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比例比例
Proportions
::比例比例

When two ratios are equal to each other, we call it a proportion . For example, the equation $\begin{aligned} \frac{10}{15} = \frac{6}{9} \end{aligned}$ is a proportion. We know it’s true because we can reduce both fractions to $\begin{aligned} \frac{2}{3} \end{aligned}$ .
::当两个比率对等时,我们把它称为比例。例如,等式1015=69是一个比例。我们知道这是真的,因为我们可以将两个分数都减少到23个。

(Check this yourself to make sure!)
:检查这个自己确定! )

We often use proportions in science and business—for example, when scaling up the size of something. We generally use them to solve for an unknown, so we use algebra and label the unknown variable $\begin{aligned} x \end{aligned}$ .
::我们经常在科学和商业中使用比例,例如,在扩大某物的大小时,我们通常使用比例来解答未知物,所以我们使用代数来标注未知变量x。

Real-World Example: Profit
::真实世界实例:利润

A small fast food chain operates 60 stores and makes $1.2 million profit every year. How much profit would the chain make if it operated 250 stores?
::一个小型快餐连锁店经营60家商店,每年赚取120万美元的利润。如果该连锁店经营250家商店,它能赚取多少利润?

First, we need to write a ratio: the ratio of profit to number of stores. That would be $\begin{aligned} \frac{$ 1, 200, 000}{60} \end{aligned}$ .
::首先,我们需要写一个比率:利润与商店数量的比率。这将是1 200 000美元。

Now we want to know how much profit 250 stores would make. If we label that profit $\begin{aligned} x \end{aligned}$ , then the ratio of profit to stores in that case is $\begin{aligned} \frac{x}{250} \end{aligned}$ .
::现在我们想知道250家商店能赚多少利润。如果我们标注利润x,那么利润与商店的比率是x250。

Since we’re assuming the profit is proportional to the number of stores, the ratios are equal and our proportion is $\begin{aligned} \frac{1, 200, 000}{60} = \frac{x}{250} \end{aligned}$ .
::由于我们假设利润与商店数量成正比, 比率是相等的, 我们的比例是1200,00060=x250。

(Note that we can drop the units – not because they are the same in the numerator and denominator, but because they are the same on both sides of the equation.)
:注意我们可以降低单位, 不是因为它们在分子和分母中是相同的, 而是因为它们在等式的两侧都是相同的。 )

To solve this equation, first we simplify the left-hand fraction to get $\begin{aligned} 20, 000 = \frac{x}{250} \end{aligned}$ . Then we multiply both sides by 250 to get $\begin{aligned} 5, 000, 000 = x \end{aligned}$ .
::为了解决这个方程, 首先我们简化左手部分, 以获得 20,000 =x250 。然后我们把两侧乘以250 以获得 5000,000 =x 。

If the chain operated 250 stores, the annual profit would be 5 million dollars.
::如果连锁店经营250家商店,年利润为500万美元。

Solve Proportions Using Cross Products
::使用交叉产品解决比例

One neat way to simplify proportions is to cross multiply. Consider the following proportion:
::简化比例的一个简洁的方法是交叉乘法。考虑以下比例 :

$\begin{aligned} \frac{16}{4} = \frac{20}{5} \end{aligned}$

If we want to eliminate the fractions, we could multiply both sides by 4 and then multiply both sides by 5. But suppose we just do both at once?
::如果我们想要消除分数, 我们可以把两边乘以4, 然后把两边乘以5。但是假设我们同时同时同时这样做呢?

$\begin{aligned} 4 \times 5 \times \frac{16}{4} & = 4 \times 5 \times \frac{20}{5} \\ 5 \times 16 & = 4 \times 20 \end{aligned}$

Now comparing this to the proportion we started with, we see that the denominator from the left hand side ends up being multiplied by the numerator on the right hand side. You can also see that the denominator from the right hand side ends up multiplying the numerator on the left hand side.
::现在把这个比作我们最初的比重, 我们可以看到左手侧的分母由右手侧的分子乘以。您也可以看到右手侧的分母最终会乘以左手侧的分子。

In effect the two denominators have multiplied across the equal sign:
::事实上,两个分母在等号之间乘以乘以:

becomes $\begin{aligned} 5 \times 16 = 4 \times 20 \end{aligned}$ .
::= 5x16=4x20。

