15.1: PROBATION
CHAPTER 15 - PROBATION AND RESTORATIVE JUSTICE
PROBATION [144]
Probation is arguably the oldest, and certainly the largest, of intermediate sanctions. Its roots stem from concepts of common law from England, like many of our other legal/correctional practices. In early American courts, a person was able to be released on their own recognizance, if they promised to be responsible citizens and pay back what they owed. In the early 1840s, John Agustus, a Boston bootmaker was regularly attending court and began to supervise these individuals as a Surety. A Surety was a person who would help these individuals in court, making sure they repaid these costs to the courts. We would consider John Augustus as the Father of Probation, for this work in the courts in Boston in the 1840s and 1850s. Augustus, pictured below, would take in many of these individuals, providing options like work and housing, to help ensure these individuals would remain crime free and pay back society. He continued this practice for nearly two decades, effectively becoming the first probation officer. For a lengthier historical discussion of probation, see the history of probation at https://probation.smcgov.org/history-probation .
Probation is a form of a suspended sentence, in that the jail or prison sentence of the convicted offender is resuspended, for the privilege of serving conditions of supervision in the community. Conditions of probation often include reporting to a probation officer, submitting random drug screens, not consorting with known felons, paying court costs, making restitution and damages, and attending AA or NA courses, among other conditions. Probation lengths vary greatly, as do the conditions of probation place on an individual. Almost all people on probation will have at least one condition of probation. Some have many conditions, depending on the seriousness of the conviction, while others are just a blanket condition that is imposed on all in that jurisdiction, or for that conviction type. Juvenile Probation Departments were within all States in the 1920s, and by the middle of the 1950s, all States had adult probation.
PROBATION OFFICERS
Probation officers usually work directly for the state or federal government, but that can be directed through local or municipal agencies. Many Counties will have a community justice level structure where probation offices operate. Within these offices, probation officers will be assigned cases (caseload), in term of the probationers, they will manage. The volume of cases in a probation officer’s caseload can vary from just a few clients if they are high need/risk clients, to several hundred probationers. It depends on the jurisdiction, the structure of the local PO office, and the abilities of the probation officers themselves.
The role of the probation officer is complex, and sometimes diametrically opposing. A PO’s primary function is to enforce compliance of individuals on probation. This is done through check-ins, random drug screenings, and enforcement of other conditions that are placed on the probationers. Additionally, the PO may go out into the field to serve warrants, do home checks for compliance, even make arrests if need be.
However, at the same time, a probation officer is trying to help individuals on probation succeed. This is done by trying to help individuals to get jobs, get schooling, enter into substance or alcohol programs, and generally support people on probation to be successful. This is why the job of the PO is complex, as they are trying to be supportive, but also having to enforce compliance. Many equate this to kind of like being a parent. Recently, there has been a movement within probation to have probation officers act more like coaches than just disciplinarians. Here is a talk about how POs can view themselves as coaches to enact positive change within individuals on probation:
The other primary function of a probation officer is to complete PSI reports on individuals going through the court process. A PSI or Pre-Sentence Investigation report is a psycho-social workup on a person headed to trial. It includes basic background information on an individual, such as age, education, relationships, physical and mental health, employment, military service, social history, and substance abuse history. It also has a detailed account of the current offense, witness or victim impact statements of the event, and prior offenses (criminal records), which are tracked across numerous agencies. Finally, the PSI also has a section that is devoted to a plan of supervision or recommendations, which are created by the PO. These usually list out the conditions of probation recommendations, if probation is to be granted. Judges use this information during sentencing discussions and hearings and will usually follow these recommendations often (around 85% of the time). Thus, many of the conditions of probation are prescribed by the PO.
INDIVIDUALS ON PROBATION
As stated, there are several million people on probation, serving various lengths of probation, and under numerous conditions or condition types. Additionally, the convictions which place individuals on probation vary, to include misdemeanors and felonies. Probationers serve their probation at the state level, and there is even federal probation. As depicted below, it is easy to see how much probation is used in the United States.
USE OF PROBATION IN THE U.S.
Probation is not a right, and it is not a suspended sentence. It is a privilege, but it most certainly comes with conditions for the suspension of the incarceration. Due to how cheap probation is, relative to jail or prison, and the ability for lower-risk individuals to maintain connections within their community, millions of people will be on probation in the United States at any given time.
Other important factors that help to decide if a person warrants probation is within the PSI and other assessments are done on the individual. If the person is basically, a prosocial person, has an education and a job, has a family, these would all be considered as ties to the community. These ties to the community could weaken or break if a person was incarcerated. Thus, providing a sanction while allowing the person to stay in the community is often the approach that is utilized within probation and other intermediate sanctions.
PROBATION SUCCESS
There are mixed reviews about probation. Recently, the Bureau of Justice Statistics (2018 report, for 2016) listed the successful completion rate at about 56%. In years past, this number has been reported higher, upwards of 65%, depending on the years 2008-2013. There are a host of reasons listed for unsuccessful completion, which include: incarcerated on a new sentence/charge, or placement for the current sentence/charge, absconding (fleeing jurisdiction), discharged to warrant or detainer, other unsatisfactory reason, death, or some other unknown or not reported reason. Unsuccessful completion can produce some different responses but can include a concept called tourniquet sentencing. Tourniquet sentencing is where the restrictions of a level of sanction are increased, due to non-compliance, in order to force compliance. If an individual on probation is not adhering to the conditions of probation, a PO can recommend a probation revocation hearing. This bench hearing can lead to an informal admonishment by a judge, an increase in the sanctions or sanction lengths, an increased level of control (moving from regular probation to intensive supervised probation), even up to placement in a secure facility (jail or prison), all depending on the infraction of the condition of probation that has been violated. Many go from regular probation to ISP, in an effort to force compliance through increased monitoring.
INTENSIVE SUPERVISED PROBATION
Intensive Supervision Probation (ISP) began in the late 1950s, early 1960s, in California. Their basic premise was to allow caseworkers (POs) to have smaller caseloads and increase the level of treatment across offenders. As stated, many promised multiple success measures. However, if an individual who was revoked because of a technical violation due to an increase in control, they were not seen as a failure. Rather, they were seen as a success because of the way the public was served by the recidivism. However, this went directly against the notion that ISPs could save money. Because of these problems, the earlier forms of ISPs may have become less popular. In the 1980s, a newer model of the ISP was created in Georgia. More emphasis was placed on the control aspect rather than on treatment. Further, less emphasis was placed on the reduction of money saved.
ISP and regular probation are similar, except for the frequency of contacts with POs, the increases in surveillance and monitoring, and usually the volume of conditions. Rather than meeting a PO once a month in regular probation, a person on ISP would likely be meeting with their PO weekly, or even more frequently. Additionally, individuals on ISP normally are submitting drug screens weekly. The increased conditions of supervision more frequently include more substance abuse treatment, either in the form of AA, NA, or some other residential or outpatient substance abuse treatment programs. Thus, the core difference is about the increased level of surveillance and control over the offender.
ISP SUCCESS
While initial praise of the newer model for its increase on control was evidenced by its rapid spread through the States, some researchers questioned their effectiveness. In one of the largest studies of ISPs, in conjunction with the RAND Corporation. They examined the effectiveness of ISPs in reducing recidivism and saving costs. In a random sample of 14 cities across 9 States, they evaluated the reductions of recidivism against a sample of regular probationers. Their findings suggested that there were higher amounts of technical violations, which were probably substance violations, but there were no significant differences between control-centered ISPs and regular probation, as far as new arrests. Moreover, when looking at outcomes over 3 years, they found that recidivism rates were slightly higher for these ISPs (39%), vs. regular probation (33%). Also, there were no substantive cost savings. Other studies have produced similar findings as to the effects of non-treatment oriented ISPs. While these findings might be better than prison recidivism rates, there were no reductions in prison overcrowding, which was also one of the intents of ISP.
ELEMENTS OF PAROLE [147]
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Learning Objective: Key Terms |
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For most of the history of probation and parole in the United States, offenders were viewed as having received a gift from the state when they were not sent to prison. Because being on probation or parole was viewed as a privilege conferred by the state, most states believed that they were under no obligation to provide probationers and parolees with the elements of due process they were afforded prior to conviction. In today’s legal landscape, the Supreme Court has intervened and now probationers and parolees enjoy some, but not all, of the protections afforded by the Constitution. Note that most of the Supreme Court decisions regarding the rights of probationers and parolees blur the distinction. That is, most of the Court’s rulings on probation issues apply to parole as well, and vice versa.
REVOCATION OF PAROLE
Implicit in the criminal justice system’s concern with parole violations is the idea that individuals on parole are entitled to retain their liberty as long as they largely abide by the conditions of parole (or probation). When parolees do fail to live up to these standards, their parole can be revoked. The first step in the parole revocation process involves answering a factual question: whether the parolee has in fact acted in violation of one or more conditions of his or her parole. Only if it is determined that the parolee did violate the conditions does the second question arise: should the parolee be recommitted to prison or should other steps be taken to protect society and improve chances of rehabilitation?
