Course: 4.2 单位使用率和等效率

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单位使用率和等效率
Myra’s junior soccer team scored 10 goals in the last 2 games. At this rate , how many goals will Myra’s team score in the next 5 games?
::Myra的初级足球队在过去两场比赛中得分10分。以这个速度计算,Myra的初级足球队在接下来的五场比赛中得分多少?

In this concept, you will learn to use and find equivalent .
::在这一概念中,你会学会使用和找到等效的。

Unit Rate
::单位费率

A unit rate is a special kind of ratio, where the second number, or the denominator , is equal to one. With a unit rate, you are comparing a quantity to one. Some common unit rates are miles per gallon, price per pound, and pay rate per hour. To find a unit rate, simplify the ratio so that you have a 1 in the denominator. You can simply divide the first number in the ratio by the second. Make sure you keep track of the units.
::单位费率是一种特殊的比例, 即第二个数字或分母等于一个。如果使用一个单位费率, 您将比较一个数量和一个数量。一些通用单位费率是每加仑英里、每磅价格和每小时工资率。要找到一个单位费率, 请简化比率, 以便您在分母中有一个。您可以简单地将第一个数字除以第二个。请确保您跟踪各个单位。

Let’s look at an example.
::让我们举个例子。

Kayla bought 5.5 pounds of apples. She paid a total of $7.15. What was the unit rate of the apples per pound?
::Kayla购买了5.5磅苹果,她总共支付了7.15美元。每磅苹果的单位价格是多少?

First, write the ratio of the price to pounds.
::首先,写下价格与磅的比率。

$\begin{aligned} \frac{price}{pounds} = \frac{$ 7.15}{5.5} \end{aligned}$
$\begin{align*}\frac{\text{price}}{\text{pounds}}=\frac{\$ 7.15}{5.5}\end{align*}$
::价格=7.155.5美元

Next, divide the price by the pounds so you can find the unit rate.
::接下来,将价格除以磅,这样你就可以找到单位费率。

$\begin{aligned} 1.3 \\ 5.5 \bar{) 7.15} \end{aligned}$
$\begin{align*}\ \ 1.3 \ \ \\5.5 \overline{\smash{\kern-0.5pt \bigr)} 7.15}\\\end{align*}$

The answer is 1.3.
::答案是1.3。

The unit price for the apples is $1.30 per pound.
::苹果的单价是每磅1.30美元。

Let’s look at another example.
::让我们再看看另一个例子。

Brian worked for 8 hours yesterday and made a total of $86. What is his pay rate?
::Brian昨天工作了8个小时总共赚了86美元

First, write the ratio of the wages to hours.
::首先,写下工资与小时的比率。

$\begin{aligned} \frac{wages}{hours} = \frac{$ 86}{8} \end{aligned}$
$\begin{align*}\frac{\text{wages}}{\text{hours}}=\frac{\$ 86}{8}\end{align*}$ Next, divide the wages by the hours so you can find the unit pay rate.

$\begin{aligned} 10.75 \\ 8 \bar{) 86} \end{aligned}$
$\begin{align*}\ \ 10.75\\ 8 \overline{\smash{ \kern-0.5pt \bigr)} 86} \ \ \ \ \ \\\end{align*}$

The answer is 10.75.
::答案是10.75。

Brian is paid $10.75 per hour.
::布赖恩每小时10.75美元

Next, let’s look at equivalent rates. Equivalent rates are rates that are equal.
::接下来,让我们看看等效率。等效率是等效率。等效率是等效率。

Let’s look at an example.
::让我们举个例子。

Find an equivalent rate for this comparison.
::为这一比较寻找等值的汇率。

$\begin{aligned} \frac{$ 2.00}{1} = \frac{x}{8} \end{aligned}$
$\begin{align*}\frac{\$ 2.00}{1}=\frac{x}{8}\end{align*}$
::2.001=x8美元

In order to find the equivalent rate for this unit rate, cross multiply.
::为了找到这一单位费率的等值率,乘以乘数。

