Problem of the Week

Problem B and Solution

Fare’s Fair!

Problem

Three brothers, Andy, Bob, and Curly, take a taxi together home from the airport.
Their homes lie along the same route;

• Andy’s is \(21\) km from the airport,

• Bob’s is \(42\) km, and

• Curly’s is \(63\) km.

If the taxi fare is $2.00 per km, try to find at least two possible fair ways for each of the three travellers to pay the driver (not including the tip)?

Solution

The total cost is \(\$2\) per km \(\times\ 63\) km \(= \$126\). Here are three possible ways to pay the driver.
Solution 1:

If each passenger pays \(\frac{1}{3}\) of the total cost, then they each pay \(\$126 \div 3 = \$42\).

Solution 2:

Three passengers travel the first \(\frac{1}{3}\) of the trip (21 km) for $42, so each pays \(\$42 \div 3 = \$14\) for that portion of the trip. So Andy pays $14.

Two passengers travel the next 21 km for $42, so each pays \(\$42\div 2 = \$21\) each for that portion of the trip plus \(\$14\) for the first portion of the trip. So Bob pays \(\$21 + \$14 = \$35\).

Only one passenger, Curly, travels the final 21 km for $42. So Curly pays \(\$14 + \$21 + \$ 42 = \$ 77\) for all three portions of the trip.
Notice the total paid is \(\$14 + \$35 + \$77 = \$126\).

Solution 3:

A third possibility is that they each pay according to their distance travelled. Andy travels 21 km, Bob travels 42 km, and Curly travels 63 km, a total of \(21 + 42 + 63 = 126\) km. Thus it would cost \(\$126 \div 126\) km \(= \$1.00\) per km per person. So Andy pays $21, Bob pays $42, and Curly pays $63.