## 1 Mar 2013

### Prize Maths Quiz: Fractions of Phi (PMQ8)

This week has been dedicated to the Liber Abaci by Fibonacci. In this book, Fibonacci introduces into Italy the ‘Indian figures’, 1 to 9, plus the 0, which the Arabs called ‘zephir’. In order to show the superiority of this numbering system compared to the Roman system, Fibonacci spends a considerable number of pages handling fractions. Trade within the Mediterranean involved a plethora of coinage, weights and measures and the wisdom of using a single number system to handle them all was not lost on the merchants of Pisa, and beyond.

A unit fraction is one where the numerator is 1 and the denominator is some other whole number. There are times when such fractions continue to be useful, especially with non-decimal units of measurement such as those based on 12, 24 or 60 units. Fibonacci shows how to handle both decimal fractions and sums of unit fractions.

For example, the circle to the left is subdivided in such a way to show that one-half is equal to a third plus a sixth. Alternatively, the other half is divided into 1/3 plus 1/9 and 1/18. Without any further calculations, the diagram itself also shows that we can express 1/3 as 1/6 + 1/9 + 1/18. And finally, we have 2/3 = 1/2 + 1/9 + 1/18.

The Question

Let us take the following three numbers from the Fibonacci sequence: 21, 34, 55. Below are two fractions formed from adjacent terms, both expressed as sums of unit fractions. With a, b, c and d being whole numbers, and a < b < c < d, find the value of (a + b + c + d). As always, you must show a valid method together with the correct answer.

During this week, we have barely scratched the surface of Fibonacci's work in mathematics. Although we have concentrated on Fibonacci’s practical and mercantile mathematics, he also wrote on number theory and geometry. We shall return to explore some of these other works in the future.

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How to Enter

Send your complete solution by email to [competition closed - email removed]. This email address shall be removed after the competition closes to avoid spam. This PMQ8 competition closes on Sunday 3 March at 23:59 GMT.

The Prize

The prizes for this PMQ8 are 3 free places in our Online Classroom for 2 months. The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received.

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