Suppose 2+11+12+23+x=(144)/(53). What is the value of x?

2021 AMC 12B Spring Problem 3

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Suppose

2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+x}}}=\dfrac{144}{53}.

What is the value of

x?

\dfrac{3}{4}

\dfrac{7}{8}

\dfrac{14}{15}

\dfrac{37}{38}

\dfrac{52}{53}

Peel off one layer at a time, starting by subtracting

2

from

\dfrac{144}{53}

After

\dfrac{144}{53}-2=\dfrac{38}{53},

take reciprocals and continue inward

Working from the outside in,

\dfrac{144}{53}-2=\dfrac{38}{53},

so the inner fraction equals

\dfrac{38}{53}.

Its reciprocal gives

1+\cfrac{1}{2+\frac{2}{3+x}}=\dfrac{53}{38},

\cfrac{1}{2+\frac{2}{3+x}}=\dfrac{15}{38}.

Then

2+\dfrac{2}{3+x}=\dfrac{38}{15},

\dfrac{2}{3+x}=\dfrac{8}{15},

giving

3+x=\dfrac{15}{4}.

Therefore

x=\dfrac{15}{4}-3=\dfrac{3}{4}.

Thus, the correct answer is A.