The second question involves the application of expertise by the parole authority in making a prediction as to the ability of the individual to live in society without committing antisocial acts. This part of the decision, too, depends on facts, and therefore it is important for the parole board to know not only that some violation was committed but also to know accurately how many and how serious the violations were. Yet this second step, deciding what to do about the violation once it is identified, is not purely factual but also predictive and discretionary.
Parole revocation is very serious for the offender. If a parolee is returned to prison, he or she usually receives no credit for the time “served” on parole. Thus, the violator may face a potential of substantial imprisonment. Revocation deprives an individual, not of the absolute liberty to which every citizen is entitled, but only of the conditional liberty properly dependent on observance of special parole restrictions. This means that the legal standards for parole revocation are not the same as a finding of guilt in criminal court.
DUE PROCESS
The liberty of a parolee, although indeterminate, includes many of the core values of unqualified liberty and its termination inflicts a “grievous loss” on the parolee and often on others. Historically, it was common for judges to speak of this problem in terms of whether the parolee’s liberty was a “right” or a “privilege.” By whatever name, the Supreme Court has determined that liberty is valuable and must be seen as within the protection of the Fourteenth Amendment. Because of this, the courts have determined that its termination calls for some orderly process, however informal.
In Morrissey v. Brewer (1972), the Supreme Court refused to write a code of procedure for parole revocation hearings; that, they said, is the responsibility of each State. In this case, the court pointed out that most States have set out procedures by legislation. The Supreme Court did establish a list of minimum due process requirements that must be followed in all revocation proceedings. They include (a) written notice of the claimed violations of parole; (b) disclosure to the parolee of evidence against him; (c) opportunity to be heard in person and to present witnesses and documentary evidence; (d) the right to confront and cross-examine adverse witnesses (unless the hearing officer specifically finds good cause for not allowing confrontation); (e) a “neutral and detached” hearing body such as a traditional parole board, members of which need not be judicial officers or lawyers; and (f) a written statement by the factfinders as to the evidence relied on and reasons for revoking parole.
Specifically, then, Morrissey held that a parolee is entitled to two hearings, one a preliminary hearing at the time of his arrest and detention to determine whether there is probable cause to believe that he has committed a violation of his parole, and the other a somewhat more comprehensive hearing prior to the making of the final revocation decision.
In Gagnon v. Scarpelli (1973), the court considered the problem of probation revocation hearings. In Scarpelli, the court stated:
Petitioner does not contend that there is any difference relevant to the guarantee of due process between the revocation of parole and the revocation of probation, nor do we perceive one. Probation revocation, like parole revocation, is not a stage of a criminal prosecution, but does result in a loss of liberty. Accordingly, we hold that a probationer, like a parolee, is entitled to a preliminary and a final revocation hearing, under the conditions specified in Morrissey v. Brewer.
In Mempa v. Rhay (1967), the Court held that a probationer is entitled to be represented by appointed counsel at a combined revocation and sentencing hearing. Reasoning that counsel is required “at every stage of a criminal proceeding where substantial rights of a criminal accused may be affected.”
THE FOURTH AMENDMENT
As with due process rights, a person’s Fourth Amendment rights are not nullified just because they are convicted of a crime. What makes probationers and parolees different than the average citizen are their conditions of release. Most states require parolees to give up their right to be free from unreasonable searches as part of their conditions. Because the parolee is giving up Fourth Amendment rights, this element is often referred to as a Fourth waiver . The rules that govern officer conduct vary from state to state. In some states, an officer must have reasonable suspicion before conducting a probation search. In many states, an officer can conduct a suspicionless search at any time, without reason to believe that the offender committed a new crime. Who may search also varies from jurisdiction to jurisdiction? Some jurisdictions only allow probation and parole officers to search without probable cause, and some extend this authority to police officers as well.
CONDITIONS OF PROBATION AND PAROLE
As previously discussed, offenders are only granted probation or parole if they agree to abide by certain, specified conditions. These can be general conditions that apply to all offenders released in a particular jurisdiction, or they can be tailored to the special needs of a particular offender. The intent of these conditions is to help ensure that the dual objectives of control and rehabilitation are met. Because of the fragmented nature of courts in the United States, there is a great deal of variability in the philosophy and practice of imposing these conditions.
The power to impose conditions of probation and parole is most often vested in the courts. Judges have immense discretion when it comes to choosing conditions. Most courts rely on community corrections officers to make suggestions, but the final say is up to the judge. This wide discretion is not, however, without bounds.
CLARITY
Recall the void for vagueness doctrine discussed in the criminal law chapter. The basis of this legal limit on the power of lawmakers is that it is fundamentally unfair when a reasonable person cannot figure out what exactly a law prohibits. The courts have viewed conditions of probation in the same light. In other words, if the offender cannot figure out what exactly is prohibited because the specification of the condition is too vague, then the condition is unconstitutional. In practice, this means that conditions of probation can vary widely in subject, purpose, and scope, but what is prohibited (or mandated) must be specified in such a way that there is no confusion as to what is required. Conditions that are crafted in vague terms such as “must live honorably” will be struck down by the courts.
REASONABLENESS
In the context of probation and parole conditions, the term reasonableness is often synonymous with realistic. The basic requirement is that the conditions set forth by the judge must be such that the offender has the ability to abide by them. If the offender is likely to fail because the conditions cannot possibly be complied with, then the condition will be deemed not reasonable by the courts. It would be unreasonable, for example, to order an indigent offender to pay $10,000 a month in restitution. Addicts have argued that it is unreasonable to expect them to refrain from drug and alcohol use because of the nature of addiction. These claims fail the vast majority of the time. Various courts have reasoned that drug use is illegal, and illegal behavior by probationers and parolees cannot be tolerated.
RELATED TO PROTECTION AND REHABILITATION
Since the major goals of probation and parole are to protect society from crime and to rehabilitate the offender, conditions of probation and parole must be reasonably related to one or both of these objectives. If a condition does not relate to these objectives, it will likely be struck down by the courts. In practice, this gives judges very wide latitude in selecting conditions that may be related to these goals. Many courts have struck down conditions of probation that were obviously intended to be “scarlet letter” punishments.
CONSTITUTIONALITY
Several courts have nullified conditions that were contrary to constitutionally protected actions. When constitutional rights are at stake, the government will usually have to establish a compelling state interest in violating the right. In other words, the appellate court will balance the interest the state has in curtailing the right with the cost to the offender. Some rights are afforded greater protection by the court than other rights. These special liberties are often referred to as fundamental rights . The freedom of the press, freedom of assembly, freedom of speech, and freedom of religion are among these fundamental rights. For example, courts have struck down conditions that required and offender to attend Sunday school on a regular basis. The court reasoned that forcing someone to participate in a church activity violated the offender’s freedom of religion. As previously discussed, Fourth Amendment rights are not nearly so well protected.
LEGAL RIGHTS
GRIFFIN V. WISCONSIN [148]
In Griffin v. Wisconsin, the court found a warrant or probable cause is not required to search the home of a person on probation provided “reasonable grounds” for a search exists. The court explained, “A State's operation of a probation system, like its operation of a school, government office or prison, or its supervision of a regulated industry, . . . presents 'special needs' beyond normal law enforcement that may justify departures from the usual warrant and probable cause requirements” “Probation, like incarceration, is a form of criminal sanction”. The Court also noted a requirement of a warrant or probable cause would interfere with the proper functioning of the ongoing [non- adversarial] supervisory relationship.
REVOCATION RIGHTS
An offer gives an offeree the power to form a contract by accepting. The Restatement (Second) of Contracts describes a number of ways that the offeree’s power to accept may end:
Methods of Termination of the Power of Acceptance [149]
- An offeree’s power of acceptance may be terminated by
- rejection or counteroffer by the offeree, or
- lapse of time, or
- revocation by the offeror, or
- death or incapacity of the offeror or offeree.
- In addition, an offeree’s power of acceptance is terminated by the non-occurrence of any condition of acceptance under the terms of the offer.
We will discuss both the common law and UCC rules governing rejection and counteroffers in the next section. For the moment, note that an offer ordinarily remains open long enough to give the offeror a reasonable opportunity to accept. An oral offer made during a face-to-face or telephone conversation expires at the end of that conversation unless the offeror has indicates a willingness to keep the offer open beyond that time. The offeror nevertheless retains the right to terminate her offer at any subsequent time unless she has also expressly agreed not to revoke it—thus creating a “firm offer.”
Recall that in order to accept an offer of a unilateral contract an offeree must tender a performance rather than a reciprocal promise. The consequences of a revocation are especially acute when an offeror revokes such an offer after the offeree has begun performing. In the following excerpt, a scholar defends the early common law rule, which required full performance for acceptance:
Suppose A says to B, “I will give you $100 if you walk across the Brooklyn Bridge,” and B walks — is there a contract? It is clear that A is not asking B for B’s promise to walk across the Brooklyn Bridge. What A wants from B is the act of walking across the bridge. When B has walked across the bridge there is a contract, and A is then bound to pay to B $100. At that moment there arises a unilateral contract. A has bartered away his volition for B’s act of walking across the Brooklyn Bridge.
