On a recent trip, Cindy drove her car 290 miles…
In solving this problem we must remember that we are given a rate in miles per gallon. It follows that:
Rate = Distance/Gallons
We are given that she drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons of gasoline, rounded to the nearest gallon. We are being asked to determine a possible range for Cindy’s rate. Thus, we really are trying to determine the maximum value for her rate (the largest distance traveled divided by the smallest number of gallons used) and the minimum value for her rate (the smallest distance traveled divided by the largest number of gallons used).
Let’s first determine the minimum and maximum distance. We know that she drove 290 miles, rounded to the nearest ten miles. Thus, we first determine what values will round to 290 when rounded to the nearest 10 miles. Let’s consider many possible values and then choose the ones that would round to 290:
281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296
From this list, we see that 281, 282, 283, and 284 would all round DOWN to 280, and so we eliminate them. Similarly, at the upper end of the list, 295 and 296 would round UP to 300, and we would likewise eliminate them. This leaves
285, 286, 287, 288, 289, 290, 291, 292, 293, 294
as the numbers that would round to 290, rounded to the nearest 10 miles. Now we choose the minimum and maximum values from this set of numbers.
Minimum = 285 miles
Maximum = 294 miles
Next, we determine the minimum and maximum number of gallons used. We know that she used 12 gallons of gas, rounded to the nearest gallon. Using the same technique as we did earlier for the miles, we obtain:
Minimum = 11.5 gallons
Maximum = 12.4 gallons
Using the formula Rate = Distance/Gallons we see that the maximum rate is the maximum miles divided by the minimum gallons, which is 294/11.5.
Similarly, the minimum rate is the minimum miles divided by the maximum gallons, or 285/12.4
We can now compare these to our answer choices:
(A) 290/12.5 and 290/11.5
(B) 295/12 and 285/11.5
(C) 285/12 and 295/12
(D) 285/12.5 and 295/11.5
(E) 295/12.5 and 285/11.5
We can see that our max and min values fall between the values given in answer choice D.