 150 km/h.
• Time to meet: 300 / 150 = 2 hours.
• Where they meet: the first truck has covered 70 x 2 = 140 km, and the second 80 x 2 = 160 km. The two distances add up to 300 km, which confirms the answer.

Moving apart: two vehicles leaving the same point in opposite directions also separate at the sum of their speeds. At 45 and 55 mph, they are 100 miles apart after 1 hour and 250 miles apart after 2.5 hours.

The trap: using the difference of the speeds instead of the sum. Ask yourself whether the gap is closing quickly (toward each other, add) or slowly (same direction, subtract).

Overtaking problems: same direction

When one vehicle chases another in the same direction, the gap closes at the difference of their speeds.

Worked example: a bus leaves a depot at 9:00 and travels at 50 km/h. A car leaves the same depot at 9:30 on the same route at 75 km/h. When does the car catch the bus?

1. Find the head start. By 9:30 the bus has traveled 50 x 0.5 = 25 km.
2. Find the closing speed. 75 - 50 = 25 km/h.
3. Divide. 25 km / 25 km/h = 1 hour after the car leaves, so 10:30.
4. Check. At 10:30 the bus has driven 1.5 hours x 50 = 75 km, and the car 1 hour x 75 = 75 km. They are level.

Trains passing objects. A train must travel its own length to pass a pole, and its own length plus the platform or bridge to clear it. A 200 m train at 72 km/h (20 m/s) passes a pole in 200 / 20 = 10 seconds, and clears a 400 m bridge in (200 + 400) / 20 = 30 seconds. The trap is forgetting the train's own length.

How speed, distance and time questions appear in job tests

Job tests rarely ask "a train leaves a station" in isolation. More often the same formula sits inside a work setting:

• Delivery and route tables. A table lists routes with distances and scheduled times, and asks which route has the highest average speed. Route A is 84 miles in 1 hour 45 minutes: 84 / 1.75 = 48 mph. Convert every time to decimal hours before comparing.
• Timetables and schedules. "A technician must reach a site 63 miles away by 14:00 and drives at an average of 42 mph. What is the latest time she can leave?" Time = 63 / 42 = 1.5 hours, so 12:30.
• Rates that are not travel. The same structure covers work rates: a machine that packs 240 boxes an hour packs 60 in 15 minutes. If you can do speed questions, you can do these. For problems that share work or output between people, see our guide to ratio questions.
• Word problems in fast mixed tests. Short, one-line questions in timed cognitive tests, where mental arithmetic and clean unit handling matter more than any formula. Practice these in our CCAT-style practice test.

Some tests allow a calculator and some do not. Your invitation email or the test's instructions page usually says which; prepare for the stricter case if it does not.

A speed method for timed tests

1. Circle the units in the question and in the options. If the question gives minutes and the answer is in hours, convert now.
2. Write the three letters D, S and T with the two numbers you know. The missing one tells you which version of the formula to use.
3. Estimate before you calculate. 135 miles in just over 2 hours must be a little under 67.5 mph. That alone can rule out two options.
4. Use the "per minute" trick when no calculator is allowed. 60 mph is 1 mile a minute, 30 mph is half a mile a minute and 45 mph is 0.75 miles a minute. So 18 miles at 45 mph takes 18 / 0.75 = 24 minutes.
5. For two moving objects, decide add or subtract before writing any numbers: toward each other or apart, add; same direction, subtract.
6. Check against the question. If it asks for a clock time, add the travel time to the start time. If it asks for an average over a round trip, make sure you used total time.

Practice this until steps 1 and 2 take a couple of seconds. Most wrong answers in this topic are set to catch people who skip them.