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## Some People Are Moving

Fifty people wish to reach a place 27 1/2 miles away. The only available transportation is a bus having a capacity of 30 people and a speed of 35 mph. The party is divided into two roughly equal groups, which start at the same time. The first group starts on foot, walking at an average rate of 4 mph. The second group rides on the bus a certain distance and then walks the rest of the way at an average rate of 3 mph. The bus returns to the first group and to carry it the rest of the way. How far should each group walk in order that all may arrive at their destination at the same time? How many hours are required for the transfer?

Note: Once you solve the above problem, can you solve the generalized version below...and also find the range-conditions for the variables under which answers are possible.

Fifty people wish to reach a place n miles away. The only available transportation is a bus having a capacity of 30 people and a speed of m mph. The party is divided into two roughly equal groups, which start at the same time. The first group starts on foot, walking at an average rate of p mph. The second group rides on the bus a certain distance and then walks the rest of the way at an average rate of q mph. The bus returns to the first group and to carry it the rest of the way. How far should each group walk in order that all may arrive at their destination at the same time? How many hours are required for the transfer?

Source: H. Rice & R. Knight's Technical Mathematics With Calculus, 1966, p. 174

Hint: Somehow d = rt sounds useful...or even, a graph of the situation sounds useful...or even, guess-and-check sounds useful...

Solution Commentary: Though I will not vouch for it, the text provided this answer: "1st group walks 4.91 miles; 2nd group walks 3.57 miles; time for entire transfer = 1.87 hr."

Do you agree? And what about the general case?