AMC 10 · 2014 · #9

Grade 7 arithmetic

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

fraction-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 2 insights

Problem

For real numbers w and z, the compound fraction built from their reciprocals satisfies

\frac{\frac{1}{w}+\frac{1}{z}}{\frac{1}{w}-\frac{1}{z}} = 2014

. Find the value of

\frac{w+z}{w-z}

Pick an answer.

(A)

-2014

(B)

$\frac{-1}{2014}$

(C)

$\frac{1}{2014}$

(D)

(E)

2014

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The given fraction has fractions hiding inside it, so Tool #7 (Identify Subproblems) splits the work: first clear the inner frac1w and frac1z so the equation becomes a clean ratio (w+z)/(z-w)=2014, then set that side by side with the target (w+z)/(w-z) and notice they differ only by a flipped denominator. Tool #6 (Guess and Check) with easy numbers like w=2, z=1 confirms the result, and Tool #3 (Eliminate Possibilities) rules out the look-alike traps 1/2014 and 2014 that appear only if you drop the sign.

1STEP 1

Clear the inner fractions

Multiply top and bottom by wz: $\frac{1}{w}$ becomes z and $\frac{1}{z}$ becomes w, so the equation reads $\frac{z+w}{z-w}$ = 2014.

(frac1w+frac1z)/(frac1w-frac1z)=((frac1w+frac1z)wz)/((frac1w-frac1z)wz)=(z+w)/(z-w)=2014

2STEP 2

Line it up with the target

Set it beside the target $\frac{w+z}{w-z}$ : the tops agree since w+z=z+w, but the bottoms are opposites, w-z=-(z-w).

w+z=z+w, w-z=-(z-w)

3STEP 3

A negated bottom flips the sign

The minus in the bottom pulls out front: $\frac{w+z}{w-z}$ = $-\frac{z+w}{z-w}$ = -2014, choice (A).

(w+z)/(w-z)=(z+w)/(-(z-w))=-(z+w)/(z-w)=-(2014)=-2014 → (A)

Answer

-2014

The target differs from the given expression only by swapping which order the denominator is subtracted, and that swap flips the sign of a fraction, so an answer of -2014 (the exact negative of 2014) is what we should expect — matching choice (A). The trap choices

\frac{1}{2014}

and 2014 come from forgetting the sign flip or wrongly taking a reciprocal; both are ruled out.

💡Key takeaway

Clear the little fractions first, then notice the denominator is just reversed — reversing a subtraction flips the sign, so the answer is the negative of 2014.

Clear the inner fractions
Line it up with the target
A negated bottom flips the sign

Clear the inner fractions

Line it up with the target

A negated bottom flips the sign

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