AMC 10 · 2004 · #15

Grade 7 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
optimizationsign-analysis convert-to-algebrasystematic-enumeration ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 1 insight
Problem
The number x can be anywhere from -4 to -2, and the number y can be anywhere from 2 to 4. Among all allowed pairs, find the largest value the expression (x+y)/x can reach.

Pick an answer.

(A)
-1
(B)
$-\frac12$
(C)
0
(D)
$\frac12$
(E)
1

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The question asks for a maximum over a range of inputs, which is exactly what the Extreme Principle (Tool #14) is for: the best value lives at a boundary corner, not somewhere in the middle. But before hunting corners, rewrite the messy fraction into a cleaner shape (Tool #15): (x+y)/x=1+y/x. That split makes the whole problem about one small piece, y/x, so it is obvious which way to push x and y. Because x ＜ 0 and y ＞ 0, the piece y/x is always negative, so maximizing means making it the least negative — the smallest y over the largest |x|. Tool #3 (Eliminate Possibilities) then confirms the corner is a true maximum by comparing it against the opposite corner, which produces the smallest listed value.

1STEP 1

Rewrite the fraction

Divide each part of the top by x: (x+y)/x = x/x + y/x = 1 + y/x. The 1 is fixed, so only y/x moves.

(x+y)/x=x/x+y/x=1+y/x
2STEP 2

Read the signs

Here x is always negative and y always positive, so y/x is always negative and 1 + y/x always falls short of 1.

x ＜ 0, y ＞ 0 → y/x ＜ 0
3STEP 3

Push to the best corner

Smallest top over largest bottom: y = 2 with x = -4 gives y/x = -1/2, so the expression is 1 - 1/2 = 1/2 — choice (D).

y=2, x=-4: 1+2/-4=1-1/2=1/2 → (D)
Answer
1/2
The result 1/2 is less than 1, which fits: since y/x is always negative, 1+y/x can never reach 1 (choice (E)), so (E) is impossible and 1/2 being just under it looks right. The opposite corner is a good sanity check for the maximum: taking the largest y=4 over the smallest |x|, x=-2, gives 1+4/-2=1-2=-1, which is choice (A) — the smallest listed value. So the expression really does swing from -1 up to 1/2, and 1/2 is the top.
💡Key takeaway

Split a hard fraction into a fixed part plus a moving part, then push the moving part to its extreme; with opposite signs on top and bottom, the quotient is smallest in size when the top is small and the bottom is big.

  • Rewrite the fraction
  • Read the signs
  • Push to the best corner

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