节: 解决问题时使用比例因数 | 4.7 解决问题时使用比例因素

章节大纲

Lifan’s driveway has a length of 24 feet. If the scale is $\begin{aligned} 2 inches : 4 feet \end{aligned}$ , what is the scale factor? In a diagram, how many inches would Lifan draw to represent his driveway?
::利凡的车道长度为24英尺。如果比例是2英寸4英尺,那么比例因素是什么? 在一张图表中,利凡将抽取多少英寸来代表他的车道?

In this concept, you will learn to use scale factors when problem solving.
::在此概念中,您将学会在解决问题时使用比例因素。

Scale Factor
::比例因数

The scale can be used to help you with scale dimensions or actual dimensions. This scale is key in problem solving.
::比例尺可用于帮助您处理大小或实际大小。这个比例尺是解决问题的关键。

If you look at the scale $\begin{aligned} 2 : 1 \end{aligned}$ , you can use this information to determine the scale factor . The scale factor is the relationship between the scale dimension and the measurement comparison between the scale measurement of the model and the actual length. In this case the scale factor is $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::如果您查看比额表2: 1, 您可以使用此信息来确定比额表系数。比额表系数是比额表尺寸与模型的比额表计量与实际长度之间的测量比较之间的关系。在这种情况下,比额表系数是12。

Let’s look at an example.
::让我们举个例子。

What is the scale factor if 3 inches is equal to 12 feet?
::如果3英寸等于12英尺,尺度因素是什么?

First, write the ratio.
::首先,写比例。

$\begin{aligned} \frac{3}{12} \end{aligned}$

Next, simplify the fraction.
::312 接下来简化分数

$\begin{aligned} \frac{3}{12} = \frac{1}{4} \end{aligned}$

The answer is $\begin{aligned} \frac{1}{4} \end{aligned}$ .
::312=14 答案是14

The scale factor is $\begin{aligned} 1 : 4 \end{aligned}$ .
::比额表系数为1:4。

Now let’s look at a problem where you are applying this information.
::现在让我们来看看您在应用此信息时遇到的问题。

If the scale dimension is 4, then you can figure out the actual dimension. Look at this proportion:
::如果天平尺寸为 4, 那么您就可以找出实际的尺寸。看看这个比例 :

$\begin{aligned} 1 : 2 = 4 : x \end{aligned}$
::1:2=4:x

First, put the proportion in fraction form.
::首先,将比例以分数形式列出。

$\begin{aligned} \frac{1}{2} = \frac{4}{x} \end{aligned}$

::12=4x 12=4x

Next, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::下一个, 交叉乘以x 解析。

$\begin{aligned} \begin{array}{rcl} \frac{1}{2} & = & \frac{4}{x} \\ 1 x & = & 2 \times 4 \\ x & = & 8 \end{array} \end{aligned}$

::12=4x1x=2x4x=8

The answer is 8.
::答案是8岁

This is the actual dimension.
::这是实际的维度。

Let’s look at a real world problem.
::让我们看看真正的世界问题。

The plans for a flower garden show that it is 6 inches wide on the plan. If the scale for the flower garden is $\begin{aligned} 1 : 12 \end{aligned}$ , what is the actual width of the flower garden?
::花卉园的计划显示,这个计划宽6英寸。如果花卉园的规模是1:12,那么花卉园的实际宽度是多少?

First, write the proportion.
::首先,写比例。

$\begin{aligned} 1 : 12 = 6 : x \end{aligned}$

::1:12=6:x

Next, put the proportion in fraction form.
::接下来,将比例以分数形式排列。

$\begin{aligned} \frac{1}{12} = \frac{6}{x} \end{aligned}$

::112=6x 112=6x

Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::然后,交叉倍数来解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{1}{12} & = & \frac{6}{x} \\ 1 x & = & 12 \times 6 \\ x & = & 72 \end{array} \end{aligned}$

::112=6x1x=12x6x=72

The answer is 72.
::答案是72。

The actual width of the flower garden is 72 inches.
::花园的实际宽度是72英寸

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Lifan and his long driveway. The scale is $\begin{aligned} 2 inches : 4 feet \end{aligned}$ and the driveway is 24 feet long.
::早些时候,你遇到了利凡和他长途车道的问题。规模是2英寸4英尺,车道是24英尺。

First, write the proportion. Note that $\begin{aligned} 2 : 4 \end{aligned}$ is the scale factor.
::首先,写下比例。请注意,2:4是比额表系数。

$\begin{aligned} 2 : 4 = x : 24 \end{aligned}$

::2:4=x:24

Next, put the proportion in fraction form.
::接下来,将比例以分数形式排列。

$\begin{aligned} \frac{2}{4} = \frac{x}{24} \end{aligned}$

::24=x24

Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::然后,交叉倍数来解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{2}{4} & = & \frac{x}{24} \\ 4 x & = & 2 \times 24 \\ 4 x & = & 48 \end{array} \end{aligned}$

