Welcome back to our series on GMAT quant rate problems. In the last article, we learned about the usefulness of “machine hours” for solving problems featuring more than two or three machines. This article will prepare you for questions that involve rates of consumption of fuel.
GMAT quant rates problems: Rates of consumption of fuel
These problems are unique in that they attach two kinds of rates to a single “machine,” or vehicle: a speed (distance per unit time) and a rate of fuel consumption.
Further complicating matters is the fact that the fuel consumption rate may be expressed in the familiar way of gallons per unit of distance (miles or kilometers) or in the unconventional way of gallons per unit of time (usually hours).
- gallons/mile or miles/gallon
- gallons/hour or hours/gallon
Note that gallons/mile and miles/gallon are simply reciprocal fractions, as are gallons/hour and hours/gallon. Since speed = distance/time, speed always has a unit in common with a rate of consumption. It is this commonality that enables us to “translate” between the two ways of expressing fuel consumption.
If fuel consumption is expressed in terms of distance, we can use the given speed to “translate” this consumption rate into terms of time. If fuel consumption is expressed in terms of time, we can use the given speed to “translate” the consumption rate in terms of distance.
Here’s an official GMAT problem that directly asks for this “translation”:
While traveling at a constant speed of 32 miles per hour, a certain motorboat consumes 24 gallons of fuel per hour. What is the fuel consumption of this boat at this speed measured in miles traveled per gallon of fuel?
(A) 2/3
(B) 3/4
(C) 4/5
(D) 4/3
(E) 3/2
The fuel consumption rate has been expressed in terms of time (gallons per hour), and we are asked to “translate” this rate to terms of distance (miles per gallon). Since the consumption rate corresponds to the given speed of 32 miles per hour, we can use the common unit of hours to perform this “translation.”
24 gallons per hour
32 miles per hour
Putting these rates together, we can say that in a single hour of travel, two things happen: 32 miles are covered, and 24 gallons of fuel are consumed. So we have a fuel consumption rate in terms of distance: 32 miles per 24 gallons! Reducing this fraction will yield the rate in terms of miles per single gallon. 32 miles / 24 gallons = 4 miles / 3 gallons or 4/3 miles per gallon. The correct answer is D.
Let’s try the same method on another official problem:
During a certain time period, Car X traveled north along a straight road at a constant rate of 1 mile per minute and used fuel at a constant rate of 5 gallons every 2 hours. During this time period, if Car X used exactly 3.75 gallons of fuel, how many miles did Car X travel?
(A) 36
(B) 37.5
(C) 40
(D) 80
(E) 90
This problem requires us to go one step further than simply “translating” from gallons per hour to gallons per mile (or miles per gallon). Once we have this fuel consumption rate in terms of distance instead of time, we must use the given consumption of 3.75 gallons to determine the number of miles traveled during this consumption.
We should start by fixing the given speed of “1 mile per minute.” Since fuel consumption is given in gallons per hour (not gallons per minute), using minutes instead of hours for speed isn’t very helpful. 1 mile per minute = 60 miles per hour.
The given fuel consumption rate is “5 gallons every 2 hours.” There are now two ways to proceed. We can multiply the speed of 60 miles per hour by 2 so that the speed will have the “2 hours” in common with the consumption rate. Or we can divide the consumption rate of 5 gallons every 2 hours by 2 so that this consumption rate will have the “per hour” in common with the speed. Here are both options.
60 miles/hour * 2 = 120 miles/2 hours
5 gallons/2 hours = 5 gallons/2 hours
60 miles/hour = 60 miles/hour
5 gallons/ 2 hours / 2 = 2.5 gallons/hour
The first option is probably easier since it avoids decimals. But either way, we can perform the division to determine the number of miles traveled per single gallon of fuel consumed. 120 miles / 5 gallons = 24 miles per gallon, and 60 miles / 2.5 gallons = 24 miles per gallon.
Since the car travels 24 miles for every gallon of gas it consumes, we can find the number of miles traveled during 3.75 gallons of fuel consumption by multiplying 24 by 3.75. 24 * 3.75 = 90, so the correct answer is E.
Here’s a final – and different – fuel consumption problem:
A car traveled 462 miles per tankful of gasoline on the highway and 336 miles per tankful of gasoline in the city. If the car traveled 6 fewer miles per gallon in the city than on the highway, how many miles per gallon did the car travel in the city?
(A) 14
(B) 16
(C) 21
(D) 22
(E) 27
This problem can be challenging because it is different from most fuel consumption problems. But a focus on the question asked by this problem can make our solving process a simple one. We want to know the fuel consumption rate in miles per gallon for city driving.
We are provided with two differences between city driving and highway driving: one in terms of miles per tankful and another in terms of miles per gallon. This is helpful! The difference between highway miles per gallon and city miles per gallon is 6.
Another way to say this is that for every gallon of gas in the tank, the car can travel 6 miles further on the highway than in the city. Let’s translate this sentence into algebra:
(# of gallons) * 6 = (highway fuel range – city fuel range)
“Fuel range” is another way to say “miles per tankful,” as given in the problem. So we can plug in those “miles per tankful” values:
(# of gallons) * 6 = 462 – 336
(# of gallons) * 6 = 126
(# of gallons) = 126 / 6 = 21
But C is not the correct answer! Remember that we were asked for the car’s miles per gallon fuel consumption rate in the city, not the number of gallons in the tank. But now we are only one step away.
336 miles / 21 gallons = 16 miles per gallon
If this division gives you trouble, you can determine the correct answer choice by using units digits. The units digit of the quotient of 336/21 must be 6, in order for the product (quotient * 21) to have the units digit of 6 that we see in the dividend of 336. So the only possible answer choice is B.
This concludes our survey of GMAT quant fuel consumption problems. Remember, when provided with both a speed and a rate of fuel consumption, multiply or divide one of these rates such that there is a “match” in the value of the unit common to both rates – whether that unit is speed or distance. Then simply reduce the fraction, and you have the fuel consumption rate in the term you need.
The next article in the series will address rate problems that incorporate geometry elements.
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Contributor: Elijah Mize (Apex GMAT Instructor)