$1 + 2 \div \left\{ 1 + 2 \div \left( 1 + \frac{1}{3} \right) \right\}$ is equal to

### Concept & Strategy This is a classic simplification problem requiring the strict application of the BODMAS/PEMDAS rule. You must resolve brackets from the innermost parentheses `()` outward to the curly braces `{}` before performing the final arithmetic outside. ### Step-by-Step Solution * **Given:** $1 + 2 \div \left\{ 1 + 2 \div \left( 1 + \frac{1}{3} \right) \right\}$ * **Step 1: Solve the innermost parentheses** * $1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}$ * The expression becomes: $1 + 2 \div \left\{ 1 + 2 \div \frac{4}{3} \right\}$ * **Step 2: Solve the division inside the curly braces** * Convert division to multiplication by inverting the fraction: $2 \div \frac{4}{3} = 2 \times \frac{3}{4}$ * $2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2}$ * The expression becomes: $1 + 2 \div \left\{ 1 + \frac{3}{2} \right\}$ * **Step 3: Solve the addition inside the curly braces** * $1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2}$ * The expression becomes: $1 + 2 \div \frac{5}{2}$ * **Step 4: Final division and addition** * Division comes before addition: $2 \div \frac{5}{2} = 2 \times \frac{2}{5} = \frac{4}{5}$ * Finally, add to $1$: $1 + \frac{4}{5} = 1 \frac{4}{5}$ ### Exam Strategy & Shortcut Write the expression out and solve it "in place" by crossing out resolved sections. Once you see $(1 + \frac{1}{3})$ is $\frac{4}{3}$, immediately flip it and multiply by $2$ to get $\frac{3}{2}$ mentally. Adding $1$ gives $\frac{5}{2}$. Flip it again and multiply by $2$ to get $\frac{4}{5}$. Adding $1$ gives $1 \frac{4}{5}$. This requires almost zero scratchpad writing. ### Common Pitfall A very common mistake is ignoring the order of operations and adding the $1 + 2$ at the very beginning to make it $3 \div \{ \dots \}$. Always remember that addition is the absolute last step unless it is contained within a bracket. ### Final Answer **Therefore, the correct answer is $1 \frac{4}{5}$.**