GMAT Quantitative: Distance and Rate Practice
This type of GMAT test question sounds like the beginning of a joke but leaves most students groaning in front of their computers: “A train leaves the station at 4:53am going east at 60mph. A second train….” These word problems are often long, confusingly worded, and just plain boring. The intimidation factor comes from not knowing how to set up the algebra. Let’s look at two must-know formulas that will help boost your test prep confidence.
The first important formula to memorize is: D = R x T. This stands for Distance = Rate x Time. It is perfectly acceptable to also think of it as Time = Distance / Rate or as Rate = Distance / Time. Usually the “Rate” is speed but it could be anything “per” anything. In a word problem, if you see the word “per” you know this is a question involving rates.
The second formula is: Average Rate = Total Distance / Total Time. This is its own special concept and you will notice that it is NOT an Average of the Speeds (which would be something like the Sum of the Speeds / the Number of Different Speeds or what we know as the Arithmetic Mean). Average Rate is a completely different concept, so do not let the common word “average” confuse you.
GMAT Rate Problems: Useful Formulas
- Rate * Time = Work (or Distance)
- Time for Two People to do a Job Together = (Product of their separate Times) / (Sum of their separate Times)
- ½ (smaller time) < Time it takes to do a job together < ½ (larger time)
- Rate of 2 or more things working together = Sum of those individual rates
- Average Speed for Entire Trip = (Total Distance) / (Total Time)
GMAT Rate Problems: Key Takeaways
- Overall, don’t let Rates worry you too much or suck up too much study time, given the fact that these problems aren’t appearing with high frequency on the official test right now. Put those Useful Formulas onto flashcards until they’re 2nd nature.
- The biggest thing that helps is Making Up My Own Number for Distance or Work, when one isn’t provided.
- To get some harder questions correct, you may find that Scaling Up Ratios or Reciprocal Thinking is the easiest way to arrive at the answer.
- It can pay to develop some chops at Approximating, especially since some of these problems might be good contenders for skipping.
Now let’s look at several GMAT practice problems:
GMAT Practice Question 1
I got in my car and drove 40 miles to see my cousin and was going 20 mph. It took me 2 hours to get there. Then, I left my cousin’s and drove another 30 miles to the store but this time went 10mph. It took me 3 hours to arrive at the store. What was my “Average Speed” for the whole trip?
GMAT Practice Question 1 Answer and Explanation
Average Speed = Total Distance / Total Time. I traveled 40 miles + 30 miles so my Total Distance was 70 miles. I drove for 2 hours + 3 hours so my Total Time was 5 hours. 70/5 = 14. My 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 we had just averaged the two speeds (10mph and 20mph) we would have gotten 15mph. Think of Average Speed as a weighted average. I spent more time in the problem going 10mph than 20mph, so it makes sense that the Average Speed would be closer to 10mph.
Be careful because the “Average of the Speeds” will often be a tempting wrong answer choice.
GMAT Practice 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?
GMAT Practice Question 2 Answer and Explanation
To find the “Average Rate” of the bus, we know we will need to find the Total Distance and the Total Time, so let’s see how we can use the D = R x T formula to find the missing info.
For the first part of the trip, we know that 30 miles = 15mph x T, so we know that T = 2 hours. For the middle part of the trip, we know that D = 10mph x 3 hours, so we know that D = 30 miles. For the last part of the trip, we know that 40 miles = R x 2 hours, so we know that R = 20mph.
Now we can find the Total Distance and the Total Time. Total Distance = 30 miles + 30 miles + 40miles = 100 miles. Total Time = 2 hours + 3 hours + 2 hours = 7 hours.
So the Average Rate = 100 miles/ 7 hours = 14.28mph.
GMAT Practice 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; her average speed rose to 14 miles per hour. If the entire run took Tracey exactly one hour to complete and she did not make any stops, how many miles is the trail one way?
GMAT Practice Question 3 Answer and Explanation
For the way up the hill, we know that D = 6mph x T.
For the way down the hill, we know that D = 14mph x T. Since we went know that the distance up the hill was the same as the distance down the hill, we can pick a number for D. Let’s choose “84” since it is a multiple of both 6 and 14. If 84 = 6mph x T, then we know that T = 14 hours. If 84 = 14mph x T, then we know that T = 6 hours.
Now we can use another formula, the Average Rate formula, to find the average speed for the WHOLE trip. Average Rate = Total Distance / Total Time
Using our Picked Number of 84, we 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 we know she didn’t “really” go 20 hours. We Picked a Number just so that we could find the ratio of the Total Distance to the Total Time in order to calculate the Average Rate of the ENTIRE journey.
Now that we have found the Average Rate for the whole trip, we can plug it in to the “DIRT” formula to find the ACTUAL distance for the entire journey.
D = R x T
D = 8.4mph x 1 hour
We 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.
GMAT Practice Question 4
A train runs over a straight route from town A to town D. It is scheduled to depart town A at 7am and arrive at town D at 2pm, with 10 minute stops in towns B and C. The train’s top speed is 60mph. The entire length of the route is 320 miles. Will the train arrive on time?
