2009 AMC 10A Problem 3

P1-10~1 minPrint

Fractions & Decimals Set up the equation/formula and compute; no special trick needed

Which of the following is equal to $1 + \frac{1}{1+\frac{1}{1+1}}$ ?

Show solution without answering

Key insight

Work from the innermost part outward: 1 + 1 = 2, then 1 + 1/2 = 3/2, flip it to get 2/3, then 1 + 2/3 = 5/3.

Solution

A fraction stacked inside a fraction is easiest to handle from the inside out, one layer at a time.

The deepest part is $1 + 1 = 2$ .
The next layer is $1 + \frac{1}{2} = \frac{3}{2}$ .
Now we need $\dfrac{1}{3/2}$ . Dividing $1$ by a fraction flips it, so this is $\frac{2}{3}$ .
The outer layer is $1 + \frac{2}{3} = \frac{5}{3}$ .

The answer is $\boxed{\textbf{(C)}\ \frac{5}{3}}$ .

Why this works

Each layer of this stacked fraction is "one plus one over the layer below it." Doing one layer at a time and writing each result as a single fraction keeps you from mixing up which number goes on top.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Work from the inside out.

The innermost piece is $1 + 1 = 2$ , so the middle denominator is
$1 + \frac{1}{2} = \frac{3}{2}.$
Its reciprocal is $\frac{2}{3}$ , so the whole expression is
$1 + \frac{2}{3} = \frac{5}{3}.$

The answer is $\boxed{\textbf{(C)}\ \frac{5}{3}}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 11: Decimals and Fractions
Prealgebra, Chapter 6: Decimals (§6.3, §6.4)