When an act is thus wanted in return for a promise, a unilateral contract is created when the act is done. It is clear that only one party is bound. B is not bound to walk across the Brooklyn Bridge, but A is bound to pay B $100 if B does so. Thus, in unilateral contracts, on one side we find merely an act, on the other side a promise.
It is plain that in the Brooklyn Bridge case as first put, what A wants from B is the act of walking across the Brooklyn Bridge. A does not ask for B’s promise to walk across the bridge and B has never given it. B has never bound himself to walk across the bridge. A, however, has bound himself to pay $100 to B, if B does so. Let us suppose that B starts to walk across the Brooklyn Bridge and has gone about one-half of the way across. At that moment A overtakes B and says to him, “I withdraw my offer.” Has B then any rights against A? Again, let us suppose that after A has said, “I withdraw my offer,” B continues to walk across the Brooklyn Bridge and completes the act of crossing.
Under these circumstances, has B any rights against A? In the first of the cases just suggested, A withdrew his offer before B had walked across the bridge. What A wanted from B, what A asked for, was the act of walking across the bridge. Until that was done, B had not given to A what A had requested. The acceptance by B of A’s offer could be nothing but the act on B’s part of crossing the bridge. It is elementary that an offeror may withdraw his offer until it has been accepted. It follows logically that A is perfectly within his rights in withdrawing his offer before B has accepted it by walking across the bridge — the act contemplated by the offeror and the offeree as the acceptance of the offer.
More recent decisions have rejected this traditional approach. Courts now protect the offeree who has begun performance by barring revocation of the offer until the offeree has had a reasonable opportunity to complete the requested performance. The Restatement (Second) of Contracts sensibly describes the resulting obligation as an option contract.
IRREVOCABLE OFFERS
The rule for unilateral contracts described in Restatement (Second) § 45 creates an implied option contract once an offeree has begun performing and gives her a reasonable time to complete performance. In other circumstances, however, parties may prefer to create an express option contract.
INTERMEDIATE SANCTIONS [150]
Community corrections as a whole has changed dramatically over the last half-century. Due to a rapid and overwhelming increase of the offender population, largely based on policy changes, we have witnessed an immense increase in the use of sanctions at the community level; this includes probation. It has only been within the most recent 10 years that we have seen a decrease in community corrections. Individuals on probation hover around 3.7 million, with another million in some form of community-level control, for a total of about 4.6 million under community supervision, probation, or parole. [1] Because of the sheer volume of these intermediate sanctions, it is important to put it in the perspective of jails and prisons. Below is a graphic to demonstrate how much we are talking about.
U.S. CORRECTIONAL CONTROL
Figure 15.3 National correctional control, 2018 [151]
This graphic does not include the volume of people in the community corrections outside of regular probation, or parole (which is about another million), but it does shed light on just how much probation is used. Therefore, it important to understand that the vast majority of individuals under correctional control aren’t even in the prisons and jails in the United States, even though that alone is a hefty number. The majority of persons under correctional control fall are in sanctions like Probation, Intensive Supervision Probation, Boot Camps/Shock Incarceration, Drug Courts, and Halfway Houses. Therefore, this section will discuss the history and effectiveness of some of the forms of intermediate sanctions we use in the United States.
As we have discussed, in the late 1970s and early 1980s, there was a fundamental shift in corrections. This is largely due to the “Nothing Works” dogma in the area of rehabilitation. Many reforms were made towards the housing of offenders. Many liked this idea because they did not trust the government’s attempts at rehabilitation. Others were pleased as well since more emphasis was placed on control. Rooted in deterrence theory, and to lesser extent incapacitation, intermediate sanctions flourished and were seen as an instant success. That is, because they promised to increase control of the growing offender population, maintain security, and do all of this at a reduced cost, they were quickly welcomed across the nation. However, by reviewing each one, we can see the problems that promising too much may have created.
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Think About It… What Would You Do? As stated before, there are three primary goals for corrections, to punish the offender, to protect society, and to rehabilitate the offender. Often the first two goals might be opposing to the last goal. Additionally, doing too much of the first one might have unfavorable on the second and third. Here is an example of how this might happen. Let’s say there is a guy, who is married, has a couple of kids, a stable blue-collar job, a house /mortgage, and is living just a little bit better than a paycheck to paycheck. We can call him the average Joe Citizen. This might even sound familiar, and you may even know him. Joe likes to hang out with his friends after his men’s softball game, having a beer and catching up on life. For all intents and purposes, Joe is a decent guy. He does not have a significant criminal record. Perhaps one misdemeanor when he was a juvenile, and a couple of speeding tickets, like tens of millions of other adults. However, one time after softball, Joe is driving home, and his wife is texting him to pick something up at the store. He looks down at his phone at the exact wrong time that someone pulls out in front of him. An accident occurs. No one is seriously injured, but the damage to both vehicles is enough to warrant a write up of the accident. This leads to police presence. At the scene, the officer smells alcohol on Joe. The officer is obligated to go through standard procedures, which results in Joe being arrested. This is not asking you to debate this action, as it is a violation, and the officer had every right to arrest Joe. The question is this – what should Joe’s punishment be? The reason to ask this is due to both the rule of law and the consequences of those laws. Joe should be punished, as he chose to drive after drinking alcohol. But would Joe’s incarceration lead to other events that may have lasting effects? Probably. This brings up the question of at what point does the level of punishment last beyond its intended point? If Joe receives a lengthy jail sentence, will he lose his job? Will he lose his family? Will this put him a greater risk of recidivating in the future? What point has the immediate action caused punishment beyond what the law stipulates is punishment? These are all valid questions. There are other alternatives out there, that still cover the concepts of punishment, monitoring, sanctioning, and control. However, these alternatives can still allow individuals to stay in the community, which this chapter will present with community corrections. |
RESTORATIVE JUSTICE [152]
The process of restorative justice programs is often linked with community justice organizations and is normally carried out within the community. Therefore, RJ is discussed here in the community corrections section. Restorative justice is a community based and trauma-informed practice used to build relationships, strengthen communities, encourage accountability, repair harm, and restore relationships when wrongdoings occur. As an intervention following wrongdoing, restorative justice works for the people who have caused harm, and the victim(s), and community members impacted. Working with a restorative justice facilitator, participants identify harms, needs, and obligations, then make a plan to repair the harm and put things as right as possible. This process, restorative justice conferencing, can also be called victim-offender dialogues. It is within this process that multiple items can occur. First, the victim can be heard within the scope of both the community and within the scope of the offense discussed. This provides the victim(s) an opportunity to express the impact on them, but also to understand what was happening from the perspective of the transgressor. At the same time, it allows the person committing the action to potentially take responsibility for the acts committed, directly to the victim(s) and to the community as a whole. This restorative process provides a level of healing that is often unique to the RJC. Pictured, the different processes that can occur during the different types of dialogues within RJC.
RESTORATIVE JUSTICE PROCESSES
RESTORATIVE JUSTICE SUCCESS
For over a quarter century, restorative justice has been demonstrated to show positive outcomes in accountability of harm, and satisfaction in the restorative justice process for both offenders and victims. This is true for adult offenders, as well as juveniles, who go through the restorative justice process. Recently, there have been questions whether there is a cognitive change that occurs in the thought process of the individuals completing a restorative justice program. There is a growing body of research that demonstrates that change in cognitive distortions that may occur through successful completion of restorative justice conferencing (RJC). This will be an area of increasing interest for practitioners, as restorative justice continues to be included in the toolkit of actions within community justice and community corrections.


1.12: Chapter 1 Review Exercises
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Chapter Review Exercises
Introduction to Whole Numbers
Use Place Value with Whole Number
In the following exercises find the place value of each digit.
Exercise \(\PageIndex{1}\)
26,915
- 1
- 2
- 9
- 5
- 6
- Answer
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- tens
- ten thousands
- hundreds
- ones
- thousands
Exercise \(\PageIndex{2}\)
359,417
- 9
- 3
- 4
- 7
- 1
Exercise \(\PageIndex{3}\)
58,129,304
- 5
- 0
- 1
- 8
- 2
- Answer
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- ten millions
- tens
- hundred thousands
- millions
- ten thousands
Exercise \(\PageIndex{4}\)
9,430,286,157
- 6
- 4
- 9
- 0
- 5
In the following exercises, name each number.
Exercise \(\PageIndex{5}\)
6,104
- Answer
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six thousand, one hundred four
Exercise \(\PageIndex{6}\)
493,068
Exercise \(\PageIndex{7}\)
3,975,284
- Answer
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three million, nine hundred seventy-five thousand, two hundred eighty-four
Exercise \(\PageIndex{8}\)
85,620,435
In the following exercises, write each number as a whole number using digits.
Exercise \(\PageIndex{9}\)
three hundred fifteen
- Answer
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315
Exercise \(\PageIndex{10}\)
sixty-five thousand, nine hundred twelve
Exercise \(\PageIndex{11}\)
ninety million, four hundred twenty-five thousand, sixteen
- Answer
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90,425,016
Exercise \(\PageIndex{12}\)
one billion, forty-three million, nine hundred twenty-two thousand, three hundred eleven
In the following exercises, round to the indicated place value.