(1) The train experiences a 30 minute delay in town B in addition to its scheduled stop.
(2) The train travels at its top speed for exactly 75% of the trip.
GMAT Practice Question 4 Answer and Explanation
This is a yes/no data sufficiency. In order to answer yes or no, we need to know the time it takes the train to make its journey. From the question, we can see that the train is supposed to take 7 hours to go from Town A to Town D with two 10-minute stops. Thus, the total travel-time of the train is 7 hours – 20 minutes = 6 hours, 40 minutes, or 6.67 hours. A train that takes the full 6.67 hours to travel 320 miles would need to travel at a speed of approx 48mph or faster to make its timetable. Remember that this is what the train is supposed to do. Let’s see how each statement affects the time-table.
With a 30 minute delay, the train’s travel-time is now 6 hours, 10 minutes, or 6.167 hours. With that delay, the train needs to travel at approx 51mph to make its timetable. However, what is missing from this statement is proof that the train actually did increase its speed to make its timetable. Just because it was possible the train arrived on time, doesn’t mean it did. Statement (1) is insufficient.
Let’s look at Statement (2). 75% of the trip is 240 miles out of the total 320 miles. This means the train has to travel the remaining 80 miles in a little less than 3 hours, an average speed of approximately 25 miles an hour in order to make the timetable. This is clearly within the train’s limits, but we do not know anything about the train’s speed for these remaining 80 miles. Statement (2) is insufficient.
Combining the two statements, we know that the train only has 2 hours and 10 minutes to travel the remaining 80 miles. This requires traveling almost 40 mph. This is possible, but we have no certainty that the train accomplished this. The answer, therefore, is (E).
GMAT Rate Problems: Going Beyond the Basics
Let’s remind ourselves that Rate is actually a ratio. It will always be expressed as “some unit of work/distance PER some unit of time”
We usually think of 30 miles per hour as 30mph, but it would behoove us to remember that this is a fraction:
R = 30 miles / 1 hour
Sometimes we are given clunky looking rates, like “Ben takes 3 minutes to stamp 7 envelopes”. If we write that as a fraction, we want to make sure time is on the bottom, as we’re used to seeing it.
Rate = 7 env / 3 min
If we wanted to know how many envelopes Ben does per hour, we could do a Unit Conversion from minutes to hours. We could solve for his Rate per minute (he makes 7/3 envelopes per minute) and then multiply by 60. But, we could also make use of the idea of Scaling Up Ratios.
If he does 7 envelopes in 3 minutes, then he’ll do ___ envelopes in 6 minutes?
14 of course. Twice as much time, twice as many envelopes.
We can use that simple logic with any sort of multiplier.
If he does 7 envelopes in 3 minutes, then he’ll do ___ envelopes in 60 minutes?
140 envelopes. Twenty times as many minutes, twenty times as many envelopes.
Writing Rates horizontally as a ratio, and then scaling them up or down as needed, is often a quicker/easier way to get to your destination.
Our second initial question:
If you could read 20% faster, then what effect would that have on how long it takes you to read the whole thing?
I was originally asking this question in the context of the 1500 word blog post and 120 words/minute reading speed. But in reality, the question doesn’t need any other information beyond the 20% faster. This is the realm of Reciprocal Thinking.
We’re all familiar with the common sense reciprocal idea that “If I could read TWICE as fast, it would only take me HALF as long to read this.” That truism is playing off the idea that that 2/1 and 1/2 are reciprocals.
In order to do this with 20% faster, we need to already be fluent with our percentage and fraction conversions.
20% more than x = 120% of x
20% less than x = 80% of x
For the GMAT, we’re supposed to get really good at switching from percents into simplified fractions. When a GMAT student sees 20%, she thinks 1/5.
1/5 more than x = 6/5 of x
1/5 less than x = 4/5 of x
So when we think about someone reading 20% faster, we think about them reading 120% as fast, or 6/5 as fast.
When I read twice as fast (2/1) it takes me 1/2 the time.
When I read 6/5 as fast, it takes me 5/6 the time.
So, If you could read 20% faster, then what effect would that have on how long it takes you to read the whole thing?
It would take you 5/6 as long. You would save 1/6 the time of what you previously had to spend.
This comes into play on tricky DS questions.
How long did it take for Kanye to drive to the White House?(1) The White House was 240 miles away(2) Had Kanye driven 30% faster, the trip would have taken 30 minutes lessThe 1st statement is insufficient, since only knowing distance doesn’t suffice. We would need to know Kanye’s rate to calculate his time. The 2nd statement doesn’t feel like enough information, but it is.
Driving 30% faster means driving 130% as fast as he actually did.
Driving 3/10 faster means driving 13/10 as fast as he actually did. Thus it will only take 10/13 as much time.
Kanye’s trip would take 3/13 less time. Since statement 2 told us it would take 30 minutes less time, we know that 30 mins = 3/13 (his actual time)
So apparently, his actual time was 130 minutes.
