During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine’s average speed for the entire trip? GMAT Explanation, Video Solution, and More Practice!

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine’s average speed for the entire trip?

A. (180−x)/2

B. (x+60)/4

C. (300−x)/5

D. 600/(115−x)

E. 12,000/(x+200)

Average speed, distance, miles per hour… Yes, it’s a GMAT work and rate question. In this case we have an average rate with variables in the question and in the answer choices. That’s a signal for considering picking numbers. Why pick numbers? To make life easier. How do I judge if picking numbers will make life easier? If the question would be a simple 1-2-3 if you had the actual numbers that’s a pretty good indicator. Here, if you just had to calculate the average speed, assuming you know how to do a weighted average, things would be smooth. Here’s another GMAT word problem from the GMAT Official Guide on which you can pick numbers to make things much easier.

Let’s get back to Francine and her trip! Let’s define the question: what was Francine’s average speed for the entire trip?

Now let’s take inventory. What numbers are we missing? Distance! Yes, I’d go ahead and pick a distance for the trip. Pick something that works well with the speeds, 40 and 60. Why not go for 240 total distance and divide that into 120 and 120 for each leg of the trip? That way you’ve got easy division.

120 miles/40mph = 3 hours

120 miles/60mph = 2 hours

Total time = 5 hours

Total distance = 240 miles

240/5 = 48mph

Just as a sanity check for the 48. We spent more time (3 hours) traveling at the slower speed (40mph) so it makes sense that the average speed would be closer to 40mph than 60mph.

OK, so, with total distance at 240 the average speed is 48. Of course, that’s not an answer. We have to plug in “x”, the total distance at an average speed of 40 miles per hour, in order to yield 48. We can use our divisibility shortcut here. Since we know the correct answer is 48 we know that the numerator has to be a multiple of 3 (because 48 is a multiple of 3). So just focus on the numerator and eliminate answer choices not divisible by 3.

A. (180−x)/2.  180-50 = 130

B. (x+60)/4.  50 + 60 = 110

C. (300−x)/5.  300-50 = 250

D. 600/(115−x) Div by 3

E. 12,000/(x+200) Div by 3

You end up with D and E as possibilities. At this point you can just work out the entire expression for both choices. You could also:

  1. Continue using divisibility. 48 has four 2’s in it. 600 only has 3. So that’s out.
  2. Use Magnitude. It should be pretty clear that D. is way too small.

So, in terms of x, what was Francine’s average speed for the entire trip? E. 12,000/(x+200)

I think picking numbers is a great way to go on this one. You can also do the algebra which could be a little faster depending on how speedy you are at setting it up but, at least in my mind, is less straight forward and I’ve seen many GMAT tutoring students screw it up. I’ll do the algebra in the diagram and in the video for your reference.

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour GMAT Explanation Diagram

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour GMAT Explanation Shortcut Diagram

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour GMAT Explanation Alt Diagram

 

Video Solution: During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine’s average speed for the entire trip?

Additional GMAT Work and Rate/Weighted Average Practice

Here’s a work and rate question from GMAT Question of the day which has the same basic structure and explains the rate t’s a bit more.

And another average rate question from the GMAT Question of the day

Here’s a very challenging average rate question from the GMAT Prep Tests. Same basic premise but on this one the follow through is tougher because it involves a quadratic.

GMAT Work and Rate question from the Official Guide to practice picking numbers. It’s a cooperative rate so the follow through is a bit different than the above “Francine” average rate but you can still use rate T’s and again pick numbers: Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours

GMAT Question of the Day – Problem Solving – Rate

A driver completed the first half of a trip at an average speed of 30 miles per hour. What speed must the driver average for the second half of the trip in order to average a speed of 36 miles per hour for the entire journey (assuming that the driver did not make any stops during the trip)?

A. 42

B. 45

C. 48

D. 50

E. 52

Answer Show

GMAT Question of the Day Solution

This is a classic GMAT rate question: average speed for a two part journey. Let’s remember that with average rate you can’t simply take the average of the rates because doing so assumes that you spent an equal amount of time driving at each rate. Usually that’s not the case so you need to add a “weight” to each of the rates before combining them. The “weight” is how much time you spent driving at that rate.

The general strategy for GMAT weighted average/rate questions is to make a T for each part of the journey. In this case you would have the two pieces and the total. The time boxes for each of the pieces sum to the total time. The distance boxes sum to the total distance. Divide the total distance by the total time to solve for the “total” rate.

