AMC 10 · 2003 · #24

Grade 8 algebra

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

sequences-arithmeticsystems-of-equationsquadratic-equations convert-to-algebra ↑ Prerequisites: sequences-arithmeticsystems-of-equations

📏 Long solution 💡 3 insights

Problem

The four expressions

x+y

x-y

xy

\frac{x}{y}

are, in that order, the first four terms of an arithmetic sequence — every step from one term to the next adds the same fixed amount. What is the fifth term?

Pick an answer.

(A)

$-\frac{15}{8}$

(B)

$-\frac{6}{5}$

(C)

(D)

$\frac{27}{20}$

(E)

$\frac{123}{40}$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

The only fact we are given is a pattern — 'arithmetic sequence' — so first turn that pattern into arithmetic (Tool #5, Tool #13): equal gaps mean each term minus the one before it equals the same constant d. The first two terms are simple, so read off d from them (Tool #4, name the gap d). That single value d then lets us write the third and fourth terms two ways — as the given expressions, and as 'previous term plus d'. Setting those equal gives two equations in x and y. Solving that little system is the core subproblem (Tool #7); once x, y, and d are known, the fifth term is just the fourth term plus d.

1STEP 1

Read the common gap off the first two terms

The gap is constant, so read it off the first two terms: $d=(x-y)-(x+y)$ , and every step adds $-2y$ .

d = (x-y)-(x+y) = -2y

2STEP 2

Write the third and fourth terms two ways

Each term is the previous one plus $-2y$ , so the third gives $x-3y=xy$ and the fourth gives $xy-2y=\frac{x}{y}$ .

x - 3y = xy (1) xy - 2y = x/y (2)

3STEP 3

Eliminate x to get an equation in y

Put $xy=x-3y$ into the fourth-term equation and clear the $y$ in the denominator: every $x$ cancels, leaving $y(5y+3)=0$ .

xy - 5y² = x → (x-3y) - 5y² = x → y(5y+3)=0

4STEP 4

Solve for y, then x, then the gap

$y=0$ is barred because $\frac{x}{y}$ must exist, so $y=-\frac{3}{5}$ ; then $x=-\frac{9}{8}$ and the gap is $d=\frac{6}{5}$ .

y = -3/5, x = -9/8, d = -2y = 6/5

5STEP 5

Add the gap to the fourth term

The fourth term is $\frac{x}{y}=\frac{75}{40}$ , so adding $\frac{6}{5}=\frac{48}{40}$ gives $\frac{123}{40}$ , choice (E).

x/y + d = 15/8 + 6/5 = 75/40 + 48/40 = 123/40 → (E)

Answer

123/40

Check that the four given expressions really are arithmetic with x=-9/8, y=-3/5. In fortieths: x+y = -45/40-24/40 = -69/40; x-y = -45/40+24/40 = -21/40; xy = 27/40; x/y = 75/40. The differences are -21/40-(-69/40) = 48/40, 27/40-(-21/40) = 48/40, and 75/40-27/40 = 48/40 — all equal to 48/40=6/5=d. The sequence is genuinely arithmetic, so extending by one more 48/40 gives 123/40, matching (E).

💡Key takeaway

Turn 'arithmetic sequence' into 'each term equals the one before plus the same gap' — that gives you equations to pin down the unknowns, and then the next term is just add the gap once more.

Read the common gap off the first two terms
Write the third and fourth terms two ways
Eliminate x to get an equation in y
Solve for y, then x, then the gap
Add the gap to the fourth term

Read the common gap off the first two terms

Write the third and fourth terms two ways

Eliminate x to get an equation in y

Solve for y, then x, then the gap

Add the gap to the fourth term

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