AMC10 2009 B

AMC10 2009 B · Q15

AMC10 2009 B · Q15. It mainly tests Linear equations, Systems of equations.

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?

当水桶装满水的三分之二时，水桶和水的总重量为a千克。当水桶装满水的一半时，总重量为b千克。用a和b表示水桶满水时的总重量（千克）？

(A) $\frac{2}{3}a + \frac{1}{3}b$ $\frac{2}{3}a + \frac{1}{3}b$

(B) $\frac{3}{2}a - \frac{1}{2}b$ $\frac{3}{2}a - \frac{1}{2}b$

(D) $\frac{3}{2}a + 2b$ $\frac{3}{2}a + 2b$

(E) 3a - 2b 3a - 2b

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): 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 $x+\frac{2}{3}y=a$ and $x+\frac{1}{2}y=b$. Hence $\frac{1}{6}y=a-b$, so $y=6a-6b$. Thus $x=b-\frac{1}{2}(6a-6b)=-3a+4b$. Finally, $x+y=3a-2b$.

答案（E）：设 $x$ 为桶的重量，$y$ 为一满桶水的重量。已知 $x+\frac{2}{3}y=a$ 且 $x+\frac{1}{2}y=b$。因此 $\frac{1}{6}y=a-b$，所以 $y=6a-6b$。于是 $x=b-\frac{1}{2}(6a-6b)=-3a+4b$。最后，$x+y=3a-2b$。

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