$\begin{aligned} \begin{array}{rcl} \frac{$ 2.00}{1} & = & \frac{x}{8} \\ (1) x & = & $ 2.00 \times 8 \\ x & = & $ 16.00 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{$ 2.00}{1} &= & \frac{x}{8} \\ (1)x &=& $2.00 \times 8 \\ x &=& $16.00 \end{array}\end{align*}$ The answer is $16.00.
::$0.001=x8(1)x=$2.00x8x=$16.00 答案是$16.00。

The rate is $\begin{align*}\frac{\$ 2.00}{1}\end{align*}$ is equivalent to the rate $\begin{align*}\frac{\$ 16.00}{8}\end{align*}$ .
::费率为2.001美元,相当于费率16.008美元。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Myra’s winning streak.
::更早之前,

Myra’s team scores on 10 goals in the last 2 games. You want to find out how many goals, at this rate, they will score in the next 5 games.
::Myra的球队在最后两场比赛的10个球中得分。您想知道,以这个速度,他们在未来5场比赛中得分有多少。

First, write an equivalent ratio.
::首先,写一个等值比例。

$\begin{aligned} \frac{10}{2} = \frac{x}{5} \end{aligned}$
$\begin{align*}\frac{10}{2} = \frac{x}{5}\end{align*}$ Next, cross multiply.
::102=x5 下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{10}{2} & = & \frac{x}{5} \\ 2 x & = & 5 \times 10 \\ 2 x & = & 50 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{10}{2} &= & \frac{x}{5} \\ 2x &=& 5 \times 10 \\ 2x &=& 50 \end{array}\end{align*}$ Then, divide by 2 to isolate $\begin{align*}x\end{align*}$ .
::102=x52x=5x102x=50 然后除以 2 以孤立 x。

$\begin{aligned} \begin{array}{rcl} 2 x & = & 50 \\ \frac{2 x}{2} & = & \frac{50}{2} \\ x & = & 25 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2x &=&50 \\ \frac{2x}{2} &= & \frac{50}{2} \\ x &=& 25 \\ \end{array}\end{align*}$ The answer is 25.
::2x=502x2=502xx=25 答案是25。

In the next 5 games, Myra’s team should score 25 goals if the team keeps their current rate.
::在接下来的五场比赛中, Myra的球队如果保持目前的速度, 应该得25个球员。

Example 2
::例2

A store sells salmon for $6.99 per pound. What is the rate for 6 pounds of salmon?
::一家商店卖鲑鱼每磅6.99美元 6磅鲑鱼的售价是多少?

First, write the ratios from the question.
::首先,从问题中写出比率。

$\begin{aligned} \frac{$ 6.99}{1} = \frac{x}{6} \end{aligned}$
$\begin{align*}\frac{\$ 6.99}{1} = \frac{x}{6}\end{align*}$
::6.991=x6美元

Next, cross multiply to find the cost of 6 pounds of salmon.
::接下来,交叉倍增以找到 6磅鲑鱼的成本。

$\begin{aligned} \begin{array}{rcl} \frac{$ 6.99}{1} & = & \frac{x}{6} \\ (1) x & = & $ 6.99 \times 6 \\ x & = & $ 41.94 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{\$ 6.99}{1} &= & \frac{x}{6} \\ (1)x &=& $6.99 \times 6 \\ x &=& $41.94 \end{array}\end{align*}$
::6.991=x6(1)x=6.99x6x=41.94美元

The answer is $41.94. Six pounds of salmon will cost $41.94.
::答案是41.94 6磅鲑鱼需要41.94美元

Example 3
::例3

Write find the unit rate for the following ratio.
::写入找到以下比率的单位费率。

$\begin{aligned} \frac{14}{7} \end{aligned}$
$\begin{align*}\frac{14}{7}\end{align*}$

First, divide the numerator by the denominator.

