Manhattan Challenge Problem of the week! - 05/21/07

Edwin is planning to drive from Boston to New Orleans. By what percent would his travel time be reduced if Edwin decides to split the driving time equally with his friend George, instead of making the trip alone?

(1) The driving distance from Boston to New Orleans is 1500 miles.
(2) George’s driving speed is 1.5 times Edwin’s driving speed.

Answer: B

OE - The question asks for the percent decrease in Edwin’s travel time. To determine this, we need to be able to find the ratio between, T1 (the travel time if Edwin drives alone) and T2 (the travel time if Edwin and George drive together). Note that we do NOT need to determine specific values for T1 and T2; we only need to find the ratio between them.

Percentage change is defined as follows: Difference/Original = (T1 - T2)/ T1 = 1 - (T2/ T1)

Ultimately, we can solve the percentage change equation above by simply determining the value of T2 /T1

Using the formula Rate × Time = Distance, we can write equations for each of the 2 possible trips

T1 = Travel time if Edwin drives alone
T2 = Travel time if Edwin and George drive together
E = Edwin’s Rate
G = George’s Rate
D = Distance of the trip

If Edwin travels alone: ET1 = D
If Edwin and George travel together: .5(E + G)T2 = D

(Since Edwin and George split the driving equally, the rate for the trip is equal to the average of Edwin and George’s individual rates).

Since both trips cover the same distance (D), we can combine the 2 equations as follows:

ET1 = .5(E + G)T2

Then, we can isolate the ratio of the times (T2/T1) as follows:

E/ .5(E + G) = T2/ T1

Now we look at the statements to see if they can help us to solve for the ratio of the times.

Statement (1) gives us a value for D, the distance, which does not help us since D is not a variable in the ratio equation above.

Statement (2) tells us that George’s rate is 1.5 times Edwin’s rate. Thus, G = 1.5E. We can substitute this information into the ratio equation above:

E/ .5(E + G) = T2/ T1 ---> E/ .5(E + 1.5E) = T2/ T1 ---> E/ .5E + .75E = T2/T1
---> E/ 1.25E = T2/ T1 ---> 1/ 1.25 = T2/ T1 ---> .8 = T2/ T1

Thus, using this ratio we can see that Edwin’s travel time for the trip will be reduced as follows:

1 - (T2/T1) = 1 - .8 = .2 ---> 20%

Statement (2) alone is sufficient to answer the question.

The correct answer is B.