This movement of denominators is known as cross multiplying . It is extremely useful in solving proportions, especially when the unknown variable is in the denominator.
::分母的这种移动被称为交叉乘法,对于解决比例问题非常有用,特别是当未知变量在分母中时。

Solving for Unknown Values
::解决未知值

1. Solve this proportion for $\begin{aligned} x \end{aligned}$ : $\begin{aligned} \frac{4}{3} = \frac{9}{x} \end{aligned}$
::1. x: 43=9x解决这一比例

Cross multiply to get $\begin{aligned} 4 x = 9 \times 3 \end{aligned}$ , or $\begin{aligned} 4 x = 27 \end{aligned}$ . Then divide both sides by 4 to get $\begin{aligned} x = \frac{27}{4} \end{aligned}$ , or $\begin{aligned} x = 6.75 \end{aligned}$ .
::交叉乘以 4x= 9x3 或 4x= 27。然后将两边除以 4 以获得 x= 274 或 x= 6. 75 。

2. Solve the following proportion for $\begin{aligned} x \end{aligned}$ : $\begin{aligned} \frac{0.5}{3} = \frac{56}{x} \end{aligned}$
::2. 解决x的下列比例:0.53=56x

Cross multiply to get $\begin{aligned} 0.5 x = 56 \times 3 \end{aligned}$ , or $\begin{aligned} 0.5 x = 168. \end{aligned}$ Then divide both sides by 0.5 to get $\begin{aligned} x = 336. \end{aligned}$
::交叉乘以0.5x=56x3,或0.5x=168。然后将两边除以0.5以获得 x=336。

Solve Real-World Problems Using Proportions
::利用比例解决现实世界问题

1. A cross-country train travels at a steady speed . It covers 15 miles in 20 minutes. How far will it travel in 7 hours assuming it continues at the same speed?
::1. 跨国列车以稳定速度行驶,在20分钟内行驶15英里,如果以同样速度行驶,在7小时内行驶多远?

We’ve done speed problems before; remember that speed is just the ratio $\begin{aligned} \frac{distance}{time} \end{aligned}$ , so that ratio is the one we’ll use for our proportion. We can see that the speed is $\begin{aligned} \frac{15 m i l e s}{20 m i n u t e s} \end{aligned}$ , and that speed is also equal to $\begin{aligned} \frac{x m i l e s}{7 h o u r s} \end{aligned}$ .
::我们以前曾遇到过速度问题; 记住速度只是距离时间的比率, 因此这个比率是我们比例所使用的比例。我们可以看到速度是15英里20分钟,而且速度也相当于x英里7小时。

To set up a proportion, we first have to get the units the same. 20 minutes is $\begin{aligned} \frac{1}{3} \end{aligned}$ of an hour, so our proportion will be $\begin{aligned} \frac{15}{\frac{1}{3}} = \frac{x}{7} \end{aligned}$ . This is a very awkward looking ratio, but since we’ll be cross multiplying, we can leave it as it is.
::要设定比例,我们首先必须让单位相同。 20分钟是每小时13分钟, 因此我们的比例将是1513=x7。这是一个非常尴尬的外观比例, 但是既然我们要交叉乘法, 我们就可以保持原样。

Cross multiplying gives us $\begin{aligned} 7 \times 15 = \frac{1}{3} x \end{aligned}$ . Multiplying both sides by 3 then gives us $\begin{aligned} 3 \times 7 \times 15 = x \end{aligned}$ , or $\begin{aligned} x = 315 \end{aligned}$ .
::交叉乘以 7x15=13x。乘以 3 乘以 3, 3x7x15=x, 或 x= 315 。

The train will travel 315 miles in 7 hours.
::火车7小时后将行驶315英里

2. In the United Kingdom, Alzheimer’s disease is said to affect one in fifty people over 65 years of age. If approximately 250000 people over 65 are affected in the UK, how many people over 65 are there in total?
::2. 在联合王国,阿尔茨海默氏病据说影响到65岁以上每五十人中就有一人,如果65岁以上65岁以上人口在联合王国受到影响的大约250 000人,那么65岁以上人口总数是多少?

