Many people have asked us to clear the confusion surrounding the various formulas of average speed. We will start with the bottom line – There is a single versatile formula for ALL average speed questions and that is

**Average Speed = Total Distance/Total Time**

No matter which formula you choose to use, it will always boil down to this one. Keeping this in mind, let’s discuss the various formulas we come across:

1. **Average Speed = (a + b)/2**

Applicable when one travels at speed a for half the time and speed b for other half of the time. In this case, average speed is the arithmetic mean of the two speeds.

2. **Average Speed = 2ab/(a + b)**

Applicable when one travels at speed a for half the distance and speed b for other half of the distance. In this case, average speed is the harmonic mean of the two speeds. On similar lines, you can modify this formula for one-third distance.

3. **Average Speed = 3abc/(ab + bc + ca)**

Applicable when one travels at speed a for one-third of the distance, at speed b for another one-third of the distance and speed c for rest of the one-third of the distance.

Note that the generic Harmonic mean formula for n numbers is

Harmonic Mean = n/(1/a + 1/b + 1/c + …)

4. You can also use weighted averages. Note that in case of average speed, the weight is always ‘time’. So in case you are given the average speed, you can find the ratio of time as

**t1/t2 = (a – Avg)/(Avg – b)**

As you already know, this is just our weighted average formula.

Now, let’s look at some simple questions where you can use these formulas.

Question 1: Myra drove at an average speed of 30 miles per hour for T hours and then at an average speed of 60 miles/hr for the next T hours. If she made no stops during the trip and reached her destination in 2T hours, what was her average speed in miles per hour for the entire trip?

(A) 40

(B) 45

(C) 48

(D) 50

(E) 55

Solution: Here, time for which Myra traveled at the two speeds is same.

Average Speed = (a + b)/2 = (30 + 60)/2 = 45 miles per hour

Answer (B)

Question 2: Myra drove at an average speed of 30 miles per hour for the first 30 miles of a trip & then at an average speed of 60 miles/hr for the remaining 30 miles of the trip. If she made no stops during the trip what was her average speed in miles/hr for the entire trip?

(A) 35

(B) 40

(C) 45

(D) 50

(E) 55

Solution: Here, distance for which Myra traveled at the two speeds is same.

Average Speed = 2ab/(a+b) = 2*30*60/(30 + 60) = 40 mph

Answer (B)

Question 3: Myra drove at an average speed of 30 miles per hour for the first 30 miles of a trip, at an average speed of 60 miles per hour for the next 30 miles and at a average speed of 90 miles/hr for the remaining 30 miles of the trip. If she made no stops during the trip, Myra’s average speed in miles/hr for the entire trip was closest to

(A) 35

(B) 40

(C) 45

(D) 50

(E) 55

Solution: Here, Myra traveled at three speeds for one-third distance each.

Average Speed = 3abc/(ab + bc + ca) = 3*30*60*90/(30*60 + 60*90 + 30*90)

Average Speed = 3*2*90/(2 + 6 + 3) = 540/11

This is a bit less than 50 so answer (D).

Question 4: Myra drove at an average speed of 30 miles per hour for some time and then at an average speed of 60 miles/hr for the rest of the journey. If she made no stops during the trip and her average speed for the entire journey was 50 miles per hour, for what fraction of the total time did she drive at 30 miles/hour?

(A) 1/5

(B) 1/3

(C) 2/5

(D) 2/3

(E) 3/5

Solution: We know the average speed and must find the fraction of time taken at a particular speed.

t1/t2 = (A2 – Aavg)/(Aavg – A1)

t1/t2 = (60 – 50)/(50 – 30) = 1/2

So out of a total of 3 parts of the journey time, she drove at 30 mph for 1 part and at 60 mph for 2 parts of the time. Fraction of the total time for which she drove at 30 mph is 1/3.

Answer (B)

Hope this sorts out some of your average speed formula confusion.

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the **GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

Hi Karishma, shouldn’t the answer to Q4 be (A) 1/5 instead of (B) 1/3? If it were B, then the average speed of the total journey would be 45m/hr and not 50m/hr as the question requires.

For example, let us say that the total distance is 90 miles. If Myra drives 1/3 of the way at 30 miles and 2/3rd of the way at 60 miles; then, the total time would be 1+ 1 = 2 hours (30/30 + 60/60). Average speed would then be 90 miles/2 hours = 45m/hr (and not 50 m/hr).

However, the application of your formula (copied below) does yield 1/3. Could you please provide a derivation of the formula?

t1/t2 = (A2 – Aavg)/(Aavg – A1)

t1/t2 = (60 – 50)/(50 – 30) = 1/2

This is not correct. She drove for 1/3rd TIME at 30 mph, not 1/3rd DISTANCE (So 30 miles at 30 mph is not correct). Considering time to be 1/3 and 2/3, you get average speed as 50 mph.

Thanks for the effort Karishma!

Thanks Karishma