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My teacher challenged us:

"Can you express the fraction $55\over 108$ as the sum of two unit fractions$?$"

I figured out that ${1\over 54} + {1\over 2} = {56\over 108}$ but I could not figure out a way to get $55\over 108$ as a sum of two unit fractions.

Any advice about how to approach this would be much appreciated.

user118494
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guestabc12345
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2 Answers2

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How about $\frac 12 + \frac 1{108}$? Finding them in the general case can be difficult. You can look up Egyptian fractions to find ways to express rationals as unit fractions, but often finding the shortest expansion is not easy.

Ross Millikan
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$\frac{55}{108}=\frac{1}{x}+\frac{1}{y}$, let $y≥x>0$, so $\frac{55}{108}>\frac{1}{x}⟹x>\frac{108}{55}$. The $x$ is integer, so $x≥2$, also rewrite the equation as form like $\frac{1}{108}=\frac{1}{55x}+\frac{1}{55y}$, and recalling $y≥x>0$, so that $55x≤108×2⟹x≤\frac{306}{55}$, hence $2≤x≤5$, this means that the possible values of $x$ are $x=2,3,4,5$，trying $x$ in such range one by one there is only one solution $\frac{55}{108}=\frac{1}{2}+\frac{1}{108}$.

xMath
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