The fixed ratio in this case is the 1 person in 50. The unknown quantity $\begin{aligned} x \end{aligned}$ is the total number of people over 65. Note that in this case we don’t need to include the units, as they will cancel between the numerator and denominator.
::这里的固定比率是50分之一的1人。未知数量x是65岁以上的总人数。请注意,在此情况下,我们不需要包括单位,因为它们会取消分子和分母之间的连接。

Our proportion is $\begin{aligned} \frac{1}{50} = \frac{250000}{x} \end{aligned}$ . Each ratio represents
::我们的比例是150=250 000x。

$\begin{aligned} \frac{people with Alzheimer's}{total people} \end{aligned}$
.
::阿尔茨海默症患者总数。

Cross multiplying, we get $\begin{aligned} 1 \cdot x = 250000 \cdot 50 \end{aligned}$ , or $\begin{aligned} x = 12, 500, 000 \end{aligned}$ .
::交叉乘法,我们得到1x=250000-50,或x=12500,000。

There are approximately 12.5 million people over the age of 65 in the UK.
::联合王国约有1 250万65岁以上的人。

Example
::示例示例示例示例

Example 1
::例1

A chemical company makes up batches of copper sulfate solution by adding 250 kg of copper sulfate powder to 1000 liters of water. A laboratory chemist wants to make a solution of identical concentration, but only needs 350 mL (0.35 liters) of solution. How much copper sulfate powder should the chemist add to the water?
::化学公司通过在1000升水中添加250公斤硫酸铜粉末,组成一批硫酸铜溶液。实验室化学家希望用同样的浓度来溶解,但只需要350毫升(0.35升)溶液。化学家应该给水添加多少硫酸铜粉?

The ratio of powder to water in the first case, in kilograms per liter, is $\begin{aligned} \frac{250}{1000} \end{aligned}$ , which reduces to $\begin{aligned} \frac{1}{4} \end{aligned}$ . In the second case, the unknown amount is how much powder to add. If we label that amount $\begin{aligned} x \end{aligned}$ , the ratio is $\begin{aligned} \frac{x}{0.35} \end{aligned}$ . So our proportion is $\begin{aligned} \frac{1}{4} = \frac{x}{0.35} \end{aligned}$ .
::第一种情况下的粉末与水的比例,按每升公斤计算,是250,1000, 降为14, 在第二种情况下,未知的量是需要添加多少粉末。如果我们标注该数量x,那么比例是x0.35。因此我们的比例是14=x0.35。

To solve for $\begin{aligned} x \end{aligned}$ , first we multiply both sides by 0.35 to get $\begin{aligned} \frac{0.35}{4} = x \end{aligned}$ , or $\begin{aligned} x = 0.0875 \end{aligned}$ .
::要解析 x, 首先我们要将两边乘以0.35, 以获得0. 354=x, 或 x=0.0875 。

The mass of copper sulfate that the chemist should add is 0.0875 kg, or 87.5 grams.
::化学家应添加的铜硫酸盐质量为0.0875公斤,即87.5克。

Review
::回顾

Solve the following proportions.
::解决以下比例。

$\begin{aligned} \frac{13}{6} = \frac{5}{x} \end{aligned}$
::136=5x

$\begin{aligned} \frac{1.25}{7} = \frac{3.6}{x} \end{aligned}$
::1.257=3.6x

$\begin{aligned} \frac{6}{19} = \frac{x}{11} \end{aligned}$
::619=x11

$\begin{aligned} \frac{1}{x} = \frac{0.01}{5} \end{aligned}$
::1x=0.015

$\begin{aligned} \frac{300}{4} = \frac{x}{99} \end{aligned}$
::3004=x99

$\begin{aligned} \frac{2.75}{9} = \frac{x}{(\frac{2}{9})} \end{aligned}$
::2.759=x(29)

$\begin{aligned} \frac{1.3}{4} = \frac{x}{1.3} \end{aligned}$
::1.34=x1.3

$\begin{aligned} \frac{0.1}{1.01} = \frac{1.9}{x} \end{aligned}$
::0.1.1.01=1.9x

$\begin{aligned} \frac{5}{36} = \frac{x}{30} \end{aligned}$
::536=x30

$\begin{aligned} \frac{10}{3} = \frac{6.9}{x} \end{aligned}$
::103=6.9x

Review (Answers)
::回顾(答复)

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::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。

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