Exercise \(\PageIndex{13}\)
Round to the nearest ten.
- 407
- 8,564
- Answer
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- 410
- 8,560
Exercise \(\PageIndex{14}\)
Round to the nearest hundred.
- 25,846
- 25,864
In the following exercises, round each number to the nearest 1. hundred 2. thousand 3. ten thousand.
Exercise \(\PageIndex{15}\)
864,951
- Answer
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- 865,000865,000
- 865,000865,000
- 860,000
Exercise \(\PageIndex{16}\)
3,972,849
Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
Exercise \(\PageIndex{17}\)
168
- Answer
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by 2,3,6
Exercise \(\PageIndex{18}\)
264
Exercise \(\PageIndex{19}\)
375
- Answer
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by 3,5
Exercise \(\PageIndex{20}\)
750
Exercise \(\PageIndex{21}\)
1430
- Answer
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by 2,5,10
Exercise \(\PageIndex{22}\)
1080
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
Exercise \(\PageIndex{23}\)
420
- Answer
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2\(\cdot 2 \cdot 3 \cdot 5 \cdot 7\)
Exercise \(\PageIndex{24}\)
115
Exercise \(\PageIndex{25}\)
225
- Answer
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3\(\cdot 3 \cdot 5 \cdot 5\)
Exercise \(\PageIndex{26}\)
2475
Exercise \(\PageIndex{27}\)
1560
- Answer
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\(2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 13\)
Exercise \(\PageIndex{28}\)
56
Exercise \(\PageIndex{29}\)
72
- Answer
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\(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\)
Exercise \(\PageIndex{30}\)
168
Exercise \(\PageIndex{31}\)
252
- Answer
-
\(2 \cdot 2 \cdot 3 \cdot 3 \cdot 7\)
Exercise \(\PageIndex{32}\)
391
In the following exercises, find the least common multiple of the following numbers using the multiples method.
Exercise \(\PageIndex{33}\)
6,15
- Answer
-
30
Exercise \(\PageIndex{34}\)
60, 75
In the following exercises, find the least common multiple of the following numbers using the prime factors method.
Exercise \(\PageIndex{35}\)
24, 30
- Answer
-
120
Exercise \(\PageIndex{36}\)
70, 84
Use the Language of Algebra
Use Variables and Algebraic Symbols
In the following exercises, translate the following from algebra to English.
Exercise \(\PageIndex{37}\)
25−7
- Answer
-
25 minus 7, the difference of twenty-five and seven
Exercise \(\PageIndex{38}\)
5\(\cdot 6\)
Exercise \(\PageIndex{39}\)
\(45 \div 5\)
- Answer
-
45 divided by 5, the quotient of forty-five and five
Exercise \(\PageIndex{40}\)
x+8
Exercise \(\PageIndex{41}\)
\(42 \geq 27\)
- Answer
-
forty-two is greater than or equal to twenty-seven
Exercise \(\PageIndex{42}\)
3n=24
Exercise \(\PageIndex{43}\)
\(3 \leq 20 \div 4\)
- Answer
-
3 is less than or equal to 20 divided by 4, three is less than or equal to the quotient of twenty and four
Exercise \(\PageIndex{44}\)
\(a \neq 7 \cdot 4\)
In the following exercises, determine if each is an expression or an equation.
Exercise \(\PageIndex{45}\)
\(6 \cdot 3+5\)
- Answer
-
expression
Exercise \(\PageIndex{46}\)
y−8=32
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
Exercise \(\PageIndex{47}\)
\(3^{5}\)
- Answer
-
243
Exercise \(\PageIndex{48}\)
\(10^{8}\)
In the following exercises, simplify
Exercise \(\PageIndex{49}\)
6+10/2+2
- Answer
-
13
Exercise \(\PageIndex{50}\)
9+12/3+4
Exercise \(\PageIndex{51}\)
\(20 \div(4+6) \cdot 5\)
- Answer
-
10
Exercise \(\PageIndex{52}\)
\(33 \div(3+8) \cdot 2\)
Exercise \(\PageIndex{53}\)
\(4^{2}+5^{2}\)
- Answer
-
41
Exercise \(\PageIndex{54}\)
\((4+5)^{2}\)
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Exercise \(\PageIndex{55}\)
9x+7 when x=3
- Answer
-
34
Exercise \(\PageIndex{56}\)
5x−4 when x=6
Exercise \(\PageIndex{57}\)
\(x^{4}\) when \(x=3\)
- Answer
-
81
Exercise \(\PageIndex{58}\)
\(3^{x}\) when \(x=3\)
Exercise \(\PageIndex{59}\)
\(x^{2}+5 x-8\) when \(x=6\)
- Answer
-
58
Exercise \(\PageIndex{60}\)
\(2 x+4 y-5\) when
\(x=7, y=8\)
Simplify Expressions by Combining Like Terms
In the following exercises, identify the coefficient of each term.
Exercise \(\PageIndex{61}\)
12n
- Answer
-
12
Exercise \(\PageIndex{62}\)
9\(x^{2}\)
In the following exercises, identify the like terms.
Exercise \(\PageIndex{63}\)
\(3 n, n^{2}, 12,12 p^{2}, 3,3 n^{2}\)
- Answer
-
12 and \(3, n^{2}\) and 3\(n^{2}\)
Exercise \(\PageIndex{64}\)
\(5,18 r^{2}, 9 s, 9 r, 5 r^{2}, 5 s\)
In the following exercises, identify the terms in each expression.
Exercise \(\PageIndex{65}\)
\(11 x^{2}+3 x+6\)
- Answer
-
\(11 x^{2}, 3 x, 6\)
Exercise \(\PageIndex{66}\)
\(22 y^{3}+y+15\)
In the following exercises, simplify the following expressions by combining like terms.
Exercise \(\PageIndex{67}\)
17a+9a
- Answer
-
26a
Exercise \(\PageIndex{68}\)
18z+9z
Exercise \(\PageIndex{69}\)
9x+3x+8
- Answer
-
12x+8
Exercise \(\PageIndex{70}\)
8a+5a+9
Exercise \(\PageIndex{71}\)
7p+6+5p−4
- Answer
-
12p+2
Exercise \(\PageIndex{72}\)
8x+7+4x−5
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the following phrases into algebraic expressions.
Exercise \(\PageIndex{73}\)
the sum of 8 and 12
- Answer
-
8+12
Exercise \(\PageIndex{74}\)
the sum of 9 and 1
Exercise \(\PageIndex{75}\)
the difference of x and 4
- Answer
-
x−4
Exercise \(\PageIndex{76}\)
the difference of x and 3
Exercise \(\PageIndex{77}\)
the product of 6 and y
- Answer
-
6y
Exercise \(\PageIndex{78}\)
the product of 9 and y
Exercise \(\PageIndex{79}\)
Adele bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let bb represent the cost of the blouse. Write an expression for the cost of the skirt.
- Answer
-
b+15
Exercise \(\PageIndex{80}\)
Marcella has 6 fewer boy cousins than girl cousins. Let g represent the number of girl cousins. Write an expression for the number of boy cousins.
Add and Subtract Integers
Use Negatives and Opposites of Integers
In the following exercises, order each of the following pairs of numbers, using < or >.
Exercise \(\PageIndex{81}\)
- 6___2
- −7___4
- −9___−1
- 9___−3
- Answer
-
- >
- <
- <
- >
Exercise \(\PageIndex{82}\)
- −5___1
- −4___−9
- 6___10
- 3___−8
In the following exercises,, find the opposite of each number.
Exercise \(\PageIndex{83}\)
- −8
- 1
- Answer
-
- 8
- −1
Exercise \(\PageIndex{84}\)
- −2
- 6
In the following exercises, simplify.
Exercise \(\PageIndex{85}\)
−(−19)
- Answer
-
19
Exercise \(\PageIndex{86}\)
−(−53)
In the following exercises, simplify.
Exercise \(\PageIndex{87}\)
−m when
- m=3
- m=−3
- Answer
-
- −3
- 3
Exercise \(\PageIndex{88}\)
−p when
- p=6
- p=−6
Simplify Expressions with Absolute Value
In the following exercises,, simplify.
Exercise \(\PageIndex{89}\)
- |7|
- |−25|
- |0|
- Answer
-
- 7
- 25
- 0
Exercise \(\PageIndex{90}\)
- |5|
- |0|
- |−19|
In the following exercises, fill in <, >, or = for each of the following pairs of numbers.
Exercise \(\PageIndex{91}\)
- −8___|−8|
- −|−2|___−2
- Answer
-
- <
- =
Exercise \(\PageIndex{92}\)
- |−3|___−|−3|
- 4___−|−4|
In the following exercises, simplify.
Exercise \(\PageIndex{93}\)
|8−4|
- Answer
-
4
Exercise \(\PageIndex{94}\)
|9−6|
Exercise \(\PageIndex{95}\)
8(14−2|−2|)
- Answer
-
80
Exercise \(\PageIndex{96}\)
6(13−4|−2|)
In the following exercises, evaluate.
