So Many Fractions.

Algebra Level 3

Evaluate:

1 4 + 1 4 + 1 + 1 4 4 4 1 4 + 1 + 1 1 + 4 = m n \Large\frac{\frac{\frac{1}{4}+1}{4}+\frac{1+\frac{1}{4}}{4}}{\frac{4}{\frac{1}{4+1}+\frac{1}{1+4}}} = \frac{m}{n}

Then find m + n m+n , where m m and n n are coprime positive integers.


The answer is 17.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Munem Shahriar
Jul 28, 2018

Simplifying the numerator:

1 4 + 1 4 + 1 4 + 1 4 = 5 4 4 + 5 4 4 = 5 4 × 1 4 + 5 4 × 1 4 = 10 16 = 5 8 \begin{aligned} \dfrac{\frac 14 + 1}4 + \dfrac{\frac 14 + 1}4 & = \dfrac{\frac 54}4 + \dfrac{\frac 54}4 \\ & = \dfrac 54 \times \frac 14 + \dfrac 54 \times \dfrac 14 \\ & = \dfrac{10}{16} \\ & = \dfrac 58 \\ \end{aligned}

Simplifying the denominator:

4 1 5 + 1 5 = 4 2 5 = 4 × 5 2 = 10 \begin{aligned} \dfrac 4{\frac15 + \frac 15} & = \dfrac 4{\frac 25} \\ & = 4 \times \dfrac 52 \\ & = 10 \\ \end{aligned}

So m n = 5 8 10 = 5 8 × 1 10 = 5 80 = 1 16 . \dfrac mn = \dfrac{\frac 58}{10} = \dfrac 58 \times \dfrac 1{10} = \dfrac 5{80} = \dfrac 1{16}. Hence m + n = 1 + 16 = 17 m + n = 1 +16 = \boxed{17}

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