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  • 猜测和检查,后向工作

    Guess and Check and Working Backwards
    ::猜测、检查和后向工作

    In this section, you will learn about the methods of Guess and Check and Working Backwards . These are very powerful strategies in problem solving and probably the most commonly used in everyday life. Let’s review our problem-solving plan.
    ::在本节中,您将了解猜疑和支票以及后向工作的方法。 这些是解决问题的非常有力的策略,可能是日常生活中最常用的策略。 让我们审视一下我们的解决问题计划。

    Step 1
    ::步骤1

    Understand the problem.
    ::了解问题所在。

    Read the problem carefully. Then list all the components and data involved, and assign your variables.
    ::仔细读取问题。 然后列出所涉及的全部组件和数据, 并指定变量 。

    Step 2
    ::步骤2

    Devise a plan – Translate
    ::制定计划 - 翻译

    Come up with a way to solve the problem. Set up an equation , draw a diagram, make a chart or construct a table.
    ::找到解决问题的方法。 设置方程式、 绘制图表、 绘制图表或构建表格 。

    Step 3
    ::步骤3

    Carry out the plan – Solve
    ::执行计划 — — 解决

    This is where you solve the equation you came up with in Step 2.
    ::这就是你解开第二步的方程式的地方

    Step 4
    ::步骤4

    Look – Check and Interpret
    ::- 检查和解释

    Check that the answer makes sense.
    ::检查答案是有道理的。

    Let’s now look at some strategies we can use as part of this plan.
    ::现在让我们来看看一些我们可用作这项计划一部分的战略。

    Develop and Use the Strategy “Guess and Check”
    ::制定和使用“指导和检查”战略

    The strategy for the method “Guess and Check” is to guess a solution and then plug the guess back into the problem to see if you get the correct answer. If the answer is too big or too small, make another guess that will get you closer to the goal, and continue guessing until you arrive at the correct solution. The process might sound long, but often you will find patterns that you can use to make better guesses along the way.
    ::方法“ Guess and Check” 的策略是猜测一个解决方案, 然后将猜想插回问题中, 看看您是否得到了正确的答案。 如果答案太大或太小, 再做一个猜想, 让你更接近目标, 继续猜想, 直到你找到正确的解决方案。 这一过程听起来可能很长, 但通常你会发现一些模式, 您可以在路上更好地猜测。

    Using the "Guess and Check" Strategy
    ::使用“ Guess and Check” 策略

    Nadia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece?
    ::Nadia拿了一条长48英寸的丝带 把它切成两块 一块是另一块的三倍

    Step 1: Understand
    ::第1步:理解

    We need to find two numbers that add up to 48. One number is three times the other number.
    ::我们需要找到两个加到48的号码 一个数字是另一个数字的三倍

    Step 2: Strategy
    ::步骤2:战略

    We guess two random numbers, one three times bigger than the other, and find the sum.
    ::我们猜两个随机数字, 一个比另一个大三倍, 然后找到数字。

    If the sum is too small we guess larger numbers, and if the sum is too large we guess smaller numbers.
    ::如果总和太小,我们猜数字会更大,如果总和太大,我们猜数字会较小。

    Then, we see if any patterns develop from our guesses.
    ::然后,我们看看是否有任何模式 发展出我们的猜测。

    Step 3: Apply Strategy/Solve
    ::步骤3:应用战略/单元

    Guess 5   and   15 5 + 15 = 20 sum is too small Guess 6   and   18 6 + 18 = 24 sum is too small

    ::Guess5和155+15=20sum太小,Guess6和186+18=24sum太小

    Our second guess gives us a sum that is exactly half of 48. What if we double that guess?
    ::我们的第二个猜想给我们的金额 正好是48的的一半 如果我们加倍这个猜想呢?

    12 + 36 = 48

    There’s our answer. The pieces are 12 and 36 inches long.
    ::这就是我们的答案。 碎片长12和36英寸。

    Step 4: Check
    ::第4步:检查

    12 + 36 = 48   The pieces add up to 48 inches . 36 = 3 ( 12 ) One piece is three times as long as the other .

