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2003 AMC 10B Problem 12

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2003 AMC 10B Problem 12AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P11-15

Al, Betty, and Clare split $1000\$1000 among them to be invested in different ways. Each begins with a different amount. At the end of one year, they have a total of $1500\$1500 dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose $100\$100 dollars. What was Al's original portion?

  1. A)

    $250

  2. B)

    $350

  3. C)

    $400

  4. D)

    $450

  5. E)

    $500

Answer

C

Key insight

Betty and Clare only matter as a combined sum (1000 - a) that doubles; the total 1500 then forces a = 400.

Solution

Let Al start with aa dollars. Betty and Clare together start with 1000−a1000 - a dollars. Since each of them doubles her money, together they end with 2(1000−a)2(1000 - a), while Al ends with a−100a - 100.

The final total is 15001500:
(a−100)+2(1000−a)=1500.(a - 100) + 2(1000 - a) = 1500.
a−100+2000−2a=1500⟹−a+1900=1500⟹a=400.a - 100 + 2000 - 2a = 1500 \quad\Longrightarrow\quad -a + 1900 = 1500 \quad\Longrightarrow\quad a = 400.

Check: Betty and Clare start with 600600 and end with 12001200; Al ends with 300300; 1200+300=15001200 + 300 = 1500.

The answer is (C) $400\boxed{\textbf{(C)}\ \$400}.

Why this works

Betty and Clare undergo the same transformation (doubling), so their individual amounts never need to be separated; only their sum enters the equations. Collapsing them into one variable turns a three-unknown system into a single linear equation. The phrase "each begins with a different amount" is irrelevant to the answer.

Alternative approach

Had Al not lost 100100, the final total would be 16001600, a gain of 600600 over the initial 10001000. All of that gain came from Betty and Clare doubling, so they started with 600600 and Al with 1000−600=4001000 - 600 = 400.

The trap

Trying to determine Betty's and Clare's individual amounts (impossible) instead of treating their combined money as one quantity.

Common mistakes

  • Trying to determine Betty's and Clare's individual amounts (impossible) instead of treating their combined money as one quantity.
  • Writing Al's final amount as 100100 or as a−100a - 100 but then doubling it along with the others.

Techniques

Set up the equation/formula and compute; no special trick needed