In this question you are given the total rate and need to work backwards in order to solve the rate for the second half of the journey. The procedure is the same. Work with the time and distance in order to calculate the rate. We are not given a number for the total distance but we can certainly pick a number so that the question comes more into focus. Pick a number that works nicely with the given information. In this case a multiple of 180 will work well with 30 and 36. Here’s an Official GMAT question on which we also use the T method: A boat traveled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then traveled the same distance downstream at an average speed of (v+3) miles per hour

GMAT Question of the Day Problem Solving Work/Rate Solution 5 Diagram

Here are more GMAT average rate practice questions:

Distance, Work and Rate – Average Rate 1

Distance, Work and Rate – Average Rate 2

For more GMAT distance, work, and rate practice visit GMAT Question of the Day.

 

 

GMAT Question of the Day – Problem Solving – Work

Tank A contains X gallons of water while Tank B contains Y gallons. Tank A has water removed at a rate of K gallons per hour and Tank B has water added at a rate of M gallons per hour. Assuming that both tanks still contain some water, in terms of X, Y, K, and M how many gallons of water will both tanks have 2 minutes from now?

A.  (x – y – m + k)/60

B. (30(x-y) – m + k)/60

C. (30(x-y) + m – k)/60

D. (30(x-y) + m – k)/30

E. (30(x-y) – m + k)/30

Answer Show

GMAT Question of the Day Solution

For GMAT work and rate questions it helps to organize the information in T’s. This way you have a place to store the numbers in a way that identifies what the numbers represent. The T’s also help you make inferences. Notice that in this question there is a unit conversion from minutes to hours. For GMAT work and rate questions it’s a great idea to double check the units before committing to an answer.

GMAT Question of the Day Problem Solving Work and Rate Solution 4 Diagram

GMAT Question of the Day – PS –  Work/Rate 4

GMAT Question of the Day – Problem Solving – Work/Rate

Worker A and Worker B working together can paint 31 walls per day. If Worker B increases his rate by 1/3 the two of them could paint 34 walls per day. How many walls can Worker A paint in a day?

A. 9

B. 15

C. 19

D. 20

E. 22

Answer Show

GMAT Question of the Day Solution

This work question may seem a slight bit out of the ordinary but you can tackle it the same way as you would tackle any other work/rate question. Make T’s to organize the information and then work your way towards what you are solving for. In this case we are looking for the rate of Worker A. We are given some information about the two people working together and some information about Worker B. Most likely we’ll be subtracting the rate of Worker B from the combined rate to get the Rate of Worker A. In this question you’re not directly given the rate of Worker B but you’re really only one step removed from figuring that out.

GMAT Question of the Day Problem Solving Work/Rate Solution Diagram

If you are still a little confused about how we figured out that the rate of Worker B is equal to 9 you might consider setting up some parallel equations. Then you can subtract out the rate of Worker A and solve for the rate of Worker B. Here’s an Official GMAT cooperative distance, work, and rate question to practice on: Pumps A, B, and C operate at their respective constant rates…

GMAT Question of the Day Problem Solving Work/Rate Solution 2 Diagram

 

Here are another bunch of cooperative work and rate questions to practice on:

Here’s a cooperative rate question from the GMAT Official Guide. It’s has the same components mixed up in a slightly different way: Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours

These are all from GMAT Question of the day:

Work and Rate 1

Work and Rate 2

Work and Rate 3

Work and Rate 4

Work and Rate 5

For more distance, work, and rate examples you can also check out our GMAT Question of the Day page

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GMAT Question of the Day – PS – Work/Rate 2

GMAT Question of the Day – Problem Solving – Work and Rate

Paul and Janice are headed directly towards each other with Paul traveling at a rate 5/7ths that of Janice. If Janice can cover the current distance between them in 120 hours and if they maintain constant speed and direction then in how many hours will they meet?

A. 70

 

B. 72

 

C. 74

 

D. 75

 

E. 77

 

Answer Show

Here’s an Official GMAT distance, work, and rate question and explanation to practice on: Pumps A, B, and C operate at their respective constant rates…

GMAT Work Rate

Here are another bunch of cooperative work and rate questions to practice on:

Here’s a GMAT Cooperative Work and Rate Question from the GMAT Official Guide. It looks different but the components are very similar and you’ll also use similar tools to solve.

And here is a very typical but tough cooperative/combined rate question from the GMAT Prep Tests 1 and 2. These are common so it’s important to know the style:

Mary passed a certain gas station on a highway while traveling west at a constant speed of 50 miles per hour. Then, 15 minutes later, Paul passed the same gas station while traveling west at a constant speed of 60 miles per hour. If both drivers maintained their speeds and both remained on the highway for at least 2 hours, how long after he passed the gas station did Paul catch up with Mary?

Here are more examples from GMAT Question of the day:

Work and Rate 1

Work and Rate 2

Work and Rate 3

Work and Rate 4

Work and Rate 5

For more distance, work, and rate examples you can also check out our GMAT Question of the Day page

GMAT Question of the Day Signup

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