    ::12+36=48 碎片加起来共48英寸36=3(12) 一块是另一块的三倍长。

    The answer checks out.
    ::答案检查出来。

    Develop and Use the Strategy “Work Backward”
    ::制定和使用 " 后向工作 " 战略

    The “Work Backward” method works well for problems where a series of operations is done on an unknown number and you’re only given the result. To use this method, start with the result and apply the operations in reverse order until you find the starting number.
    ::“后向工作”方法对于一系列操作以未知数字完成,而您只得到结果的问题效果良好。 要使用这种方法,请从结果开始,在找到起始数字之前按相反顺序应用操作。

    Using the "Work Backward" Strategy 
    ::使用“向后工作”战略

    Anne has a certain amount of money in her bank account on Friday morning. During the day she writes a check for $24.50, makes an ATM withdrawal of $80 and deposits a check for $235. At the end of the day she sees that her balance is $451.25. How much money did she have in the bank at the beginning of the day?
    ::安妮在星期五早上的银行帐户里有一定数额的钱。在她写24.50美元的支票的当天,她提取了80美元的自动取款机,并存入了235美元的支票。 在最后一天,她看到自己的余额是451.25美元。她第一天在银行里有多少钱?

    Step 1: Understand
    ::第1步:理解

    We need to find the money in Anne’s bank account at the beginning of the day on Friday.
    ::我们必须在周五当天初在安妮的银行账户中找到这笔钱。

    She took out $24.50 and $80 and put in $235.
    ::她拿出了24.50美元和80美元 然后放进235美元

    She ended up with $451.25 at the end of the day.
    ::她最后在一天结束时有451.25美元。

    Step 2: Strategy
    ::步骤2:战略

    We start with an unknown amount, do some operations, and end up with a known amount.
    ::我们从一个未知的数量开始, 做一些操作, 最后有一个已知的数量。

    We need to start with the result and apply the operations in reverse.
    ::我们需要从结果开始,反向运用行动。

    Step 3: Apply Strategy/Solve
    ::步骤3:应用战略/单元

    Start with $451.25. Subtract $235, add $80, and then add $24.50.
    ::从451.25美元开始,减235美元,再加80美元,再加24.50美元。

    451.25 − 235 + 80 + 24.50 = 320.75

    Anne had $320.75 in her account at the beginning of the day on Friday.
    ::安妮在周五那天的第一天 在她的账户里有320.75美元

    Step 4: Check
    ::第4步:检查

    Anne starts with     $ 320.75 She writes a check for   $ 24.50. $ 320.75 − 24.50 = $ 296.25 She withdraws   $ 80.     $ 296.25 − 80 = $ 216.25 She deposits   $ 235. $ 216.25 + 235 = $ 451.25

    ::Anne以320.75美元开始,她写了一张24.50美元的支票,320.75-24.50美元=296.25美元,她退款80美元。 296.25美元-80美元=216.25美元,她存款235.216.25美元+235美元=451.25美元。

    The answer checks out.
    ::答案检查出来。

    Plan and Compare Alternative Approaches to Solving Problems
    ::计划和比较解决问题的替代办法

    Most word problems can be solved in more than one way. Often one method is more straightforward than others, but which method is best can depend on what kind of problem you are facing.
    ::多数单词问题可以不止一种方式解决。 一种方法通常比其他方法简单,但哪种方法最好取决于你面临的问题是什么。

    Comparing Strategies 
    ::比较战略

    Nadia’s father is 36. He is 16 years older than four times Nadia’s age. How old is Nadia?
    ::Nadia的父亲36岁,他16岁,比Nadia大四倍,Nadia多大?

