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=1/x, and a fraction flips, (a/b)⁻¹=b/a. Every layer takes a reciprocal, then adds 1.

x⁻¹ = 1/x, (a/b)⁻¹ = b/a

2STEP 2

Innermost block

Deepest parentheses first: 2+1=3, so 3⁻¹=1/3, then 1/3+1=4/3.

(2+1)⁻¹+1 = 3⁻¹+1 = 1/3 + 1 = 4/3

3STEP 3

Middle block

Apply the block to 4/3: flip to (4/3)⁻¹=3/4, then 3/4+1=7/4.

(4/3)⁻¹+1 = 3/4 + 1 = 7/4

4STEP 4

Outer block

Apply it once more to 7/4: flip to (7/4)⁻¹=4/7, then 4/7+1=11/7.

(7/4)⁻¹+1 = 4/7 + 1 = 11/7

5STEP 5

Match the choice

The whole expression equals 11/7, which is choice (B).

(((2+1)⁻¹+1)⁻¹+1)⁻¹+1 = 11/7 → (B)

Answer

11/7

The value should be a little more than 1, since the final step adds 1 to a positive reciprocal 4/7 that is less than 1. Indeed 11/7 ≈ 1.57, which lands between the smallest choice 5/8 = 0.625 and the largest 15/8 = 1.875. The denominator 7 also makes sense: the second-to-last value was 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: 1/3+1=4/3, then 3/4+1=7/4, then 4/7+1=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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