Fractions

Algebra Level 2

1 2 + 1 2 + 1 2 + 1 2 = A B \huge \frac{1}{ 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}} }=\frac{A}{B}

A A and B B are coprime positive integers. Find the value of B − 2 A B-2A .

3 9 5 4

Zeeshan Ali by Zeeshan Ali , 25, Pakistan

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1 solution

Zeeshan Ali
Dec 20, 2015

1 2 + 1 2 + 1 2 + 1 2 \frac{1}{ 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}} } = 1 2 + 1 2 + 1 5 2 =\frac{1}{ 2+\frac{1}{2+\frac{1}{\frac{5}{2}}} } = 1 2 + 1 2 + 2 5 =\frac{1}{ 2+\frac{1}{2+\frac{2}{5}} } = 1 2 + 1 12 5 =\frac{1}{ 2+\frac{1}{\frac{12}{5}} } = 1 2 + 5 12 =\frac{1}{ 2+\frac{5}{12} } = 1 29 12 =\frac{1}{\frac{29}{12} } = 12 29 =\frac{12}{29} hence 29 − 2 × 12 29 - 2\times12 = 5 \boxed{5}

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