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?

$\dfrac{2}{3}a + \dfrac{1}{3}b$

$\dfrac{3}{2}a - \dfrac{1}{2}b$

$\dfrac{3}{2}a + b$

$\dfrac{3}{2}a + 2b$

$3a - 2b$

Answer: E

Concepts:system of equations substitution

Difficulty rating: 1370

Solution:

Let $x$ be the bucket's weight and $y$ the weight of a full bucket of water. Then $x + \dfrac{2}{3}y = a$ and $x + \dfrac{1}{2}y = b.$

Subtracting gives $\dfrac{1}{6}y = a - b,$ so $y = 6a - 6b$ and $x = b - \dfrac{1}{2}y = 4b - 3a.$ The full weight is $x + y = 3a - 2b.$

Thus, the correct answer is E.

2009 AMC 12B Problem 8

Answer: E

Difficulty rating: 1370

Solution:

Problem 8 in Other Years