Exercise \(\PageIndex{97}\)
1. |x| when x=−28
- Answer
-
- 28
- 15
Exercise \(\PageIndex{98}\)
- ∣y∣ when y=−37
- |−z| when z=−24
Add Integers
In the following exercises, simplify each expression.
Exercise \(\PageIndex{99}\)
−200+65
- Answer
-
−135
Exercise \(\PageIndex{100}\)
−150+45
Exercise \(\PageIndex{101}\)
2+(−8)+6
- Answer
-
0
Exercise \(\PageIndex{102}\)
4+(−9)+7
Exercise \(\PageIndex{103}\)
140+(−75)+67
- Answer
-
132
Exercise \(\PageIndex{104}\)
−32+24+(−6)+10
Subtract Integers
In the following exercises, simplify.
Exercise \(\PageIndex{105}\)
9−3
- Answer
-
6
Exercise \(\PageIndex{106}\)
−5−(−1)
Exercise \(\PageIndex{107}\)
- 15−6
- 15+(−6)
- Answer
-
- 9
- 9
Exercise \(\PageIndex{108}\)
- 12−9
- 12+(−9)
Exercise \(\PageIndex{109}\)
- 8−(−9)
- 8+9
- Answer
-
- 17
- 17
Exercise \(\PageIndex{110}\)
- 4−(−4)
- 4+4
In the following exercises, simplify each expression.
Exercise \(\PageIndex{111}\)
10−(−19)
- Answer
-
29
Exercise \(\PageIndex{112}\)
11−(−18)
Exercise \(\PageIndex{113}\)
31−79
- Answer
-
−48
Exercise \(\PageIndex{114}\)
39−81
Exercise \(\PageIndex{115}\)
−31−11
- Answer
-
−42
Exercise \(\PageIndex{116}\)
−32−18
Exercise \(\PageIndex{117}\)
−15−(−28)+5
- Answer
-
18
Exercise \(\PageIndex{118}\)
71+(−10)−8
Exercise \(\PageIndex{119}\)
−16−(−4+1)−7
- Answer
-
-20
Exercise \(\PageIndex{120}\)
−15−(−6+4)−3
Multiply Integers
In the following exercises, multiply.
Exercise \(\PageIndex{121}\)
−5(7)
- Answer
-
−35
Exercise \(\PageIndex{122}\)
−8(6)
Exercise \(\PageIndex{123}\)
−18(−2)
- Answer
-
36
Exercise \(\PageIndex{124}\)
−10(−6)
Divide Integers
In the following exercises, divide.
Exercise \(\PageIndex{125}\)
\(-28 \div 7\)
- Answer
-
-4
Exercise \(\PageIndex{126}\)
\(56 \div(-7)\)
Exercise \(\PageIndex{127}\)
\(-120 \div(-20)\)
- Answer
-
6
Exercise \(\PageIndex{128}\)
\(-200 \div 25\)
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Exercise \(\PageIndex{129}\)
−8(−2)−3(−9)
- Answer
-
43
Exercise \(\PageIndex{130}\)
−7(−4)−5(−3)
Exercise \(\PageIndex{131}\)
\((-5)^{3}\)
- Answer
-
−125
Exercise \(\PageIndex{132}\)
\((-4)^{3}\)
Exercise \(\PageIndex{133}\)
\(-4 \cdot 2 \cdot 11\)
- Answer
-
−88
Exercise \(\PageIndex{134}\)
\(-5 \cdot 3 \cdot 10\)
Exercise \(\PageIndex{135}\)
\(-10(-4) \div(-8)\)
- Answer
-
-5
Exercise \(\PageIndex{136}\)
\(-8(-6) \div(-4)\)
Exercise \(\PageIndex{137}\)
31−4(3−9)
- Answer
-
55
Exercise \(\PageIndex{138}\)
24−3(2−10)
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Exercise \(\PageIndex{139}\)
x+8 when
- x=−26
- x=−95
- Answer
-
- −18
- −87
Exercise \(\PageIndex{140}\)
y+9 when
- y=−29
- y=−84
Exercise \(\PageIndex{141}\)
When b=−11, evaluate:
- b+6
- −b+6
- Answer
-
- −5
- 17
Exercise \(\PageIndex{142}\)
When c=−9, evaluate:
- c+(−4)c+(−4)
- −c+(−4)
Exercise \(\PageIndex{143}\)
\(p^{2}-5 p+2\) when
\(p=-1\)
- Answer
-
8
Exercise \(\PageIndex{144}\)
\(q^{2}-2 q+9\) when \(q=-2\)
Exercise \(\PageIndex{145}\)
\(6 x-5 y+15\) when \(x=3\) and \(y=-1\)
- Answer
-
38
Exercise \(\PageIndex{146}\)
\(3 p-2 q+9\) when \(p=8\) and \(q=-2\)
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Exercise \(\PageIndex{147}\)
the sum of −4 and −17, increased by 32
- Answer
-
(−4+(−17))+32;11
Exercise \(\PageIndex{148}\)
- the difference of 15 and −7
- subtract 15 from −7
Exercise \(\PageIndex{149}\)
the quotient of −45 and −9
- Answer
-
\(\frac{-45}{-9} ; 5\)
Exercise \(\PageIndex{150}\)
the product of −12 and the difference of c and d
Use Integers in Applications
In the following exercises, solve.
Exercise \(\PageIndex{151}\)
Temperature The high temperature one day in Miami Beach, Florida, was 76°. That same day, the high temperature in Buffalo, New York was −8°. What was the difference between the temperature in Miami Beach and the temperature in Buffalo?
- Answer
-
84 degrees
Exercise \(\PageIndex{152}\)
Checking Account Adrianne has a balance of −$22 in her checking account. She deposits $301 to the account. What is the new balance?
Visualize Fractions
Find Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
Exercise \(\PageIndex{153}\)
\(\frac{1}{4}\)
- Answer
-
\(\frac{2}{8}, \frac{3}{12}, \frac{4}{16}\) answers may vary
Exercise \(\PageIndex{154}\)
\(\frac{1}{3}\)
Exercise \(\PageIndex{155}\)
\(\frac{5}{6}\)
- Answer
-
\(\frac{10}{12}, \frac{15}{18}, \frac{20}{24}\) answers may vary
Exercise \(\PageIndex{156}\)
\(\frac{2}{7}\)
Simplify Fractions
In the following exercises, simplify.
Exercise \(\PageIndex{157}\)
\(\frac{7}{21}\)
- Answer
-
\(\frac{1}{3}\)
Exercise \(\PageIndex{158}\)
\(\frac{8}{24}\)
Exercise \(\PageIndex{159}\)
\(\frac{15}{20}\)
- Answer
-
\(\frac{3}{4}\)
Exercise \(\PageIndex{160}\)
\(\frac{12}{18}\)
Exercise \(\PageIndex{161}\)
\(-\frac{168}{192}\)
- Answer
-
\(-\frac{7}{8}\)
Exercise \(\PageIndex{162}\)
\(-\frac{140}{224}\)
Exercise \(\PageIndex{163}\)
\(\frac{11 x}{11 y}\)
- Answer
-
\(\frac{x}{y}\)
Exercise \(\PageIndex{164}\)
\(\frac{15 a}{15 b}\)
Multiply Fractions
In the following exercises, multiply.
Exercise \(\PageIndex{165}\)
\(\frac{2}{5} \cdot \frac{1}{3}\)
- Answer
-
\(\frac{2}{15}\)
Exercise \(\PageIndex{166}\)
\(\frac{1}{2} \cdot \frac{3}{8}\)
Exercise \(\PageIndex{167}\)
\(\frac{7}{12}\left(-\frac{8}{21}\right)\)
- Answer
-
\(-\frac{2}{9}\)
Exercise \(\PageIndex{168}\)
\(\frac{5}{12}\left(-\frac{8}{15}\right)\)
Exercise \(\PageIndex{169}\)
\(-28 p\left(-\frac{1}{4}\right)\)
- Answer
-
7p
Exercise \(\PageIndex{170}\)
\(-51 q\left(-\frac{1}{3}\right)\)
Exercise \(\PageIndex{172}\)
\(\frac{14}{5}(-15)\)
- Answer
-
−42
Exercise \(\PageIndex{173}\)
\(-1\left(-\frac{3}{8}\right)\)
Divide Fractions
In the following exercises, divide
Exercise \(\PageIndex{174}\)
\(\frac{1}{2} \div \frac{1}{4}\)
- Answer
-
2
Exercise \(\PageIndex{175}\)
\(\frac{1}{2} \div \frac{1}{8}\)
Exercise \(\PageIndex{176}\)
\(-\frac{4}{5} \div \frac{4}{7}\)
- Answer
-
\(-\frac{7}{5}\)
Exercise \(\PageIndex{177}\)
\(-\frac{3}{4} \div \frac{3}{5}\)
Exercise \(\PageIndex{178}\)
\(\frac{5}{8} \div \frac{a}{10}\)
- Answer
-
\(\frac{25}{4 a}\)
Exercise \(\PageIndex{179}\)
\(\frac{5}{6} \div \frac{c}{15}\)
Exercise \(\PageIndex{180}\)
\(\frac{7 p}{12} \div \frac{21 p}{8}\)
- Answer
-
\(\frac{2}{9}\)
Exercise \(\PageIndex{181}\)
\(\frac{5 q}{12} \div \frac{15 q}{8}\)
Exercise \(\PageIndex{182}\)
\(\frac{2}{5} \div(-10)\)
- Answer
-
\(-\frac{1}{25}\)
Exercise \(\PageIndex{183}\)
\(-18 \div-\left(\frac{9}{2}\right)\)
In the following exercises, simplify.
