Section: 涉及费率和单位分析的解决问题 | 4.17 涉及费率和单位分析的解决问题

Section outline

Joanne is training for the City to Surf marathon in Sydney, Australia. She knows she can run a 30 kilometer marathon in about 5 hours. How fast is Joanne going per minute ?
::Joanne在澳洲悉尼的Surf马拉松训练她知道5小时内可以跑30公里马拉松 Joanne每分钟跑多快?

In this concept, you will learn to solve problems involving rates and unit analysis .
::在此概念中,您将学会解决涉及费率和单位分析的问题。

Unit Rate
::单位费率

A rate often refers to speed or a rate could refer to the amount of money someone makes per hour .
::兑换率通常是指速度,或兑换率可指某人每小时挣钱的数额。

When you talk about a unit rate , you look at comparing a rate to 1. An example of a unit rate would be 65 miles per hour. A key word when working with unit rate is the word “per”.
::当您谈论单位费率时,请查看对单位费率与1的费率的比较。单位费率的示例是每小时65英里。在使用单位费率时,关键词是“每个”一词。

Let’s look at an example.
::让我们举个例子。

Jeff makes $150.00 an hour as a consultant. What is his rate per minute?
::Jeff当顾问每小时赚150美元他每分钟的费率是多少?

First, use units to convert hours to minutes.
::首先,使用单位将小时转换为分钟。

$\begin{aligned} \begin{array}{rcl} # min & = & 1 hr \times \frac{60 min}{1 hr} \\ # min & = & 60 min \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \# \ \text{min} &=& 1 \ \text{hr} \times \frac{60 \ \text{min}}{1 \ \text{hr}} \\ \# \ \text{min} &=& 60 \ \text{min} \end{array}\end{align*}$
::# 分钟=1 小时x60 分钟1 小时= 分钟=60 分钟

Next, convert $\begin{align*}\frac{\$ 150}{\text{hr}}\end{align*}$ into $\begin{align*}\frac{\$}{\text{min}}\end{align*}$ .
::接下来,把150美元兑换成美元。

$\begin{aligned} \begin{array}{rcl} \frac{$ 150.00}{1 hr} & = & \frac{$ 150.00}{60 min} \\ \frac{$ 150.00}{60 min} & = & $ 2.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{\$150.00}{1 \ \text{hr}} &=& \frac{\$ 150.00}{60 \ \text{min}} \\ \frac{\$150.00}{ 60 \ \text{min} }&=& \$ 2.5 \end{array}\end{align*}$
::150.001 小时= 150.0060 分钟 150.0060 分钟= 2.5美元

The answer is 2.5.
::答案是2.5

Jeff makes $\begin{align*}\frac{\$ 2.50}{\text{min}}\end{align*}$ .
::杰夫赚了2.50美分

Unit analysis is when you look at how to measure individual units in different measurement amounts and it is used to convert units of measurement.
::单位分析是指当你研究如何用不同的计量数量衡量单个单位时,它被用来转换计量单位。

Let’s look at an example.
::让我们举个例子。

Juanita worked for 18 hours last week. She made $116.00 at the end of the week. Juanita was sure that her manager had made a mistake and that she should have made more money. Juanita makes $9.00 per hour. Did Juanita make the correct amount of money or was there a mistake?
::Juanita上周工作了18小时,在周末她赚了116美元,Juanita确信她的经理犯了错误,她应该赚更多的钱,Juanita每小时挣9美元。 Juanita是赚了正确数额的钱还是弄错了?

First, find out how much Juanita should have made if she made.
::首先查查Juanita如果做了应该赚多少钱

$\begin{aligned} \begin{array}{rcl} pay & = & \frac{$ 9.00}{1 hr} \times 18 hr \\ pay & = & $ 162 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{pay} &=& \frac{ \$9.00}{1 \text{ hr}} \times 18 \ \text{hr} \\ \text{pay} &=& \$162 \end{array}\end{align*}$
::工资=9.001美元 hrx18 小时工资= 162美元

Next, determine if Juanita was not paid correctly.
::接下来,确定Juanita是否没有得到正确的报酬。

Juanita was paid $116 but should have been paid $162.
::向Juanita支付了116美元,但本应支付162美元。

There was a mistake in her pay by $46.
::她的薪水有误46美元

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Joanne’s marathon training.
::Joanne在马拉松训练时遇到问题。

Joanne can run 30 km in 5 hours but wants to know her rate per minute.
::Joanne5小时可以跑30公里但想知道她每分钟的速度

First, use units to convert hours to minutes.
::首先,使用单位将小时转换为分钟。

$\begin{aligned} \begin{array}{rcl} # min & = & 5 hr \times \frac{60 min}{1 hr} \\ # min & = & 300 min \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \# \ \text{min} &=& 5 \text{ hr} \times \frac{60 \ \text{min}}{1 \ \text{hr}}\\ \# \ \text{min} &=& 300\ \text{min} \end{array}\end{align*}$
::# 分钟= 5 小时x60 分钟1 小时= 分钟= 300 分钟

