AMC 10 · 2017 · #4

Grade 7 arithmetic

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

fraction-arithmeticlinear-equations-two-var convert-to-algebraguess-and-check ↑ Prerequisites: linear-equations-two-var

📏 Short solution 💡 1 insight

Problem

Two nonzero real numbers x and y satisfy

\frac{3x+y}{x-3y}

=-2. Using only that fact, find the numerical value of the second expression

\frac{x+3y}{3x-y}

and match it to one of the five choices.

Pick an answer.

(A)

-3

(B)

-1

(C)

(D)

(E)

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Tool #4 (Introduce a Variable): the given equation is a single relation between the two unknowns x and y. Clearing the fraction turns it into a plain linear equation, and solving that equation reveals how x and y are tied together. Once that tie is known, the target expression collapses to a number. Tool #6 (Guess and Check) lets us pick a concrete pair (x,y) that fits and read the answer directly. Tool #3 (Eliminate Possibilities) matches the result to the five choices.

1STEP 1

Clear the fraction

Since x-3y ≠ 0, multiply both sides by it to clear the fraction: 3x+y=-2(x-3y).

\frac{3x+y}{x-3y}

=-2 → 3x+y = -2(x-3y)

2STEP 2

Find how x and y relate

Expand and collect like terms: 3x+y=-2x+6y becomes 5x=5y, so x=y.

3x+y=-2x+6y → 5x=5y → x=y

3STEP 3

Substitute into the target

Put x=y into $\frac{x+3y}{3x-y}$ : it becomes $\frac{4y}{2y}$ , and since y ≠ 0 the y cancels to 2.

\frac{x+3y}{3x-y}

\frac{y+3y}{3y-y}

\frac{4y}{2y}

4STEP 4

Match the choice

The target expression equals 2, which is choice (D).

\frac{x+3y}{3x-y}

=2 → (D)

Answer

Test a concrete pair. Take x=y=1: the given expression is

\frac{3(1)+1}{1-3(1)}

\frac{4}{-2}

=-2, which matches the condition, and the target is

\frac{1+3(1)}{3(1)-1}

\frac{4}{2}

=2. Trying x=y=2 gives the same target value 2, confirming the answer does not depend on the actual size of x and y, only on x=y.

💡Key takeaway

Clear the fraction first: $\frac{3x+y}{x-3y}$ =-2 means 3x+y=-2(x-3y), which boils down to x=y, and then $\frac{x+3y}{3x-y}$ = $\frac{4y}{2y}$ =2, choice (D).

Clear the fraction
Find how x and y relate
Substitute into the target
Match the choice

Clear the fraction

Find how x and y relate

Substitute into the target

Match the choice

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