25 Feb 2013

Math-e-Monday: A Merchant from Pisa

This week I am reading the Liber Abaci (The Book of Calculations) by Leonardo Pisano, better known as Fibonacci. The book, published in 1202, contains the famous problem of rabbits breeding in such a way as to generate the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 and so on. However, the Liber Abaci also contains a wealth of other problems, and this week I shall post a small selection of questions inspired by those found in the book.

The question below is fairly easy and can be done by a lower secondary, or even upper primary, student. Interestingly, Fibonacci did not have a well-developed algebra, so his solution is a numerical recipe or algorithm. As part of Fibonacci's promotion of the Hindu-Arabic numerals, he has numerous questions involving fractions. Many such questions are designed to show the ease with which fractions can be manipulated using his 'new' system of numbers compared with using Roman numerals. This is one such question.




The Question

A merchant from Pisa sets off to Lucca with a certain amount of money. In Lucca he doubles his money but also has expenses of 20 denari. He then travels to Florence where, again, he doubles his money but the trip costs him 20 denari. Finally, he heads back to Pisa where he doubles his money again but pays 20 denari in expenses. He ends up with nothing.

How many denari did he start off with?



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