Next, convert 30 $\begin{align*}\frac{\text{km}}{\text{hr}}\end{align*}$ into $\begin{align*}\frac{\text{km}}{\text{min}}\end{align*}$ .
::接下来,将30公里转换成公里。

$\begin{aligned} \begin{array}{rcl} \frac{30 km}{1 hr} & = & \frac{30 km}{300 min} \\ \frac{30 km}{300 min} & = & 0.10 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{30 \ \text{km}}{1 \ \text{hr}} &=& \frac{30 \ \text{km}}{300 \ \text{min}} \\ \frac{30 \ \text{km}}{300 \ \text{min}} &=& 0.10 \end{array}\end{align*}$
::30公里1 小时=30公里300 分钟30公里300 分钟300 分钟=0.10

The answer is 0.10.
::答案是0.10。

Joanne runs at a speed of $\begin{align*}0.1 \frac{\text{km}}{\text{min}}\end{align*}$ .
::乔安以0.1kmmin的速度运行

Example 2
::例2

Jesse has a car that holds 14 gallons of gasoline. During the first week of the month, gasoline cost $2.75 per gallon . During the second week of the month, gasoline cost $2.50 per gallon. How much was the total cost for the 28 gallons of gasoline?
::杰西有一辆汽车,装有14加仑汽油。在每月第一周,汽油每加仑2.75美元。在每月第二周,汽油每加仑2.50美元。28加仑汽油的总成本是多少?

First, start by writing a variable expression to work on this problem. You know that the number of gallons of gasoline does not change. Therefore, let $\begin{align*}x\end{align*}$ represent the number of gallons of gasoline.
::首先, 首先写一个可变表达式来解决这个问题。您知道加仑汽油的数量没有变化。因此, 请让 x 代表加仑汽油的数量。

$\begin{aligned} Total cost = 2.75 x + 2.50 x \end{aligned}$
$\begin{align*}\text{Total cost} = 2.75x+2.50x\end{align*}$
::费用总额=2.75x+2.50x

Next, substitute 14 for our variable $\begin{align*}x\end{align*}$ .
::接下来,换14 取代我们的变数x。

$\begin{aligned} \begin{array}{rcl} Total cost & = & 2.75 (14) + 2.50 (14) \\ Total cost & = & 38.5 + 35 \\ Total cost & = & 73.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{Total cost} &=& 2.75(14)+2.50(14) \\ \text{Total cost} &=& 38.5+35 \\ \text{Total cost} &=& 73.5 \end{array} \end{align*}$
::总成本=2.75(14)+2.50(14)总成本=38.5+35

The answer is 73.5.
::答案是73.5

Therefore, Jess paid $73.50 in gasoline for the two weeks.
::因此,Jess支付了这两周的汽油73.50美元。

Example 3
::例3

Ten apples cost $3.99. What is the cost for one apple?
::十个苹果要3.99美元,一个苹果要多少钱?

First, set up a proportion.
::首先,设定一个比例。

$\begin{aligned} \frac{$ 3.99}{10 apples} = \frac{x}{1 apple} \end{aligned}$
$\begin{align*} \frac{\$3.99}{10 \ \text{apples}} = \frac{x}{1 \ \text{apple}}\end{align*}$
::3.9910 苹果=x1苹果

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{3.99}{10} & = & \frac{x}{1} \\ 10 x & = & 1 \times 3.99 \\ 10 x & = & 3.99 \end{array} \end{aligned}$
$\begin{align*} \begin{array}{rcl} \frac{3.99}{10} &=& \frac{x}{1} \\ 10 x &=& 1 \times 3.99 \\ 10x &=& 3.99 \end{array}\end{align*}$
::3.9910=x110x=1x3.9910x=3.99

Then, divide both sides by 10 to solve for $\begin{align*}x\end{align*}$ .
::然后把两边除以10 解决x

$\begin{aligned} \begin{array}{rcl} 10 x & = & 3.99 \\ \frac{10 x}{10} & = & \frac{3.99}{10} \\ x & = & 0.399 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 10x &=& 3.99 \\ \frac{10x}{10} &=& \frac{3.99}{10} \\ x &=& 0.399 \end{array}\end{align*}$
::10x=3.9910x10=3.9910x=0.399

The answer is 0.399.
::答案是0.399

One apple costs $0.40 or 40¢.
::一个苹果费用为0.40美元或40美元。

Example 4
::例4

Fifteen gallons of gasoline costs $45.00. How much is it per gallon?
::15加仑汽油价格45美元,每加仑多少?

