AMC 10 · 2014 · #9
Grade 7 arithmeticFor real numbers w and z, w1−z1w1+z1=2014. What is w−zw+z?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: For real numbers $w$ and $z$, a compound fraction built from $\frac1w$ and $\frac1z$ equals $2014$. Using that fact, find the value of $\dfrac{w+z}{w-z}$.
Givens: $\cfrac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014$; $w$ and $z$ are real numbers that sit in denominators, so both are nonzero; Answer choices: (A) $-2014$, (B) $\dfrac{-1}{2014}$, (C) $\dfrac{1}{2014}$, (D) $1$, (E) $2014$
Unknowns: The value of $\dfrac{w+z}{w-z}$
Understand
Restated: For real numbers $w$ and $z$, a compound fraction built from $\frac1w$ and $\frac1z$ equals $2014$. Using that fact, find the value of $\dfrac{w+z}{w-z}$.
Givens: $\cfrac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014$; $w$ and $z$ are real numbers that sit in denominators, so both are nonzero; Answer choices: (A) $-2014$, (B) $\dfrac{-1}{2014}$, (C) $\dfrac{1}{2014}$, (D) $1$, (E) $2014$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #6 Guess and Check, #3 Eliminate Possibilities
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 $\dfrac{w+z}{z-w}=2014$, then set that side by side with the target $\dfrac{w+z}{w-z}$ 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 $\tfrac{1}{2014}$ and $2014$ that appear only if you drop the sign.
Execute — Answer: A
7.NS.A.2 Step 1 Clear the inner fractions
- Multiply the top and bottom of the big fraction by $wz$.
- That is the same nonzero amount on top and bottom, so the fraction's value is unchanged, but the inner pieces collapse: $\frac1w\cdot wz = z$ and $\frac1z\cdot wz = w$.
- The numerator $\frac1w+\frac1z$ becomes $z+w$, and the denominator $\frac1w-\frac1z$ becomes $z-w$.
💡 Scaling a fraction's top and bottom by the same nonzero number keeps its value but wipes out the tiny inner fractions.
7.EE.A.1 Step 2 Line it up with the target
- The equation now says $\dfrac{z+w}{z-w}=2014$, and the target is $\dfrac{w+z}{w-z}$.
- Compare piece by piece.
- The numerators are identical because addition can be written in either order: $w+z=z+w$.
- The denominators are opposites, because reversing a subtraction negates it: $w-z=-(z-w)$.
💡 Flipping the order of a subtraction just flips its sign, so $w-z$ is exactly the negative of $z-w$.
7.NS.A.2 Step 3 A negated bottom flips the sign
- Putting the negative denominator into the target turns it into the negative of the known ratio: dividing by $-(z-w)$ instead of $z-w$ pulls a minus sign out front, $\dfrac{a}{-b}=-\dfrac{a}{b}$.
- So $\dfrac{w+z}{w-z}=\dfrac{z+w}{-(z-w)}=-\dfrac{z+w}{z-w}=-2014$.
- That is choice (A).
💡 A minus sign in the denominator is the same as a minus sign in front of the whole fraction.
7.NS.A.2 Multiply the top and bottom of the big fraction by $wz$. That is the same nonzer 7.EE.A.1 The equation now says $\dfrac{z+w}{z-w}=2014$, and the target is $\dfrac{w+z}{w- 7.NS.A.2 Putting the negative denominator into the target turns it into the negative of t Review
Reasonableness: 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 $\tfrac{1}{2014}$ and $2014$ come from forgetting the sign flip or wrongly taking a reciprocal; both are ruled out.
Alternative: Guess and Check with concrete numbers. Pick $w=2$ and $z=1$: then $\frac1w+\frac1z=\tfrac32$ and $\frac1w-\frac1z=-\tfrac12$, so the compound fraction is $\tfrac{3/2}{-1/2}=-3$ — not $2014$, but its structure is all we need. The target $\dfrac{w+z}{w-z}=\dfrac{3}{1}=3$ is the exact negative of $-3$. So in general the target is the negative of the given ratio, confirming $-2014$.
CCSS standards used (min grade 7)
7.NS.A.2Apply and extend understanding of multiplication and division of rational numbers (Scaling the compound fraction's numerator and denominator by $wz$ to clear the inner fractions, and using $\dfrac{a}{-b}=-\dfrac{a}{b}$ to flip the sign at the end.)7.EE.A.1Apply properties of operations to add, subtract, factor, and expand linear expressions (Recognizing $w+z=z+w$ and $w-z=-(z-w)$ to match the given ratio to the target expression.)
⭐ 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 little fractions first, then notice the denominator is just reversed — reversing a subtraction flips the sign, so the answer is the negative of $2014$.
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