    Solution
    ::解决方案

    This problem can be solved with either of the strategies you learned in this section. Let’s solve it using both strategies.
    ::这个问题可以通过你在本节所学的任何一种策略来解决。 让我们用两种策略来解决这个问题。

    Guess and Check Method
    ::猜测和检查方法

    Step 1: Understand
    ::第1步:理解

    We need to find Nadia’s age.
    ::我们需要找到Nadia的年龄。

    We know that her father is 16 years older than four times her age, or 4 × ( Nadia's age ) + 16 .
    ::我们知道她的父亲年龄比她大4倍,大16岁,或者4x(Nadia的年龄)+16岁。

    We know her father is 36 years old.
    ::我们知道她父亲36岁

    Step 2: Strategy
    ::步骤2:战略

    We guess a random number for Nadia’s age.
    ::我们猜想Nadia年龄的随机数字。

    We multiply the number by 4 and add 16 and check to see if the result equals 36.
    ::我们将数字乘以4,再加16,然后检查结果是否等于36。

    If the answer is too small, we guess a larger number, and if the answer is too big, we guess a smaller number.
    ::如果答案太小,我们猜数字会更大, 如果答案太大,我们猜数字会较小。

    We keep guessing until we get the answer to be 36.
    ::我们一直在猜测 直到我们得到的答案是36。

    Step 3: Apply strategy/Solve
    ::步骤3:应用战略/单元

    Guess Nadia's age 10 4 ( 10 ) + 16 = 56 too big for her father's age Guess a smaller number 9   4 ( 9 ) + 16 = 52 still too big

    ::猜猜 Nadia 的年龄 104(10)+16=56 对她父亲的年龄来说太大了

    Guessing 9 for Nadia’s age gave us a number that is 16 years too great to be her father’s age. But notice that when we decreased Nadia’s age by one, her father’s age decreased by four. That suggests that we can decrease our final answer by 16 years if we decrease our guess by 4 years.
    ::纳迪亚的9岁估计给了我们一个16岁太高而不能成为她父亲年龄的数字。 但请注意,当我们把纳迪亚的年龄降低1岁时,她父亲的年龄降低了4岁。 这意味着如果我们把猜测降低4岁,我们就能把最后答案降低16岁。

    4 years less than 9 is 5. 4 ( 5 ) + 16 = 36 , which is the right age.
    ::9岁以下4岁是5岁,4岁(5)+16=36岁,这是正确的年龄。

    Answer: Nadia is 5 years old.
    ::回答:Nadia5岁。

    Step 4: Check
    ::第4步:检查

    Nadia is 5 years old. Her father’s age is 4 ( 5 ) + 16 = 36 . This is correct. The answer checks out.
    ::Nadia 5岁,父亲年龄为4(5)+16=36。这是对的,答案是肯定的。

    You will be asked to try the Work Backward method in the Guided Practice section.
    ::将要求您在“指导实践”部分尝试“后向工作”方法。

    Example
    ::示例示例示例示例

    Example 1
    ::例1

    Nadia’s father is 36. He is 16 years older than four times Nadia’s age. Determine Nadia's age by using the Work Backward method.
    ::Nadia的父亲是36岁,他16岁,比Nadia大4倍。 使用“后向工作”方法确定Nadia的年龄。

    Work Backward Method
    ::向后工作方法

    Step 1: Understand
    ::第1步:理解

    We need to find Nadia’s age.
    ::我们需要找到Nadia的年龄。

    We know her father is 16 years older than four times her age, or 4 × Nadia's age + 16 .
    ::我们知道她父亲大16岁,比她大4倍,或4xNadia的16岁。

    We know her father is 36 years old.
    ::我们知道她父亲36岁

    Step 2: Strategy
    ::步骤2:战略

    To get from Nadia’s age to her father’s age, we multiply Nadia’s age by four and add 16.
    ::为了从Nadia的年龄到她父亲的年龄,我们把Nadia的年龄乘以4,再加16。

    Working backwards means we start with the father’s age, subtract 16 and divide by 4.
    ::反向工作意味着我们从父亲的年龄开始,减去16岁,除以4。

    Step 3: Apply Strategy/Solve
    ::步骤3:应用战略/单元

    Start with the father's age 36 Subtract   16   36 − 16 = 20 Divide by   4   20 ÷ 4 = 5

