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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