Question

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?

$\dfrac23 a+\dfrac13 b$

$\dfrac32 a-\dfrac12 b$

$\dfrac32 a+b$

$\dfrac32 a+2b$

$3a-2b$

Answer: E

Concepts:system of equations linear equation

Difficulty rating: 1310

Solution:

Let $x$ be the bucket's weight and $y$ the weight of a full load of water. Then $x+\dfrac23 y=a,\qquad x+\dfrac12 y=b.$

Subtracting gives $\dfrac16 y=a-b,$ so $y=6a-6b,$ and $x=b-\dfrac12 y=4b-3a.$ The full bucket weighs $\begin{aligned} x+y &= (4b-3a)+(6a-6b) \\ &= 3a-2b. \end{aligned}$

Thus, the correct answer is E.

2009 AMC 10B Problem 15

Answer: E

Difficulty rating: 1310

Solution:

Problem 15 in Other Years