# SAT Math: Average Speed (Not the “Average” of the Speeds)!

One of the most challenging concepts on the SAT Math test is average rate, also called average speed. Often found in complex word problems, this type of question is one many students are less familiar with, so don’t get nervous if you don’t know how to approach it.

Review these important equations and look at how this concept appears on the SAT.

### Distance = Rate x Time

The first important formula to memorize is d = rt. This stands for distance = rate x time. Many students find it helpful to remember this formula as the “DIRT” formula (Distance Is Rate × Time). It is equally acceptable to think of it as time = distance ÷ rate or as rate = distance ÷ time because these are simply rearranged versions. Often, the rate is a speed, but it could be any “something per something.” In a word problem, if you see the word “per,” you know this is a question involving rates.

### Average Rate = Total Distance ÷ Total Time

The second formula is average rate = total distance ÷ total time. This is its own special concept, and you should notice that it is not an average of the speeds. Average rate is completely different. Look at an example question:

Question #1

Ariella drove 40 miles to see her cousin at a speed of 20 mph. The trip took Ariella 2 hours. Then, Ariella drove from her cousin’s house another 30 miles to the store at a speed of 10 mph. It took Ariella 3 hours to arrive at the store. What was Ariella’s average speed for the trip?

average rate = total distance / total time.

Ariella traveled 40 miles + 30 miles, so her total distance was 70 miles. She drove for 2 hours + 3 hours, so her total time was 5 hours. 70 ÷ 5 = 14.

Her average speed for the whole trip was 14 mph.

The average speed in this problem is 14 mph, which is different from the average of the speeds. If you just average the two speeds (10 mph and 20 mph), you would get 15 mph. Instead, think of average speed as a weighted average. Because Ariella spent more time in the problem going 10 mph than 20 mph, it makes sense that the average speed would be closer to 10 mph. Be wary of trap answers on these questions.

The next question will require the use of both the “average rate” formula and the “DIRT” formula.

Question #2

Marion spent all day on a sightseeing trip in Tuscany. First she boarded the bus which went 15mph through a 30 mile section of the countryside. The bus then stopped for lunch in Florence before continuing on a 3 hour tour of the city’s sights at speed of 10mph. Finally, the bus left the city and drove 40 miles straight back to the hotel. Marion arrived back at her hotel exactly 2 hours after leaving Florence. What was the bus’s average rate for the entire journey?

To find the average rate of the bus, you will need to find the total distance and the total time, so use the d = rt formula to find the missing info.

For the first part of the trip, you know that 30 miles = 15 mph × t, so you know that t = 2 hours.

For the middle part of the trip, you know that d = 10 mph × 3 hours, so you know that d = 30 miles. For the last part of the trip, you know that 40 miles = r × 2 hours, so you know that r = 20 mph.

Now you can find the total distance and the total time.

total distance = 30 miles + 30 miles + 40 miles = 100 miles

total time = 2 hours + 3 hours + 2 hours = 7 hours

So the average rate = 100 miles ÷ 7 hours = 14.28 mph

Try out one more challenging question:

Question #3

Tracey ran to the top of a steep hill at an average pace of 6 miles per hour. She took the exact same trail back down. To her relief, the descent was much faster, and her average speed rose to 14 miles per hour. If the entire run took Tracey exactly 1 hour to complete and she did not make any stops, what is the length of the trail, in miles, one way?

For the way up the hill, you know that d = 6 mph × t.

For the way down the hill, you know that d = 14 mph × t. Since you know that the distance up the hill was the same as the distance down the hill, you can pick a number for d. One option is “84,” since it is a multiple of both 6 and 14. If 84 = 6 mph × t, then you know that t = 14 hours. If 84 = 14 mph × t, then you know that t = 6 hours.

Now you can use another formula, the average rate formula, to find the average speed for the whole trip.

average rate = total distance ÷ total time

Using your picked number of 84, you know that the total distance traveled would be 168 miles. The total time is 14 hours + 6 hours = 20 hours. So the average rate = 168 miles ÷ 20 hours = 8.4 mph.

It doesn’t matter that Tracey didn’t really go 168 miles or that she didn’t really go 20 hours. The picked number, 84, just helps you find the ratio of the total distance to the total time in order to calculate the average rate of the entire journey.

Now that you have found the average rate for the whole trip, you can plug it into the “DIRT” formula to find the actual distance for the entire journey.

d = r × t

d = 8.4 mph × 1 hour

You know that t = 1 hour because the problem told us so. Therefore, the actual distance for the entire trip was 8.4 miles. The problem asks how many miles the trail was one way. 8.4 ÷ 2 = 4.2. The answer to the question is 4.2 miles.

You could also solve this problem in other ways, including using a system of equations and substitution, but it’s nice to know that you can pick a number for the distance traveled and use it to find the average rate for the whole journey. Be on the lookout for those trips where the distance there and back is the same.

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