    ::从父亲的年龄开始36 16 分数 36 - 16= 20Divide 4 204=5

    Answer Nadia is 5 years old.
    ::回答Nadia是5岁。

    Step 4: Check
    ::第4步:检查

    Nadia is 5 years old. Her father’s age is 4 ( 5 ) + 16 = 36 . This is correct. The answer checks out.
    ::Nadia 5岁,父亲年龄为4(5)+16=36。这是对的,答案是肯定的。

    You see that in this problem, the “Work Backward” strategy is more straightforward than the Guess and Check method. The Work Backward method always works best when we know the result of a series of operations, but not the starting number. In the next chapter, you will learn algebra methods based on the Work Backward method.
    ::在这个问题中, “ 向后工作” 战略比 Guess and Check 方法更直截了当。 当我们知道一系列操作的结果时, 向后工作方法总是效果最好, 而不是起始数字。 在下一章中, 您将学习基于“ 向后工作” 方法的代数方法 。

    Review 
    ::回顾

    1. Nadia is at home and Peter is at school which is 6 miles away from home. They start traveling towards each other at the same time. Nadia is walking at 3.5 miles per hour and Peter is skateboarding at 6 miles per hour. When will they meet and how far from home is their meeting place?
      ::Nadia在家,Peter在离家6英里远的学校上学,他们同时开始彼此往来。Nadia每小时步行3.5英里,彼得每小时滑板6英里。他们何时会聚,会址离家多远?
    2. Peter bought several notebooks at Staples for $2.25 each; then he bought a few more notebooks at Rite-Aid for $2 each. He spent the same amount of money in both places and he bought 17 notebooks in all. How many notebooks did Peter buy in each store?
      ::彼得在Staples购买了几本笔记本,每本2.25美元;然后他又在Rite-Aid购买了几本笔记本,每本2美元。他在两个地方都花了同样的钱,总共买了17本笔记本。 彼得在每家商店都买了多少笔记本?
    3. Andrew took a handful of change out of his pocket and noticed that he was only holding dimes and quarters in his hand. He counted and found that he had 22 coins that amounted to $4. How many quarters and how many dimes does Andrew have?
      ::安德鲁从口袋里拿出了几块零钱,注意到他手里只有零钱和零钱。他计算并发现他有22个硬币,价值为4美元。安德鲁有多少个季度和多少个硬币?
    4. Anne wants to put a fence around her rose bed that is one and a half times as long as it is wide. She uses 50 feet of fencing. What are the dimensions of the garden?
      ::Anne想在她的玫瑰床四周加一道栅栏,这是宽度的1.5倍。她用50英尺的栅栏。花园的尺寸是多少?
    5. Peter is outside looking at the pigs and chickens in the yard. Nadia is indoors and cannot see the animals. Peter gives her a puzzle. He tells her that he can see 13 heads and 36 feet and asks her how many pigs and how many chickens are in the yard. Help Nadia find the answer.
      ::Peter在外面看着院子里的猪和鸡。 Nadia在室内,看不到动物。 Peter给了她一个谜题。 他告诉她,他可以看到13头和36英尺。 他问她院子里有多少猪和多少只鸡。 帮助Nadia找到答案。
    6. Andrew invests $8000 in two types of accounts: a savings account that pays 5.25% interest per year and a more risky account that pays 9% interest per year. At the end of the year he has $450 in interest from the two accounts. Find the amount of money invested in each account.
      ::安德鲁在两类账户中投资8000美元:一个储蓄账户,每年支付5.25%的利息,另一个风险账户,每年支付9%的利息。 年底,他有两个账户的450美元利息。 找到每个账户的投资金额。
    7. 450 tickets are sold for a concert: balcony seats for $35 each and orchestra seats for $25 each. If the total box office take is $13,000, how many of each kind of ticket were sold?
      ::450张票被卖为音乐会:每张35美元的阳台座位和每张25美元的管弦乐队座位。 如果拳击办公室的总票价是13 000美元,那么每张票的售价是多少?
    8. There is a bowl of candy sitting on our kitchen table. One morning Nadia takes one-sixth of the candy. Later that morning Peter takes one-fourth of the candy that’s left. That afternoon, Andrew takes one-fifth of what’s left in the bowl and finally Anne takes one-third of what is left in the bowl. If there are 16 candies left in the bowl at the end of the day, how much candy was there at the beginning of the day?
      ::我们厨房桌上有一大碗糖果。 一天早上,娜迪雅拿走了六分之一的糖果。 当天早上,彼得拿走了剩下四分之一的糖果。 当天下午,安德鲁拿走了碗里剩下五分之一的糖果,最后安妮拿走了碗里剩下三分之一的糖果。 如果一天结束时碗里还剩下16只糖果,那么一天开始时有多少糖果?
    9. Nadia can completely mow the lawn by herself in 30 minutes. Peter can completely mow the lawn by himself in 45 minutes. How long does it take both of them to mow the lawn together?
      ::彼得可以在45分钟内独自修剪草坪,他们需要多长时间才能一起修剪草坪?
    10. Three monkeys spend a day gathering coconuts together. When they have finished, they are very tired and fall asleep. The following morning, the first monkey wakes up. Not wishing to disturb his friends, he decides to divide the coconuts into three equal piles. There is one left over, so he throws this odd one away, helps himself to his share, and goes home. A few minutes later, the second monkey awakes. Not realizing that the first has already gone, he too divides the coconuts into three equal heaps. He finds one left over, throws the odd one away, helps himself to his fair share, and goes home. In the morning, the third monkey wakes to find that he is alone. He spots the two discarded coconuts, and puts them with the pile, giving him a total of twelve coconuts.
      1. How many coconuts did the first two monkeys take?
        ::前两只猴子吃了多少椰子?
      2. How many coconuts did the monkeys gather in all?
        ::猴子们聚集了多少椰子?