Exercise \(\PageIndex{184}\)
\(\frac{\frac{2}{3}}{\frac{8}{9}}\)
- Answer
-
\(\frac{3}{4}\)
Exercise \(\PageIndex{185}\)
\(\frac{\frac{4}{5}}{\frac{8}{15}}\)
Exercise \(\PageIndex{186}\)
\(\frac{-\frac{9}{10}}{3}\)
- Answer
-
\(-\frac{3}{10}\)
Exercise \(\PageIndex{187}\)
\(\frac{2}{\frac{5}{8}}\)
Exercise \(\PageIndex{188}\)
\(\frac{\frac{r}{5}}{\frac{s}{3}}\)
- Answer
-
\(\frac{3 r}{5 s}\)
Exercise \(\PageIndex{189}\)
\(\frac{-\frac{x}{6}}{-\frac{8}{9}}\)
Simplify Expressions Written with a Fraction Bar
In the following exercises, simplify.
Exercise \(\PageIndex{190}\)
\(\frac{4+11}{8}\)
- Answer
-
\(\frac{15}{8}\)
Exercise \(\PageIndex{191}\)
\(\frac{9+3}{7}\)
Exercise \(\PageIndex{192}\)
\(\frac{30}{7-12}\)
- Answer
-
-6
Exercise \(\PageIndex{193}\)
\(\frac{15}{4-9}\)
Exercise \(\PageIndex{194}\)
\(\frac{22-14}{19-13}\)
- Answer
-
\(\frac{4}{3}\)
Exercise \(\PageIndex{195}\)
\(\frac{15+9}{18+12}\)
Exercise \(\PageIndex{196}\)
\(\frac{5 \cdot 8}{-10}\)
- Answer
-
-4
Exercise \(\PageIndex{197}\)
\(\frac{3 \cdot 4}{-24}\)
Exercise \(\PageIndex{198}\)
\(\frac{15 \cdot 5-5^{2}}{2 \cdot 10}\)
- Answer
-
\(\frac{5}{2}\)
Exercise \(\PageIndex{199}\)
\(\frac{12 \cdot 9-3^{2}}{3 \cdot 18}\)
Exercise \(\PageIndex{200}\)
\(\frac{2+4(3)}{-3-2^{2}}\)
- Answer
-
-2
Exercise \(\PageIndex{201}\)
\(\frac{7+3(5)}{-2-3^{2}}\)
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
Exercise \(\PageIndex{202}\)
the quotient of c and the sum of d and 9.
- Answer
-
\(\frac{c}{d+9}\)
Exercise \(\PageIndex{203}\)
the quotient of the difference of h and k , and −5.
Add and Subtract Fractions
Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
Exercise \(\PageIndex{204}\)
\(\frac{4}{9}+\frac{1}{9}\)
- Answer
-
\(\frac{5}{9}\)
Exercise \(\PageIndex{205}\)
\(\frac{2}{9}+\frac{5}{9}\)
Exercise \(\PageIndex{206}\)
\(\frac{y}{3}+\frac{2}{3}\)
- Answer
-
\(\frac{y+2}{3}\)
Exercise \(\PageIndex{207}\)
\(\frac{7}{p}+\frac{9}{p}\)
Exercise \(\PageIndex{208}\)
\(-\frac{1}{8}+\left(-\frac{3}{8}\right)\)
- Answer
-
\(-\frac{1}{2}\)
Exercise \(\PageIndex{209}\)
\(-\frac{1}{8}+\left(-\frac{5}{8}\right)\)
In the following exercises, subtract.
Exercise \(\PageIndex{210}\)
\(\frac{4}{5}-\frac{1}{5}\)
- Answer
-
\(\frac{3}{5}\)
Exercise \(\PageIndex{211}\)
\(\frac{4}{5}-\frac{3}{5}\)
Exercise \(\PageIndex{212}\)
\(\frac{y}{17}-\frac{9}{17}\)
- Answer
-
\(\frac{y-9}{17}\)
Exercise \(\PageIndex{213}\)
\(\frac{x}{19}-\frac{8}{19}\)
Exercise \(\PageIndex{214}\)
\(-\frac{8}{d}-\frac{3}{d}\)
- Answer
-
\(-\frac{11}{d}\)
Exercise \(\PageIndex{215}\)
\(-\frac{7}{c}-\frac{7}{c}\)
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
Exercise \(\PageIndex{216}\)
\(\frac{1}{3}+\frac{1}{5}\)
- Answer
-
\(\frac{8}{15}\)
Exercise \(\PageIndex{217}\)
\(\frac{1}{4}+\frac{1}{5}\)
Exercise \(\PageIndex{218}\)
\(\frac{1}{5}-\left(-\frac{1}{10}\right)\)
- Answer
-
\(\frac{3}{10}\)
Exercise \(\PageIndex{219}\)
\(\frac{1}{2}-\left(-\frac{1}{6}\right)\)
Exercise \(\PageIndex{220}\)
\(\frac{2}{3}+\frac{3}{4}\)
- Answer
-
\(\frac{17}{12}\)
Exercise \(\PageIndex{221}\)
\(\frac{3}{4}+\frac{2}{5}\)
Exercise \(\PageIndex{222}\)
\(\frac{11}{12}-\frac{3}{8}\)
- Answer
-
\(\frac{13}{24}\)
Exercise \(\PageIndex{223}\)
\(\frac{5}{8}-\frac{7}{12}\)
Exercise \(\PageIndex{224}\)
\(-\frac{9}{16}-\left(-\frac{4}{5}\right)\)
- Answer
-
\(\frac{19}{80}\)
Exercise \(\PageIndex{225}\)
\(-\frac{7}{20}-\left(-\frac{5}{8}\right)\)
Exercise \(\PageIndex{226}\)
\(1+\frac{5}{6}\)
- Answer
-
\(\frac{11}{6}\)
Exercise \(\PageIndex{227}\)
\(1-\frac{5}{9}\)
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
Exercise \(\PageIndex{228}\)
\(\frac{\left(\frac{1}{5}\right)^{2}}{2+3^{2}}\)
- Answer
-
\(\frac{1}{275}\)
Exercise \(\PageIndex{229}\)
\(\frac{\left(\frac{1}{3}\right)^{2}}{5+2^{2}}\)
Exercise \(\PageIndex{230}\)
\(\frac{\frac{2}{3}+\frac{1}{2}}{\frac{3}{4}-\frac{2}{3}}\)
- Answer
-
14
Exercise \(\PageIndex{231}\)
\(\frac{\frac{3}{4}+\frac{1}{2}}{\frac{5}{6}-\frac{2}{3}}\)
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
Exercise \(\PageIndex{232}\)
\(x+\frac{1}{2}\) when
- \(x=-\frac{1}{8}\)
- \(x=-\frac{1}{2}\)
- Answer
-
- \(\frac{3}{8}\)
- \(0\)
Exercise \(\PageIndex{233}\)
\(x+\frac{2}{3}\) when
- \(x=-\frac{1}{6}\)
- \(x=-\frac{5}{3}\)
Exercise \(\PageIndex{234}\)
4\(p^{2} q\) when \(p=-\frac{1}{2}\) and \(q=\frac{5}{9}\)
- Answer
-
\(\frac{5}{9}\)
Exercise \(\PageIndex{235}\)
5\(m^{2} n\) when \(m=-\frac{2}{5}\) and \(n=\frac{1}{3}\)
Exercise \(\PageIndex{236}\)
\(\frac{u+v}{w}\) when
\(u=-4, v=-8, w=2\)
- Answer
-
-6
Exercise \(\PageIndex{237}\)
\(\frac{m+n}{p}\) when
\(m=-6, n=-2, p=4\)
Decimals
Name and Write Decimals
In the following exercises, write as a decimal.
Exercise \(\PageIndex{238}\)
Eight and three hundredths
- Answer
-
8.03
Exercise \(\PageIndex{239}\)
Nine and seven hundredths
Exercise \(\PageIndex{240}\)
One thousandth
- Answer
-
0.001
Exercise \(\PageIndex{241}\)
Nine thousandths
In the following exercises, name each decimal.
Exercise \(\PageIndex{242}\)
7.8
- Answer
-
seven and eight tenths
Exercise \(\PageIndex{243}\)
5.01
Exercise \(\PageIndex{244}\)
0.005
- Answer
-
five thousandths
Exercise \(\PageIndex{245}\)
0.381
Round Decimals
In the following exercises, round each number to the nearest
- hundredth
- tenth
- whole number.
