AMC 10 · 2010 · #6

Grade 6 arithmetic

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

function-evaluationfraction-arithmeticorder-of-operations identify-subproblems ↑ Prerequisites: function-evaluation

📏 Medium solution 💡 1 insight

Problem

For positive numbers x and y a made-up operation is defined by ♠(x,y) = x -

\frac{1}{y}

. Find the value of ♠(2,♠(2,2)), where one ♠ is sitting inside another.

Pick an answer.

(A)

$\ \dfrac{2}{3}$

(B)

$\ 1$

(C)

$\ \dfrac{4}{3}$

(D)

$\ \dfrac{5}{3}$

(E)

$\ 2$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The operation is nested, so the outer ♠ depends on a number we do not have yet. Split it into the inner subproblem ♠(2,2) first, plug that result into the outer ♠, and match the final value to one of the five choices.

1STEP 1

Read the rule, spot the inner piece

The rule says "first number minus one over the second", and here the second slot is itself ♠(2,2), so that comes first.

♠(2,♠(2,2)) needs ♠(2,2) first

2STEP 2

Compute the inner spade

Put x=2 and y=2 into the rule: $2-\frac{1}{2}=\frac{4}{2}-\frac{1}{2}$ leaves $\frac{3}{2}$ .

♠(2,2) = 2 - 1/2 = 4/2 - 1/2 = 3/2

3STEP 3

Set up the outer spade

Now it is ♠(2, $\frac{3}{2}$ ): dividing 1 by $\frac{3}{2}$ flips the fraction, so the amount to subtract is $\frac{2}{3}$ .

♠(2,3/2) = 2 - 1/ 3/2 = 2 - 2/3

4STEP 4

Finish the subtraction

Write 2 as $\frac{6}{3}$ , so $\frac{6}{3}-\frac{2}{3}$ gives $\frac{4}{3}$ , which is choice (C).

2 - 2/3 = 6/3 - 2/3 = 4/3

Answer

4/3

The final value 4/3 is a little more than 1, which fits: each ♠ starts at 2 and subtracts a positive amount less than 2, so the results should stay between 0 and 2. Both 3/2 and 4/3 land in that range, and 4/3 is exactly choice (C).

💡Key takeaway

When one operation sits inside another, solve the inside first, and remember that one over a fraction just flips it.

Read the rule, spot the inner piece
Compute the inner spade
Set up the outer spade
Finish the subtraction

Read the rule, spot the inner piece

Compute the inner spade

Set up the outer spade

Finish the subtraction

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