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:
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.
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:
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.