      ::三只猴子在一天的时间内聚集椰子。当他们完成椰子的时候,他们非常疲倦,而且睡着。第二天早晨,第一只猴子醒来。他不想打扰他的朋友们,但他决定把椰子分成三个相等的堆。剩下一个,他把这个奇怪的堆子扔了出去,他把这个奇怪的堆子放在一边,他把自己的份里,然后回家。几分钟后,第二只猴子醒了。不知道第一个已经过去,他就把椰子分成三块。他发现一个剩下的,把那个奇怪的堆子扔掉,把他的份子扔掉,然后回家。早上,第三个猴子醒来发现自己是孤独的。他把两个被抛弃的椰子放在一块上,给他一个12个椰子。头两个猴子吃了多少椰子?猴子聚集了多少椰子?
    11. Two prime numbers have a product of 51. What are the numbers?
      ::两个质数是51的产物 数字是多少?
    12. Two prime numbers have a product of 65. What are the numbers?
      ::两个质数是65的产物 数字是多少?
    13. The square of a certain positive number is eight more than twice the number. What is the number?
      ::某个正数的平方是数字的8倍以上。 数字是多少?
    14. Is 91 prime? (Hint: if it’s not prime, what are its prime factors?)
      :sad提示:如果它不是最佳的,那么其主要因素是什么? )
    15. Is 73 prime?
      ::73元是黄金吗?
    16. Alison’s school day starts at 8:30, but today Alison wants to arrive ten minutes early to discuss an assignment with her English teacher. If she is also giving her friend Sherice a ride to school, and it takes her 12 minutes to get to Sherice’s house and another 15 minutes to get to school from there, at what time does Alison need to leave her house?
      ::艾莉森的上学日从8:30开始,但今天艾莉森想提前10分钟和她英语老师讨论一项任务。 如果她也载她的朋友谢里克去学校,她需要12分钟才能到谢里克家,还要15分钟才能到学校,那么艾莉森什么时候才能离开家?
    17. At her retail job, Kelly gets a raise of 10% every six months. After her third raise, she now makes $13.31 per hour. How much did she make when she first started out?
      ::Kelly在零售业每半年加薪10%。第三期加薪后,她现在每小时挣13.31美元。当她刚起步时,她赚了多少钱?
    18. Three years ago, Kevin’s little sister Becky had her fifth birthday. If Kevin was eight when Becky was born, how old is he now?
      ::三年前,凯文的小妹妹贝基有五岁生日。 如果说贝基出生时凯文八岁,那么他现在几岁了?
    19. A warehouse is full of shipping crates; half of them are headed for Boston and the other half for Philadelphia. A truck arrives to pick up 20 of the Boston-bound crates, and then another truck carries away one third of the Philadelphia-bound crates. An hour later, half of the remaining crates are moved onto the loading dock outside. If there are 40 crates left in the warehouse, how many were there originally?
      ::仓库里装满了货箱;其中一半运往波士顿,另一半前往费城。一辆卡车到达时将20个运往波士顿的货箱运走,另一辆卡车将三分之一运往费城的货箱运走。一小时后,一半剩下的货箱搬到外面的装货码头。如果仓库里还有40个箱子,那么原来有多少箱?