Exercise \(\PageIndex{246}\)
5.7932
- Answer
-
- 5.79
- 5.8
- 6
Exercise \(\PageIndex{247}\)
3.6284
Exercise \(\PageIndex{248}\)
12.4768
- Answer
-
- 12.48
- 12.5
- 12
Exercise \(\PageIndex{249}\)
25.8449
Add and Subtract Decimals
In the following exercises, add or subtract.
Exercise \(\PageIndex{250}\)
18.37+9.36
- Answer
-
27.73
Exercise \(\PageIndex{251}\)
256.37−85.49
Exercise \(\PageIndex{252}\)
15.35−20.88
- Answer
-
−5.53
Exercise \(\PageIndex{253}\)
37.5+12.23
Exercise \(\PageIndex{254}\)
−4.2+(−9.3)
- Answer
-
−13.5
Exercise \(\PageIndex{255}\)
−8.6+(−8.6)
Exercise \(\PageIndex{256}\)
100−64.2
- Answer
-
35.8
Exercise \(\PageIndex{257}\)
100−65.83
Exercise \(\PageIndex{258}\)
2.51+40
- Answer
-
42.51
Exercise \(\PageIndex{259}\)
9.38+60
Multiply and Divide Decimals
In the following exercises, multiply.
Exercise \(\PageIndex{260}\)
(0.3)(0.4)
- Answer
-
0.12
Exercise \(\PageIndex{261}\)
(0.6)(0.7)
Exercise \(\PageIndex{262}\)
(8.52)(3.14)
- Answer
-
26.7528
Exercise \(\PageIndex{263}\)
(5.32)(4.86)
Exercise \(\PageIndex{264}\)
(0.09)(24.78)
- Answer
-
2.2302
Exercise \(\PageIndex{265}\)
(0.04)(36.89)
In the following exercises, divide.
Exercise \(\PageIndex{266}\)
\(0.15 \div 5\)
- Answer
-
0.03
Exercise \(\PageIndex{267}\)
\(0.27 \div 3\)
Exercise \(\PageIndex{268}\)
\(\$ 8.49 \div 12\)
- Answer
-
$0.71
Exercise \(\PageIndex{269}\)
\(\$ 16.99 \div 9\)
Exercise \(\PageIndex{270}\)
\(12 \div 0.08\)
- Answer
-
150
Exercise \(\PageIndex{271}\)
\(5 \div 0.04\)
Convert Decimals, Fractions, and Percents
In the following exercises, write each decimal as a fraction.
Exercise \(\PageIndex{272}\)
0.08
- Answer
-
\(\frac{2}{25}\)
Exercise \(\PageIndex{273}\)
0.17
Exercise \(\PageIndex{274}\)
0.425
- Answer
-
\(\frac{17}{40}\)
Exercise \(\PageIndex{275}\)
0.184
Exercise \(\PageIndex{276}\)
1.75
- Answer
-
\(\frac{7}{4}\)
Exercise \(\PageIndex{277}\)
0.035
In the following exercises, convert each fraction to a decimal.
Exercise \(\PageIndex{278}\)
\(\frac{2}{5}\)
- Answer
-
0.4
Exercise \(\PageIndex{279}\)
\(\frac{4}{5}\)
Exercise \(\PageIndex{280}\)
\(-\frac{3}{8}\)
- Answer
-
−0.375
Exercise \(\PageIndex{281}\)
\(-\frac{5}{8}\)
Exercise \(\PageIndex{282}\)
\(\frac{5}{9}\)
- Answer
-
\(0 . \overline{5}\)
Exercise \(\PageIndex{283}\)
\(\frac{2}{9}\)
Exercise \(\PageIndex{284}\)
\(\frac{1}{2}+6.5\)
- Answer
-
7
Exercise \(\PageIndex{285}\)
\(\frac{1}{4}+10.75\)
In the following exercises, convert each percent to a decimal.
Exercise \(\PageIndex{286}\)
5%
- Answer
-
0.05
Exercise \(\PageIndex{287}\)
9%
Exercise \(\PageIndex{288}\)
40%
- Answer
-
0.4
Exercise \(\PageIndex{289}\)
50%
Exercise \(\PageIndex{290}\)
115%
- Answer
-
1.15
Exercise \(\PageIndex{291}\)
125%
In the following exercises, convert each decimal to a percent.
Exercise \(\PageIndex{292}\)
0.18
- Answer
-
18%
Exercise \(\PageIndex{293}\)
0.15
Exercise \(\PageIndex{294}\)
0.009
- Answer
-
0.9%
Exercise \(\PageIndex{295}\)
0.008
Exercise \(\PageIndex{296}\)
1.5
- Answer
-
150%
Exercise \(\PageIndex{297}\)
2.2
The Real Numbers
Simplify Expressions with Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{298}\)
\(\sqrt{64}\)
- Answer
-
8
Exercise \(\PageIndex{299}\)
\(\sqrt{144}\)
Exercise \(\PageIndex{300}\)
\(-\sqrt{25}\)
- Answer
-
-5
Exercise \(\PageIndex{301}\)
\(-\sqrt{81}\)
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
Exercise \(\PageIndex{302}\)
- 9
- 8.47
- Answer
-
- \(\frac{9}{1}\)
- \(\frac{847}{100}\)
Exercise \(\PageIndex{303}\)
- −15
- 3.591
In the following exercises, list the
- rational numbers,
- irrational numbers.
Exercise \(\PageIndex{304}\)
\(0.84,0.79132 \ldots, 1 . \overline{3}\)
- Answer
-
- \(0.84,1.3\)
- \(0.79132 \ldots\)
Exercise \(\PageIndex{305}\)
\(2.3 \overline{8}, 0.572,4.93814 \ldots\)
In the following exercises, identify whether each number is rational or irrational.
Exercise \(\PageIndex{306}\)
- \(\sqrt{121}\)
- \(\sqrt{48}\)
- Answer
-
- rational
- irrational
Exercise \(\PageIndex{307}\)
- \(\sqrt{56}\)
- \(\sqrt{16}\)
In the following exercises, identify whether each number is a real number or not a real number.
Exercise \(\PageIndex{308}\)
- \(\sqrt{-9}\)
- \(-\sqrt{169}\)
- Answer
-
- not a real number
- real number
Exercise \(\PageIndex{309}\)
- \(\sqrt{-64}\)
- \(-\sqrt{81}\)
In the following exercises, list the
- whole numbers,
- integers,
- rational numbers,
- irrational numbers,
- real numbers for each set of numbers.
Exercise \(\PageIndex{310}\)
\(-4,0, \frac{5}{6}, \sqrt{16}, \sqrt{18}, 5.2537 \ldots\)
- Answer
-
- \(0, \sqrt{16}\)
- \(-4,0, \sqrt{16}\)
- \(-4,0, \frac{5}{6}, \sqrt{16}\)
- \(\sqrt{18}, 5.2537 \ldots\)
- \(-4,0, \frac{5}{6}, \sqrt{16}, \sqrt{18}, 5.2537 \ldots\)
Exercise \(\PageIndex{311}\)
\(-\sqrt{4}, 0 . \overline{36}, \frac{13}{3}, 6.9152 \ldots, \sqrt{48}, 10 \frac{1}{2}\)
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
Exercise \(\PageIndex{312}\)
\(\frac{2}{3}, \frac{5}{4}, \frac{12}{5}\)
- Answer
-
Exercise \(\PageIndex{313}\)
\(\frac{1}{3}, \frac{7}{4}, \frac{13}{5}\)
Exercise \(\PageIndex{314}\)
\(2 \frac{1}{3},-2 \frac{1}{3}\)
- Answer
-
Exercise \(\PageIndex{315}\)
\(1 \frac{3}{5},-1 \frac{3}{5}\)
In the following exercises, order each of the following pairs of numbers, using < or >.
Exercise \(\PageIndex{316}\)
−1___\(-\frac{1}{8}\)
- Answer
-
<
Exercise \(\PageIndex{317}\)
\(-3 \frac{1}{4}\)___−4
Exercise \(\PageIndex{318}\)
\(-\frac{7}{9}\) ___ \(\frac{4}{9}\)
- Answer
-
>
Exercise \(\PageIndex{319}\)
\(-2\) ___ \(\frac{19}{8}\)
Locate Decimals on the Number Line
In the following exercises, locate on the number line.
Exercise \(\PageIndex{320}\)
0.3
- Answer
-
Exercise \(\PageIndex{321}\)
−0.2
Exercise \(\PageIndex{322}\)
−2.5
- Answer
-
Exercise \(\PageIndex{323}\)
2.7
In the following exercises, order each of the following pairs of numbers, using < or >.
Exercise \(\PageIndex{324}\)
0.9___0.6
- Answer
-
>
Exercise \(\PageIndex{325}\)
0.7___0.8
Exercise \(\PageIndex{326}\)
−0.6___−0.59
- Answer
-
>
Exercise \(\PageIndex{327}\)
−0.27___−0.3
Properties of Real Numbers
Use the Commutative and Associative Properties
In the following exercises, use the Associative Property to simplify.
