AMC 10 · 2013 · #8

Grade 8 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

polynomial-factoringzero-product-propertyfraction-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: polynomial-factoring

📏 Medium solution 💡 2 insights

Problem

Two different nonzero numbers give the same value in one expression. Find their product.

Pick an answer.

(A)

$\frac{1}{4}$

(B)

$\frac{1}{2}$

(C)

(D)

(E)

How to solve

Strategy Organize Information in More Ways

Trying to solve for x alone leads to a quadratic whose right side is still unknown, and that goes nowhere. Rearranging the whole equation is the better move. Apart from the equation itself, the only fact available is that the two numbers differ, so the goal is to rewrite the equation until the block x-y appears as a factor. Once it does, 'distinct' becomes usable and the product is forced. That argument only shows what xy must be, so a concrete test pair (tool #6) and a backwards run of the argument (tool #11) then confirm such pairs really exist. A second, structurally different route names the shared value with a letter (tool #4) and uses the theory of quadratics; it appears in the review.

1STEP 1

Clear the two fractions

Clearing the fractions leaves a polynomial.

x²y + 2y = xy² + 2x

2STEP 2

Move everything to one side

Everything moves to one side.

x²y - xy² + 2y - 2x = xy(x-y) - 2(x-y) = 0

3STEP 3

Factor, then spend the word 'distinct'

Being different kills one factor.

(x-y)(xy-2) = 0, x - y ≠ 0 → xy - 2 = 0

4STEP 4

Check that such a pair exists

Such a pair really exists.

1+2/1 = 3 = 2+2/2, 1 · 2 = 2

5STEP 5

Run the argument backwards

The product is 2, choice (D).

y = 2/x → y + 2/y = 2/x + x = x + 2/x, xy = 2 → (D)

Answer

Two separate numeric tests land on the same product. The pair (1,2) gives 3=3, and the pair (4,1/2) gives 4+1/2 = 9/2 and 1/2+4 = 9/2; both have product 2, so no other listed value can be correct. There is also a structural check: replace the 2 in the problem by any nonzero constant k, and the identical grouping gives (x-y)(xy-k)=0, hence xy=k. The answer should therefore be exactly the constant printed in the problem, which is 2, matching (D). The wrong choices fit predictable slips: (E) 4 is 2², from squaring somewhere, and (A) 1/4 and (B) 1/2 come from inverting the fraction and reporting 1/xy instead of xy. A single numeric test kills all of them.

💡Key takeaway

Push everything to one side and the factor x-y appears; because the two numbers are different you are allowed to divide it away, and what remains says xy=2.

Clear the two fractions
Move everything to one side
Factor, then spend the word 'distinct'
Check that such a pair exists
Run the argument backwards

Clear the two fractions

Move everything to one side

Factor, then spend the word 'distinct'

Check that such a pair exists

Run the argument backwards

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