92. Arbitration of exchange optional

The lesson, step by step

1. Warm-up

   Ask: If 1 apple = 2 pears and 1 pear = 3 plums, how many plums is 1 apple? (6.) That chain is today's whole idea.

2. The idea

   Read:

   Owing to the constant variation of exchange, it is sometimes advantageous to draw through an intermediate point, or points, in place of drawing directly. This is called Circular Exchange.

   The process of finding the proportional exchange between two places by means of Circular Exchange, is called the Arbitration of Exchange.

3. Work the book's example

   A merchant wishes to remit $2240 to Lisbon: is it more profitable to buy a bill directly on Lisbon, at 1 milreis = $1.10; or to remit through London and Paris, at £1 = $4.88 = 25.20 francs, to Lisbon, 1 milreis = 5.5 francs?

   Direct: $2240 ÷ 1.10 = 2036.364 milreis. Through London and Paris: $2240 × 25.20 ÷ (4.88 × 5.5) = 2103.129 milreis. The roundabout way gains 66.765 milreis.

4. The chain rule

   Read:

   Form a series of equations, expressing the conditions of the question;

   and the reason for its name:

   The first is sometimes called the chain rule, because each equation and the one following, as well as the last and first, are connected as in a chain

5. Guided practice

   Set up the Memphis problem from the book as a chain: $1.015 in Memphis = $1 in Chicago; $.99 in Chicago = $1 in Detroit; $.995 in Detroit = $1 in New York. Multiply and divide to get the value of $8400 sent that way.

6. Independent practice and wrap-up

   Your student tries one or two of the examples and checks the printed answers. Ask: Why would anyone take the long way round? (When the rates make it cheaper.)