$\begin{aligned} 2 \\ 7 \bar{) 14} \end{aligned}$
$\begin{align*}\ \ 2\\ 7 \overline{ \smash{ \kern-0.5pt \bigr)} 14}\\\end{align*}$
::首先,将分子除以分母。 27)14

The answer is 2.
::答案是2。

The unit rate is $\begin{align*}\frac{2}{1}\end{align*}$ .
::单位费率为21。

Example 4
::例4

Write find the unit rate for the following ratio.
::写入找到以下比率的单位费率。

$\begin{aligned} \frac{36}{12} \end{aligned}$
$\begin{align*}\frac{36}{12}\end{align*}$

First, divide the numerator by the denominator.
::首先,将分子除以分母。

$\begin{aligned} 3 \\ 12 \bar{) 36} \end{aligned}$
$\begin{align*}\ \ 3\\12 \overline{ \smash{\kern-0.5pt \bigr)} 36}\\\end{align*}$

The answer is 3.
::答案是3

The unit rate is $\begin{align*}\frac{3}{1}\end{align*}$ .
::单位费率为31。

Example 5
::例5

Write find the unit rate for the following ratio.
::写入找到以下比率的单位费率。

$\begin{aligned} \frac{48}{8} \end{aligned}$
$\begin{align*}\frac{48}{8}\end{align*}$

First, divide the numerator by the denominator.
::首先,将分子除以分母。

$\begin{aligned} 6 \\ 8 \bar{) 48} \end{aligned}$
$\begin{align*}\ \ 6\\ 8 \overline{\smash{\kern-0.5pt \bigr)} 48}\\\end{align*}$

The answer is 6. Therefore the unit rate is $\begin{align*}\frac{6}{1}\end{align*}$ .
::答案是6,因此单位费率是61。

Review
::回顾

Use what you have learned about ratios to solve each problem.
::使用你学到的关于比率的知识来解决每一个问题。

In Kyle’s drawer, there are 14 pairs of white socks and 8 pairs of black socks
::凯尔抽屉里有14双白袜子和8双黑袜子

1. Without simplifying, write the ratio of black socks to white socks.
::1. 在不简化的情况下,写出黑袜子与白袜子的比例。

2. Without simplifying, write the ratio of black socks to total socks.
::2. 在不简化的情况下,写下黑袜子与全部袜子的比例。

3. Without simplifying, write the ratio of white socks to total socks.
::3. 在不简化的情况下,将白袜子与全部袜子的比例写成。

4. Simplify your answer for number 1.
::4. 简化对1号的答复。

5. Simplify your answer for number 2.
::5. 简化第2号的答复。

6. Simplify your answer for number 3.
::6. 简化对3号的答复。

There are 150 apartments in the Gray building. Of these, 60 are rented and the rest are owned. There are 65 apartments in the Black building. Of these, 45 are rented and the rest are owned. Simplify each answer.
::灰色大楼有150套公寓,其中60套是租用的,其余为自有,黑色大楼有65套,其中45套是租用的,其余为自有。

7. What is the ratio of rented to owned in the Gray building?
::7. Gray大楼租用与拥有的比率是多少?

8. What is the ratio of rented to owned in the Black building?
::8. 租用在黑大楼内拥有的比率是多少?

9. Write the ratio of rented to total apartments in the Gray building.
::9. 写下Gray大楼租用公寓与总公寓的比例。

10. Write the ratio of rented to total apartments in the Black building.
::10. 写下黑色大楼租用公寓与总公寓的比例。

11. Write the ratio of owned to total apartments in the Gray building.
::11. 写出Gray大楼自有公寓与总公寓的比例。

12. Write the ratio of owned to total apartments in the Black building.
::12. 写下黑大楼内自有公寓与总公寓的比率。

13. Holly works at a library re-shelving books. She re-shelved 960 books in 4 hours. What is Holly’s rate of re-shelving in books per hour?
::13. Holly在图书馆的藏书馆工作,在4小时内重新整理960本书。 Holly每小时重新整理书籍的速度是多少?

14. Sam bought 9.5 pounds of peaches to make a pie. The peaches cost $15.39. What was the unit rate of the peaches?
::14. Sam购买了9.5磅的桃子做馅饼,桃子价格为15.39美元,桃子单位价格是多少?

15. Don can wrap 8 presents in an hour. How long will it take Don to wrap 12 presents?
::15. Don一小时后可以打包8件礼物,Don需要多长时间才能打包12件礼物?

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。

Section outline

General

单位使用率和等效率

Unit Rate ::单位费率

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)

Unit Rate
::单位费率

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)