AMC12 2009 A

AMC12 2009 A · Q3

AMC12 2009 A · Q3. It mainly tests Fractions.

What number is one third of the way from $\frac14$ to $\frac34$?

从 $\frac14$ 到 $\frac34$ 的三分之一处的数是多少？

(A) $\frac{1}{3}$ $\frac{1}{3}$

(B) $\frac{5}{12}$ $\frac{5}{12}$

(D) $\frac{7}{12}$ $\frac{7}{12}$

(E) $\frac{2}{3}$ $\frac{2}{3}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

We can rewrite the two given fractions as $\frac 3{12}$ and $\frac 9{12}$. (We multiplied all numerators and denominators by $3$.) Now it is obvious that the interval between them is divided into three parts by the fractions $\boxed{\frac 5{12}}$ and $\frac 7{12}$. The number we seek can be obtained as a weighted average of the two endpoints, where the closer one has weight $2$ and the further one $1$. We compute: \[\dfrac{ 2\cdot\frac 14 + 1\cdot\frac 34 }3 = \dfrac{ \frac 54 }3 = \boxed{\dfrac 5{12}}\]

我们可以把两个分数改写为 $\frac 3{12}$ 和 $\frac 9{12}$。（将分子分母都乘以 $3$。）这样就很明显，区间被分数 $\boxed{\frac 5{12}}$ 和 $\frac 7{12}$ 分成了三等份。所求的数也可以看作两个端点的加权平均，其中较近的端点权重为 $2$，较远的端点权重为 $1$。计算： \[\dfrac{ 2\cdot\frac 14 + 1\cdot\frac 34 }3 = \dfrac{ \frac 54 }3 = \boxed{\dfrac 5{12}}\]

Topics

AR.001 Fractions