::24=x244x=2x244x=48

Then, divide both sides by 4 to solve for $\begin{aligned} x \end{aligned}$ .
::然后把两边除以4 解决x

$\begin{aligned} \begin{array}{rcl} 4 x & = & 48 \\ \frac{4 x}{4} & = & \frac{48}{4} \\ x & = & 12 \end{array} \end{aligned}$

::4x=484x4=484x4=484x=12

The answer is 12.
::答案是12岁

The scale dimension of Lifan’s driveway is 12 inches.
::Lifan车道的大小为12英寸。

Example 2
::例2

Find the missing actual dimension if the scale factor is $\begin{aligned} 2^{''} : 3^{'} \end{aligned}$ and the scale measurement is $\begin{aligned} 6^{''} \end{aligned}$ .
::如果比额表因数为 2:3,而比额表的计量为 6,则查找缺失的实际维度。

First, write the proportion.
::首先,写比例。

$\begin{aligned} 2 : 3 = 6 : x \end{aligned}$

::2:3=6:x

Next, put the proportion in fraction form.
::接下来,将比例以分数形式排列。

$\begin{aligned} \frac{2}{3} = \frac{6}{x} \end{aligned}$

::23=6x 23=6x

Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::然后,交叉倍数来解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{2}{3} & = & \frac{6}{x} \\ 2 x & = & 3 \times 6 \\ 2 x & = & 18 \end{array} \end{aligned}$

::23=6x2x=3x62x=18

Then, divide both sides by 2 to solve for $\begin{aligned} x \end{aligned}$ .
::然后将两边除以 2 解决 x 。

$\begin{aligned} \begin{array}{rcl} 2 x & = & 18 \\ \frac{2 x}{2} & = & \frac{18}{2} \\ x & = & 9 \end{array} \end{aligned}$

::2x=182x2=182x=9

The answer is 9.
::答案是9。

The actual dimension is 9 feet.
::实际尺寸是9英尺

Example 3
::例3

Find the missing actual dimension if the scale factor is $\begin{aligned} {\frac{1}{4}}^{''} : 4^{'} \end{aligned}$ and the scale measurement is $\begin{aligned} 8^{''} \end{aligned}$ .
::如果比额表系数为 14:4 和比额表计量为 8,则查找缺失的实际维度。

First, write the proportion.
::首先,写比例。

$\begin{aligned} \frac{1}{4} : 4 = 8 : x \end{aligned}$

::14:4=8:x

Next, put the proportion in fraction form.
::接下来,将比例以分数形式排列。

$\begin{aligned} \begin{array}{rcl} \frac{\frac{1}{4}}{4} & = & \frac{8}{x} \\ \frac{1}{16} & = & \frac{8}{x} \end{array} \end{aligned}$

::144=8x116=8x

Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::然后,交叉倍数来解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{1}{16} & = & \frac{8}{x} \\ 1 x & = & 8 \times 16 \\ x & = & 128 \end{array} \end{aligned}$

::116=8x1x=8x16x=128

The answer is 128.
::答案是128个

The actual dimension is 128 feet.
::实际尺寸为128英尺。

Example 4
::例4

Find the missing actual dimension if the scale factor is $\begin{aligned} {\frac{1}{4}}^{''} : 4^{'} \end{aligned}$ and the scale measurement is $\begin{aligned} 12^{'} \end{aligned}$ .
::如果比额表系数为14:4.,而比额表计量为12,则查找缺失的实际维度。

First, write the proportion.
::首先,写比例。

$\begin{aligned} \frac{1}{4} : 4 = x : 12 \end{aligned}$

::14:4=x:12 14:4=x:12

Next, put the proportion in fraction form.
::接下来,将比例以分数形式排列。

$\begin{aligned} \begin{array}{rcl} \frac{\frac{1}{4}}{4} & = & \frac{x}{12} \\ \frac{1}{16} & = & \frac{x}{12} \end{array} \end{aligned}$

::144=x12116=x12

Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::然后,交叉倍数来解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{1}{16} & = & \frac{x}{12} \\ 16 x & = & 1 \times 12 \\ 16 x & = & 12 \end{array} \end{aligned}$

::116=x1216x=1x1216x=12

Then, divide both sides by 16 to solve for $\begin{aligned} x \end{aligned}$ .
::然后,将双方除以16 解决x。

$\begin{aligned} \begin{array}{rcl} 16 x & = & 12 \\ \frac{16 x}{16} & = & \frac{12}{16} \\ x & = & \frac{12}{16} \\ x & = & \frac{3}{4} \end{array} \end{aligned}$

