AMC 10 · 2018 · #1

Grade 8 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

fraction-arithmeticorder-of-operationsexponents identify-subproblemspattern-recognition ↑ Prerequisites: fraction-arithmeticorder-of-operations

📏 Medium solution 💡 1 insight

Problem

Evaluate the nested expression (((2+1)⁻¹+1)⁻¹+1)⁻¹+1, where the ⁻¹ exponent means 'take the reciprocal.' Work from the innermost parentheses outward and match the result to one of the five answer choices.

Pick an answer.

(A)

$\frac58$

(B)

$\frac{11}7$

(C)

$\frac85$

(D)

$\frac{18}{11}$

(E)

$\frac{15}8$

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Identify Subproblems): the expression is three identical nested blocks of the form 'reciprocal, then +1.' Evaluate the innermost block to a single fraction, then feed that fraction into the next block, and again into the last block. Tool #5 (Look for a Pattern) keeps each layer mechanical — flip the fraction, add one. Tool #3 (Eliminate Possibilities) matches the final fraction to the five choices.

1STEP 1

Decode the notation

The ⁻¹ exponent means reciprocal: x⁻¹= $\frac{1}{x}$ , and a fraction flips, ( $\frac{a}{b}$ )⁻¹= $\frac{b}{a}$ . Every layer takes a reciprocal, then adds 1.

x⁻¹ =

\frac{1}{x}

, (

\frac{a}{b}

)⁻¹ =

\frac{b}{a}

2STEP 2

Innermost block

Deepest parentheses first: 2+1=3, so 3⁻¹= $\frac{1}{3}$ , then $\frac{1}{3}$ +1= $\frac{4}{3}$ .

(2+1)⁻¹+1 = 3⁻¹+1 =

\frac{1}{3}

+ 1 =

\frac{4}{3}

3STEP 3

Middle block

Apply the block to $\frac{4}{3}$ : flip to ( $\frac{4}{3}$ )⁻¹= $\frac{3}{4}$ , then $\frac{3}{4}$ +1= $\frac{7}{4}$ .

(

\frac{4}{3}

)⁻¹+1 =

\frac{3}{4}

+ 1 =

\frac{7}{4}

4STEP 4

Outer block

Apply it once more to $\frac{7}{4}$ : flip to ( $\frac{7}{4}$ )⁻¹= $\frac{4}{7}$ , then $\frac{4}{7}$ +1= $\frac{11}{7}$ .

(

\frac{7}{4}

)⁻¹+1 =

\frac{4}{7}

+ 1 =

\frac{11}{7}

5STEP 5

Match the choice

The whole expression equals $\frac{11}{7}$ , which is choice (B).

(((2+1)⁻¹+1)⁻¹+1)⁻¹+1 =

\frac{11}{7}

→ (B)

Answer

\frac{11}{7}

The value should be a little more than 1, since the final step adds 1 to a positive reciprocal

\frac{4}{7}

that is less than 1. Indeed

\frac{11}{7}

≈ 1.57, which lands between the smallest choice

\frac{5}{8}

= 0.625 and the largest

\frac{15}{8}

= 1.875. The denominator 7 also makes sense: the second-to-last value was

\frac{7}{4}

, and flipping it put the 7 on the bottom.

💡Key takeaway

A -1 exponent just flips a number over, so peel the nested parentheses one layer at a time — flip, add one, repeat: $\frac{1}{3}$ +1= $\frac{4}{3}$ , then $\frac{3}{4}$ +1= $\frac{7}{4}$ , then $\frac{4}{7}$ +1= $\frac{11}{7}$ , which is (B).

Decode the notation
Innermost block
Middle block
Outer block
Match the choice

Decode the notation

Innermost block

Middle block

Outer block

Match the choice

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