    20. Gerald is a bus driver who takes over from another bus driver one day in the middle of his route. He doesn’t pay attention to how many passengers are on the bus when he starts driving, but he does notice that three passengers get off at the next stop, a total of eight more get on at the next three stops, two get on and four get off at the next stop, and at the stop after that, a third of the passengers get off.
      1. If there are now 14 passengers on the bus, how many were there when Gerald first took over the route?
        ::如果现在巴士上有14名乘客 杰拉尔德刚接管路线时 有多少乘客?
      2. If half the passengers who got on while Gerald was driving paid the full adult fare of $1.50, and the other half were students or seniors who paid a discounted fare of $1.00, how much cash was in the bus’s fare box at the beginning of Gerald’s shift if there is now $73.50 in it?
        ::如果在Gerald开车时乘坐的乘客有一半付了成人全额1.50美元,而另一半是学生或老年人付了折扣价1.0美元,那么在Gerald轮班开始时,公车票箱里有多少现金,如果现在有73.50美元的话?
      3. When Gerald took over the route, all the passengers currently on the bus had paid full fare. However, some of the passengers who had previously gotten on and off the bus were students or seniors who had paid the discounted fare. Based on the amount of money that was in the cash box, if 28 passengers had gotten on the bus and gotten off before Gerald arrived (in addition to the passengers who had gotten on and were still there when he arrived), how many of those passengers paid the discounted fare?
        ::当Gerald接管路线时,目前乘坐公共汽车的所有乘客都付了全额票价,然而,以前往返公共汽车的乘客中有些是支付折扣票价的学生或老年人,根据现金箱中的钱数,如果28名乘客上了公共汽车并在Gerald到达之前下车(除了上过车并在到达时仍然在车位上的乘客之外),有多少乘客支付了折扣票价?
      4. How much money would currently be in the cash box if all the passengers throughout the day had paid the full fare?  
        ::如果每天所有乘客都付满票价,现金箱中目前将有多少钱?

      ::Gerald是一位公共汽车司机,他一天从另一辆公共汽车司机那里接过来,一天在他行驶的途中从另一辆公共汽车司机那里接过来。他不注意当他开始开车时车上有多少乘客。他不注意在公共汽车上有多少乘客,但他注意到有三名乘客在下一站下车,在下一站总共有8名乘客下车,在接下来的3站总共有8名乘客下车,2名上车,4名下车,在下一站,之后有三分之一的乘客下车。如果公共汽车上现在有14名乘客,在杰拉尔德刚上下班时有多少乘客?如果在杰拉尔德开车时有一半乘客在车上坐满150美元,而另一半是学生或老年人在汽车上付满成人1美元票,而另一半是学生或高年级学生或高年级学生在下一站下车,在杰拉尔德上班开始时有多少现金?当杰拉尔德走过这条路时,目前所有乘客都付满票价时,有些乘客是学生或高年级学生,他们付了折扣票价。根据乘客的金额计算,现在多少是多少现金,谁到满票?

    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 。

    Texas Instruments Resources
    ::得克萨斯州工具资源

    In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See .
    ::在CK-12得克萨斯州仪器代数I FlexBook资源中,有图表计算活动,旨在补充本章某些经验教训的目标。