AMC12 2012 A

AMC12 2012 A · Q5

AMC12 2012 A · Q5. It mainly tests Ratios & proportions, Basic counting (rules of product/sum).

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?

一份水果沙拉由蓝莓、覆盆子、葡萄和樱桃组成。这份水果沙拉共有 $280$ 块水果。覆盆子的数量是蓝莓的两倍，葡萄的数量是樱桃的三倍，樱桃的数量是覆盆子的四倍。水果沙拉中有多少块樱桃？

(A) 8 8

(B) 16 16

(D) 64 64

(E) 96 96

Answer

Correct choice: (D)

正确答案：（D）

Solution

So let the number of blueberries be $b,$ the number of raspberries be $r,$ the number of grapes be $g,$ and finally the number of cherries be $c.$ Observe that since there are $280$ pieces of fruit, \[b+r+g+c=280.\] Since there are twice as many raspberries as blueberries, \[2b=r.\] The fact that there are three times as many grapes as cherries implies, \[3c=g.\] Because there are four times as many cherries as raspberries, we deduce the following: \[4r=c.\] Note that we are looking for $c.$ So, we try to rewrite all of the other variables in terms of $c.$ The third equation gives us the value of $g$ in terms of $c$ already. We divide the fourth equation by $4$ to get that $r=\frac{c}{4}.$ Finally, substituting this value of $r$ into the first equation provides us with the equation $b=\frac{c}{8}$ and substituting yields: \[\frac{c}{4}+\frac{c}{8}+3c+c=280\] Multiply this equation by $8$ to get: \[2c+c+24c+8c=8\cdot 280,\] \[35c=8\cdot 280,\] \[c=64.\] \[\boxed{D}\]

设蓝莓的数量为 $b,$ 覆盆子的数量为 $r,$ 葡萄的数量为 $g,$ 樱桃的数量为 $c.$ 注意到共有 $280$ 块水果， \[b+r+g+c=280.\] 由于覆盆子的数量是蓝莓的两倍， \[2b=r.\] 葡萄的数量是樱桃的三倍意味着， \[3c=g.\] 因为樱桃的数量是覆盆子的四倍，我们得到： \[4r=c.\] 注意我们要找的是 $c.$ 因此尝试把其他变量都用 $c$ 表示。第三个方程已经给出了用 $c$ 表示的 $g$。将第四个方程除以 $4$ 得到 $r=\frac{c}{4}.$ 最后，把这个 $r$ 的值代入第一个方程可得 $b=\frac{c}{8}$，并代入得到： \[\frac{c}{4}+\frac{c}{8}+3c+c=280\] 将该方程乘以 $8$ 得： \[2c+c+24c+8c=8\cdot 280,\] \[35c=8\cdot 280,\] \[c=64.\] \[\boxed{D}\]

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