AMC 10 · 2015 · #7

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

function-evaluationfraction-arithmeticorder-of-operations identify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 2 insights

Problem

A new operation is defined by a◆ b = a-

\frac{1}{b}

("a minus the reciprocal of b"). Using this rule, compute ((1◆2)◆3)-(1◆(2◆3)).

Pick an answer.

(A)

$-\dfrac{7}{30}$

(B)

$-\dfrac{1}{6}$

(C)

(D)

$\dfrac{1}{6}$

(E)

$\dfrac{7}{30}$

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression stacks the new operation inside itself, so Tool #7 (Identify Subproblems) is the natural fit: evaluate each small ◆ from the innermost parentheses outward, finishing the left grouping and the right grouping separately before the final subtraction. Tool #3 (Eliminate Possibilities) handles the built-in trap — a student who assumes ◆ is associative would expect the two groupings to cancel and pick (C) 0; carrying the exact fractions instead shows they differ, leaving only one matching choice.

1STEP 1

Evaluate the innermost left piece

Inside the left group, apply the recipe to 1◆2: a=1, b=2 gives 1- $\frac{1}{2}$ = $\frac{1}{2}$ .

1◆2 = 1-

\frac{1}{2}

\frac{1}{2}

2STEP 2

Finish the left grouping

Now apply ◆3: $\frac{1}{2}$ - $\frac{1}{3}$ over common denominator 6 gives $\frac{3}{6}$ - $\frac{2}{6}$ = $\frac{1}{6}$ for the left group.

(

\frac{1}{2}

)◆3 =

\frac{1}{2}

\frac{1}{3}

\frac{3}{6}

\frac{2}{6}

\frac{1}{6}

3STEP 3

Evaluate the innermost right piece

Now the right group's inside: 2◆3 with a=2, b=3 gives 2- $\frac{1}{3}$ = $\frac{6}{3}$ - $\frac{1}{3}$ = $\frac{5}{3}$ .

2◆3 = 2-

\frac{1}{3}

\frac{6}{3}

\frac{1}{3}

\frac{5}{3}

4STEP 4

Finish the right grouping

Then 1◆( $\frac{5}{3}$ ): the reciprocal of $\frac{5}{3}$ is $\frac{3}{5}$ , so 1- $\frac{3}{5}$ = $\frac{2}{5}$ for the right group.

1◆

\frac{5}{3}

= 1-

\frac{1}{\frac{5}{3}}

= 1-

\frac{3}{5}

\frac{2}{5}

5STEP 5

Subtract the two groupings

Subtract: $\frac{1}{6}$ - $\frac{2}{5}$ = $\frac{5}{30}$ - $\frac{12}{30}$ = - $\frac{7}{30}$ — the groups did not cancel, so (C) 0 is wrong and (A) is right.

\frac{1}{6}

\frac{2}{5}

\frac{5}{30}

\frac{12}{30}

= -

\frac{7}{30}

→ (A)

Answer

\frac{7}{30}

The left grouping

\frac{1}{6}

≈0.167 is smaller than the right grouping

\frac{2}{5}

=0.4, so their difference should be a small negative number — and -

\frac{7}{30}

≈-0.233 fits. Choice (C) 0 would require the operation to be associative, but ◆ subtracts a reciprocal and is not, so a nonzero answer is expected. The magnitude

\frac{7}{30}

is small, ruling out any large value, and the sign is negative, pinning down (A) over its positive twin (E).

💡Key takeaway

A made-up symbol is just a recipe: work the innermost parentheses first, and remember that moving the parentheses can change the answer.

Evaluate the innermost left piece
Finish the left grouping
Evaluate the innermost right piece
Finish the right grouping
Subtract the two groupings

Evaluate the innermost left piece

Finish the left grouping

Evaluate the innermost right piece

Finish the right grouping

Subtract the two groupings

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