::16x=1216x16=1216x=1216x=1216x=1216x=34

The answer is $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::答案是34岁

The actual dimension is $\begin{aligned} \frac{3}{4} \end{aligned}$ inches.
::实际尺寸为34英寸。

Example 5
::例5

Find the missing actual dimension if the scale factor is $\begin{aligned} {\frac{1}{4}}^{''} : 4^{'} \end{aligned}$ and the scale measurement is $\begin{aligned} 16^{''} \end{aligned}$ .
::如果比额表系数为 14:4 和比额表衡量为 16,则查找缺失的实际维度。

First, write the proportion.
::首先,写比例。

$\begin{aligned} \frac{1}{4} : 4 = 16 : x \end{aligned}$

::14:4=16:x

Next, put the proportion in fraction form.
::接下来,将比例以分数形式排列。

$\begin{aligned} \begin{array}{rcl} \frac{\frac{1}{4}}{4} & = & \frac{16}{x} \\ \frac{1}{16} & = & \frac{16}{x} \end{array} \end{aligned}$

::144=16x116=16x

Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::然后,交叉倍数来解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{1}{16} & = & \frac{16}{x} \\ 1 x & = & 16 \times 16 \\ x & = & 256 \end{array} \end{aligned}$

::116=16x1x=16x16x=256

The answer is 256.
::答案是256个

The actual dimension is 256 feet.
::实际尺寸为256英尺。

Review
::回顾

Figure out each scale factor.
::找出每个比额表系数。

1. $\begin{aligned} \frac{2 inches}{8 feet} \end{aligned}$
::1. 2英寸8英尺

2. $\begin{aligned} \frac{13 inches}{12 feet} \end{aligned}$
::2. 13英寸12英尺

3. $\begin{aligned} \frac{6 inches}{24 feet} \end{aligned}$
::3. 6英寸24英尺

4.

$\begin{aligned} \frac{11 inches}{33 feet} \end{aligned}$
::11 英寸33英尺

5. $\begin{aligned} \frac{16 inches}{32 feet} \end{aligned}$
::5. 16英寸32英尺

6. $\begin{aligned} \frac{18 inches}{36 feet} \end{aligned}$
::6. 1836英寸36英尺

7. $\begin{aligned} \frac{6 inches}{48 feet} \end{aligned}$
::7. 6 英吋48英尺

8. $\begin{aligned} \frac{6 inches}{12 feet} \end{aligned}$
::8. 6英寸12英尺

Solve each problem.
::解决每一个问题。

9. A rectangle has a width of 2 inches. A similar rectangle has a width of 9 inches. What scale factor could be used to convert the larger rectangle to the smaller rectangle?
::9. 矩形宽度为2英寸,类似的矩形宽度为9英寸,可将较大的矩形转换成较小的矩形时可使用何种比例因素?

10. A drawing of a man is 4 inches high. The actual man is 64 inches tall. What is the scale factor for the drawing?
::10. 男子的绘画高4英寸,实际男子高64英寸,绘画的尺度因素是什么?

11. A map has a scale of $\begin{aligned} 1 inch = 4 feet \end{aligned}$ . What is the scale factor of the map?
::11. 地图的尺度为1英寸=4英尺,地图的尺度因数是多少?

12. A drawing of a box has dimensions that are 2 inches, 3 inches, and 5 inches. The dimensions of the actual box will be $\begin{aligned} 3 \frac{1}{4} \end{aligned}$ times the dimensions in the drawing. What are the dimensions of the actual box?
::12. 框图画的尺寸为2英寸、3英寸和5英寸,实际框的尺寸是图画尺寸的314倍。

13. A room has a length of 10 feet. Hadley is drawing a scale drawing of the room, using the scale factor $\begin{aligned} \frac{1}{50} \end{aligned}$ . How long will the room be in Hadley’s drawing?
::13. 一个房间的长度为10英尺,Hadley正在用150号比例系数绘制房间的平面图。

14. The distance from Anna’s room to the kitchen is 15 meters. Anna is making a diagram of her house using the scale factor of $\begin{aligned} \frac{1}{150} \end{aligned}$ . What will be the distance on the diagram from Anna’s room to the kitchen?
::14. 从安娜的房间到厨房的距离是15米,安娜正在用1150的尺度系数绘制她家的图表。从安娜的房间到厨房的距离是多少?

15. On a map of Cameron’s town, his house is 9 inches from his school. If the scale of the map is $\begin{aligned} \frac{1}{400} \end{aligned}$ , what is the actual distance, in feet, from Cameron’s house to his school?
::15. 在卡梅伦镇的地图上,他的房子离他的学校只有9英寸,如果地图的尺码是1400,那么从卡梅伦的家到他的学校的实际距离是多少?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

章节大纲

Scale Factor ::比例因数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Scale Factor
::比例因数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源