AMC 10 Step by Step
Log inSign up

2015 AMC 10B Problem 7

P1-10~2 minPrint
FunctionsFractions & DecimalsSet up the equation/formula and compute; no special trick needed

Consider the operation "minus the reciprocal of," defined by a⋄b=a−1ba\diamond b=a-\frac{1}{b} . What is ((1⋄2)⋄3)−(1⋄(2⋄3))((1\diamond2)\diamond3)-(1\diamond(2\diamond3)) ?

Log in to save your progress.
Show solution without answering

Key insight

Work from the inside parentheses out. The left side is 1/2 - 1/3 = 1/6, the right side is 1 - 3/5 = 2/5, and 1/6 - 2/5 = -7/30.

Solution

The rule is: a⋄ba \diamond b means "take aa, then subtract 11 over bb". Work from the innermost parentheses outward.

Left part, (1⋄2)⋄3(1 \diamond 2) \diamond 3.

  1. 1⋄2=1−12=121 \diamond 2 = 1 - \dfrac12 = \dfrac12.
  2. 12⋄3=12−13=36−26=16\dfrac12 \diamond 3 = \dfrac12 - \dfrac13 = \dfrac36 - \dfrac26 = \dfrac16.

Right part, 1⋄(2⋄3)1 \diamond (2 \diamond 3).

  1. 2⋄3=2−13=532 \diamond 3 = 2 - \dfrac13 = \dfrac53.
  2. 1⋄53=1−15/31 \diamond \dfrac53 = 1 - \dfrac{1}{5/3}. One over 53\dfrac53 is 35\dfrac35 (flip the fraction), so this is 1−35=251 - \dfrac35 = \dfrac25.

Subtract.

  1. 16−25=530−1230=−730\dfrac16 - \dfrac25 = \dfrac{5}{30} - \dfrac{12}{30} = -\dfrac{7}{30}.

The answer is (A) −730\boxed{\textbf{(A)}\ -\frac{7}{30}}.

Why this works

A made-up operation is just a formula, so plug in carefully one layer at a time. The two groupings give different results, which is the point of the problem: for this operation the parentheses matter. Keep every step as an exact fraction, and remember that one over a fraction is that fraction flipped.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Work from the innermost parentheses outward.

Left expression: 1⋄2=1−12=121 \diamond 2 = 1 - \tfrac12 = \tfrac12, then
12⋄3=12−13=16.\tfrac12 \diamond 3 = \tfrac12 - \tfrac13 = \tfrac16.

Right expression: 2⋄3=2−13=532 \diamond 3 = 2 - \tfrac13 = \tfrac53, then
1⋄53=1−15/3=1−35=25.1 \diamond \tfrac53 = 1 - \frac{1}{5/3} = 1 - \tfrac35 = \tfrac25.

Difference:
16−25=530−1230=−730.\tfrac16 - \tfrac25 = \tfrac{5}{30} - \tfrac{12}{30} = -\tfrac{7}{30}.

The answer is (A) −730\boxed{\textbf{(A)}\ -\frac{7}{30}}.

Review in the AoPS books

Similar problems

Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.