2009 AMC 10B Problems/Problem 15

The following problem is from both the 2009 AMC 10B #15 and 2009 AMC 12B #8, so both problems redirect to this page.

Problem

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

$\mathrm{(A)}\ \frac23a + \frac13b\qquad \mathrm{(B)}\ \frac32a - \frac12b\qquad \mathrm{(C)}\ \frac32a + b\qquad \mathrm{(D)}\ \frac32a + 2b\qquad \mathrm{(E)}\ 3a - 2b$

Solution

Let $x$ be the weight of the bucket and let $y$ be the weight of the water in a full bucket. Then we are given that $x + \frac 23y = a$ and $x + \frac 12y = b$ . Hence $\frac 16y = a-b$ , so $y = 6a-6b$ . Thus $x = b - \frac 12 (6a-6b) = -3a + 4b$ . Finally $x + y = \boxed {3a-2b}$ . The answer is $\mathrm{(E)}$ .

2009 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 10 Problems and Solutions

2009 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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2009 AMC 10B Problems/Problem 15

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