First, set up a proportion.
::首先,设定一个比例。

$\begin{aligned} \frac{$ 45.00}{15 gallons} = \frac{x}{1 gallon} \end{aligned}$
$\begin{align*}\frac{\$45.00}{15 \ \text{gallons}} = \frac{x}{1 \ \text{gallon}}\end{align*}$
::45.0015加仑=x1加仑

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{45}{15} & = & \frac{x}{1} \\ 15 x & = & 1 \times 45 \\ 15 x & = & 45 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{45}{15} &=& \frac{x}{1} \\ 15 x &=& 1 \times 45 \\ 15x &=& 45 \end{array}\end{align*}$
::4515=x115x=1x4515x=45

Then, divide both sides by 15 to solve for $\begin{align*}x\end{align*}$ .
::然后把两边除以15 解决x

$\begin{aligned} \begin{array}{rcl} 15 x & = & 45 \\ \frac{15 x}{15} & = & \frac{45}{15} \\ x & = & 3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 15x &=& 45 \\ \frac{15x}{15} &=& \frac{45}{15} \\ x &=& 3 \end{array}\end{align*}$
::15x=4515x15=4515x3

The answer is 3.
::答案是3

One gallon of gas costs $3.00.
::一加仑的煤气要三美元

Example 5
::例5

Two tickets to a ballgame costs $111.50. What is the cost for one ticket?
::两张球赛的票价111.5美元一张球赛的票价是多少?

First, set up a proportion.
::首先,设定一个比例。

$\begin{aligned} \frac{$ 111.50}{2 tickets} = \frac{x}{1 ticket} \end{aligned}$
$\begin{align*} \frac{\$111.50}{2 \ \text{tickets}} = \frac{x}{1 \ \text{ticket}}\end{align*}$
::111.502 张票=x1张票

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{111.50}{2} & = & \frac{x}{1} \\ 2 x & = & 1 \times 111.50 \\ 2 x & = & 111.50 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{111.50}{2} &=& \frac{x}{1} \\ 2x &=& 1 \times 111.50 \\ 2x &=& 111.50 \\ \end{array}\end{align*}$
::111. 502=x12x=1x111.1.502x=111.50

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

$\begin{aligned} \begin{array}{rcl} 2 x & = & 111.50 \\ \frac{2 x}{2} & = & \frac{111.50}{2} \\ x & = & 55.75 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2x &=& 111.50 \\ \frac{2x}{2} &=& \frac{111.50}{2} \\ x &=& 55.75 \end{array}\end{align*}$
::2x=111.502x2=111.502x55.75

The answer is 55.75.
::答案是55.75。

One ticket to the game is $55.75
::一场比赛的票价是55.75美元

Review
::回顾

Use what you have learned to solve each problem.
::利用你学到的东西解决每个问题

Peter runs at a rate of 10 kilometers per hour. How many kilometers will he cover in 8 hours?
::彼得以每小时10公里的速度跑步他八小时后要覆盖多少公里?

A cheetah can run at a speed of 60 miles per hour. What is his distance after 6 hours?
::猎豹可以以每小时60英里的速度跑动 6小时后他的距离是多少?

What is the distance formula?
::距离公式是什么?

If a car travels at a rate of 65 miles per hour for 30 minutes, how far will it travel?
::如果一辆汽车以每小时65英里的速度行驶30分钟,它会行驶多远?

A train travels at a rate of 50 miles per hour. If it needs to travel 320 miles, how many minutes will it take?
::一列火车以每小时50英里的速度行驶,如果需要行驶320英里,需要多少分钟?

A car travels 65 mph for 12 hours. How many miles will it travel?
::一辆65英里长的汽车飞行12小时飞行多少英里?

A bus traveled 300 miles at an average speed of 50 miles per hour. How long did this trip take the bus?
::一辆公共汽车以每小时50英里的平均速度行驶300英里。

A car traveled at an average speed of 40 miles per hour through a construction zone. If the car traveled 20 miles at this rate, how many hours did it take to travel the 20 miles?
::一辆汽车以每小时40英里的平均速度通过一个建筑区,如果汽车以这种速度行驶20英里,需要多少小时才能行驶20英里?

What is the formula for velocity?
::速度的公式是什么?

What is the velocity of an object that travels 500 miles in 2.5 hours?
::一个在2.5小时内飞行500英里的物体的速度是多少?

If an object has a velocity of 125 miles per hour, how long will it take to travel 4,375 miles?
::如果一个物体的速度是每小时125英里, 飞行4 375英里需要多长时间?

If an object has a velocity of 7 kilometers per minute, how far will it travel in 2 hours?
::如果一个物体的速度为每分钟7公里,它2小时后会走多远?

If an object has a velocity of 4 meters per second, how many kilometers will it travel in 2 days?
::如果一个物体的速度为每秒4米,它两天内要行走多少公里?

The formula for density is $\begin{align*}D=\frac{m}{v}\end{align*}$ where $\begin{align*}D\end{align*}$ represents the density of an object, $\begin{align*}m\end{align*}$ represents the mass of the object, and $\begin{align*}v\end{align*}$ represents the volume of the object. What is the density of a brick that weighs 9 grams and has a volume of $\begin{align*}36cm^3?\end{align*}$
::密度的公式是 D=mv , D 代表对象的密度, m 代表对象的质量, v 代表对象的体积。重9克、体积为36厘米3的砖块的密度是多少?

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

Section outline

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)

Resources ::资源

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)

Resources
::资源