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2009 AMC 10B Problem 15

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2009 AMC 10B Problem 15AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P11-15

When a bucket is two-thirds full of water, the bucket and water weigh aa kilograms. When the bucket is one-half full of water the total weight is bb kilograms. In terms of aa and bb , what is the total weight in kilograms when the bucket is full of water?

  1. A)

    23a+13b\frac23a + \frac13b

  2. B)

    32a−12b\frac32a - \frac12b

  3. C)

    32a+b\frac32a + b

  4. D)

    32a+2b\frac32a + 2b

  5. E)

    3a−2b3a - 2b

Answer

E

Key insight

Subtracting the two weighings isolates one-sixth of the water, so full water is 6(a - b), and the empty bucket is b minus half of that.

Solution

Let the empty bucket weigh BB and a full load of water weigh WW. The two weighings say
B+23W=a,B+12W=b.B + \tfrac23 W = a, \qquad B + \tfrac12 W = b.
Subtracting the second from the first: (23−12)W=16W=a−b\left(\tfrac23 - \tfrac12\right)W = \tfrac16 W = a - b, so W=6(a−b)W = 6(a - b).

Then B=b−12W=b−3(a−b)=4b−3aB = b - \tfrac12 W = b - 3(a - b) = 4b - 3a.

The full bucket weighs
B+W=(4b−3a)+(6a−6b)=3a−2b.B + W = (4b - 3a) + (6a - 6b) = 3a - 2b.

The answer is (E) 3a−2b\boxed{\textbf{(E)}\ 3a - 2b}.

Why this works

Two measurements of "fixed part plus a fraction of a variable part" form a linear system whose difference isolates the variable part. Once one unknown is known the other follows by back-substitution. Notice the answer is a linear combination of aa and bb with coefficients summing to 11, as it must be for a weight.

Alternative approach

Choose convenient numbers: an empty bucket of weight 00 and 66 kg of water give a=4a = 4, b=3b = 3, full weight 66. Only choice (E) gives 3⋅4−2⋅3=63 \cdot 4 - 2 \cdot 3 = 6; the others give 113,92,9,12\tfrac{11}{3}, \tfrac92, 9, 12.

The trap

Solving for the water alone (6a - 6b) and forgetting to add the bucket's own weight, or mixing up which equation has 2/3.

Common mistakes

  • Solving for the water alone (6a - 6b) and forgetting to add the bucket's own weight, or mixing up which equation has 2/3.
  • Guessing a "reasonable-looking" choice like (B) without testing it on a concrete example.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer