Simplify

Concept: To simplify a continued (nested) fraction, work from the innermost level outward: reduce the 'whole number + fraction' expression at each level to a single fraction. One level up, that collapsed fraction sits in a denominator position -- divide by it (equivalently, multiply by its reciprocal) to evaluate the fraction at that level, then add the result to the whole number there. Repeat this collapsing step by step until only the outermost expression is left.

1. Innermost level: 1 + 4/5 = 9/5.

2. Substitute into the next level up and simplify: 2 + 3/(9/5) = 2 + (3 x 5)/9 = 2 + 5/3 = 11/3.

3. Substitute into the next level up and simplify: 1 + 2/(11/3) = 1 + (2 x 3)/11 = 1 + 6/11 = 17/11.

4. Substitute into the outermost level and simplify: 1 + 1/(17/11) = 1 + 11/17 = 28/17 = 1 11/17.

Cross-check: Converting 1 11/17 to a decimal gives 28/17 = 1.647 (approx). Evaluating the same continued fraction directly in decimals from the bottom up -- 1 + 4/5 = 1.8, then 3/1.8 = 1.667 (approx) so 2 + 1.667 = 3.667, then 2/3.667 = 0.545 (approx) so 1 + 0.545 = 1.545, then 1/1.545 = 0.647 (approx) so 1 + 0.647 = 1.647 -- gives the same value, confirming the fraction-based working above.

Answer: The expression simplifies to 1 11/17.