Suppose 2+\cfrac11+\cfrac12+\cfrac23+x=(144)/(53). What is the value of x?

2021 AMC 12B Spring Problem 3 — Solution

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

What is the value of $x?$

$\dfrac{3}{4}$

$\dfrac{7}{8}$

$\dfrac{14}{15}$

$\dfrac{37}{38}$

$\dfrac{52}{53}$

Working from the outside in, $\dfrac{144}{53}-2=\dfrac{38}{53},$ so the inner fraction equals $\dfrac{38}{53}.$

Its reciprocal gives $1+\cfrac{1}{2+\frac{2}{3+x}}=\dfrac{53}{38},$ so $\cfrac{1}{2+\frac{2}{3+x}}=\dfrac{15}{38}.$

Then $2+\dfrac{2}{3+x}=\dfrac{38}{15},$ so $\dfrac{2}{3+x}=\dfrac{8}{15},$ giving $3+x=\dfrac{15}{4}.$

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

Thus, the correct answer is A.