Exercise \(\PageIndex{328}\)
−12(4m)
- Answer
-
−48m
Exercise \(\PageIndex{329}\)
30\(\left(\frac{5}{6} q\right)\)
Exercise \(\PageIndex{330}\)
(a+16)+31
- Answer
-
a+47
Exercise \(\PageIndex{331}\)
(c+0.2)+0.7
In the following exercises, simplify.
Exercise \(\PageIndex{332}\)
6y+37+(−6y)
- Answer
-
37
Exercise \(\PageIndex{333}\)
\(\frac{1}{4}+\frac{11}{15}+\left(-\frac{1}{4}\right)\)
Exercise \(\PageIndex{334}\)
\(\frac{14}{11} \cdot \frac{35}{9} \cdot \frac{14}{11}\)
- Answer
-
\(\frac{35}{9}\)
Exercise \(\PageIndex{335}\)
\(-18 \cdot 15 \cdot \frac{2}{9}\)
Exercise \(\PageIndex{336}\)
\(\left(\frac{7}{12}+\frac{4}{5}\right)+\frac{1}{5}\)
- Answer
-
1\(\frac{7}{12}\)
Exercise \(\PageIndex{337}\)
(3.98d+0.75d)+1.25d
Exercise \(\PageIndex{338}\)
11x+8y+16x+15y
- Answer
-
27x+23y
Exercise \(\PageIndex{339}\)
52m+(−20n)+(−18m)+(−5n)
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
Exercise \(\PageIndex{340}\)
- \(\frac{1}{3}\)
- 5.1
- \(-14\)
- \(-\frac{8}{5}\)
- Answer
-
- \(-\frac{1}{3}\)
- \(-5.1\)
- -14
- \(-\frac{8}{5}\)
Exercise \(\PageIndex{341}\)
- \(-\frac{7}{8}\)
- \(-0.03\)
- 17
- \(\frac{12}{5}\)
In the following exercises, find the multiplicative inverse of each number.
Exercise \(\PageIndex{342}\)
- \(10\)
- \(-\frac{4}{9}\)
- 0.6
- Answer
-
- \(\frac{1}{10}\)
- \(-\frac{9}{4}\)
- \(\frac{5}{3}\)
Exercise \(\PageIndex{343}\)
- \(-\frac{9}{2}\)
- -7
- 2.1
Use the Properties of Zero
In the following exercises, simplify.
Exercise \(\PageIndex{344}\)
83\(\cdot 0\)
- Answer
-
0
Exercise \(\PageIndex{345}\)
\(\frac{0}{9}\)
Exercise \(\PageIndex{346}\)
\(\frac{5}{0}\)
- Answer
-
undefined
Exercise \(\PageIndex{347}\)
\(0 \div \frac{2}{3}\)
In the following exercises, simplify.
Exercise \(\PageIndex{348}\)
43+39+(−43)
- Answer
-
39
Exercise \(\PageIndex{349}\)
(n+6.75)+0.25
Exercise \(\PageIndex{350}\)
\(\frac{5}{13} \cdot 57 \cdot \frac{13}{5}\)
- Answer
-
57
Exercise \(\PageIndex{351}\)
\(\frac{1}{6} \cdot 17 \cdot 12\)
Exercise \(\PageIndex{352}\)
\(\frac{2}{3} \cdot 28 \cdot \frac{3}{7}\)
- Answer
-
8
Exercise \(\PageIndex{353}\)
\(9(6 x-11)+15\)
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.
Exercise \(\PageIndex{354}\)
7(x+9)
- Answer
-
7x+63
Exercise \(\PageIndex{355}\)
9(u−4)
Exercise \(\PageIndex{356}\)
−3(6m−1)
- Answer
-
−18m+3
Exercise \(\PageIndex{357}\)
−8(−7a−12)
Exercise \(\PageIndex{358}\)
\(\frac{1}{3}(15 n-6)\)
- Answer
-
5n−2
Exercise \(\PageIndex{359}\)
\((y+10) \cdot p\)
Exercise \(\PageIndex{360}\)
(a−4)−(6a+9)
- Answer
-
−5a−13
Exercise \(\PageIndex{361}\)
4(x+3)−8(x−7)
Systems of Measurement
1.1 Define U.S. Units of Measurement and Convert from One Unit to Another
In the following exercises, convert the units. Round to the nearest tenth.
Exercise \(\PageIndex{362}\)
A floral arbor is 7 feet tall. Convert the height to inches.
- Answer
-
84 inches
Exercise \(\PageIndex{363}\)
A picture frame is 42 inches wide. Convert the width to feet.
Exercise \(\PageIndex{364}\)
Kelly is 5 feet 4 inches tall. Convert her height to inches.
- Answer
-
64 inches
Exercise \(\PageIndex{365}\)
A playground is 45 feet wide. Convert the width to yards.
Exercise \(\PageIndex{366}\)
The height of Mount Shasta is 14,179 feet. Convert the height to miles.
- Answer
-
2.7 miles
Exercise \(\PageIndex{367}\)
Shamu weights 4.5 tons. Convert the weight to pounds.
Exercise \(\PageIndex{368}\)
The play lasted \(1\frac{3}{4}\) hours. Convert the time to minutes.
- Answer
-
105 minutes
Exercise \(\PageIndex{369}\)
How many tablespoons are in a quart?
Exercise \(\PageIndex{370}\)
Naomi’s baby weighed 5 pounds 14 ounces at birth. Convert the weight to ounces.
- Answer
-
94 ounces
Exercise \(\PageIndex{371}\)
Trinh needs 30 cups of paint for her class art project. Convert the volume to gallons.
Use Mixed Units of Measurement in the U.S. System.
In the following exercises, solve.
Exercise \(\PageIndex{372}\)
John caught 4 lobsters. The weights of the lobsters were 1 pound 9 ounces, 1 pound 12 ounces, 4 pounds 2 ounces, and 2 pounds 15 ounces. What was the total weight of the lobsters?
- Answer
-
10 lbs. 6 oz.
Exercise \(\PageIndex{373}\)
Every day last week Pedro recorded the number of minutes he spent reading. The number of minutes were 50, 25, 83, 45, 32, 60, 135. How many hours did Pedro spend reading?
Exercise \(\PageIndex{374}\)
Fouad is 6 feet 2 inches tall. If he stands on a rung of a ladder 8 feet 10 inches high, how high off the ground is the top of Fouad’s head?
- Answer
-
15 feet
Exercise \(\PageIndex{375}\)
Dalila wants to make throw pillow covers. Each cover takes 30 inches of fabric. How many yards of fabric does she need for 4 covers?
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Exercise \(\PageIndex{376}\)
Donna is 1.7 meters tall. Convert her height to centimeters.
- Answer
-
170 centimeters
Exercise \(\PageIndex{377}\)
Mount Everest is 8,850 meters tall. Convert the height to kilometers.
Exercise \(\PageIndex{378}\)
One cup of yogurt contains 488 milligrams of calcium. Convert this to grams.
- Answer
-
0.488 grams
Exercise \(\PageIndex{379}\)
One cup of yogurt contains 13 grams of protein. Convert this to milligrams.
Exercise \(\PageIndex{380}\)
Sergio weighed 2.9 kilograms at birth. Convert this to grams.
- Answer
-
2,900 grams
Exercise \(\PageIndex{381}\)
A bottle of water contained 650 milliliters. Convert this to liters.
Use Mixed Units of Measurement in the Metric System
In the following exerices, solve.
Exercise \(\PageIndex{382}\)
Minh is 2 meters tall. His daughter is 88 centimeters tall. How much taller is Minh than his daughter?
- Answer
-
1.12 meter
Exercise \(\PageIndex{383}\)
Selma had a 1 liter bottle of water. If she drank 145 milliliters, how much water was left in the bottle?
Exercise \(\PageIndex{384}\)
One serving of cranberry juice contains 30 grams of sugar. How many kilograms of sugar are in 30 servings of cranberry juice?
- Answer
-
0.9 kilograms
Exercise \(\PageIndex{385}\)
One ounce of tofu provided 2 grams of protein. How many milligrams of protein are provided by 5 ounces of tofu?
Convert between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Exercise \(\PageIndex{386}\)
Majid is 69 inches tall. Convert his height to centimeters.
- Answer
-
175.3 centimeters
Exercise \(\PageIndex{387}\)
A college basketball court is 84 feet long. Convert this length to meters.
Exercise \(\PageIndex{388}\)
Caroline walked 2.5 kilometers. Convert this length to miles.
- Answer
-
1.6 miles
Exercise \(\PageIndex{389}\)
Lucas weighs 78 kilograms. Convert his weight to pounds.
Exercise \(\PageIndex{390}\)
Steve’s car holds 55 liters of gas. Convert this to gallons.
- Answer
-
14.6 gallons
Exercise \(\PageIndex{391}\)
A box of books weighs 25 pounds. Convert the weight to kilograms.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
Exercise \(\PageIndex{392}\)
95° Fahrenheit
- Answer
-
35° C
Exercise \(\PageIndex{393}\)
23° Fahrenheit
Exercise \(\PageIndex{394}\)
20° Fahrenheit
- Answer
-
–6.7° C
Exercise \(\PageIndex{395}\)
64° Fahrenheit
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.
Exercise \(\PageIndex{396}\)
30° Celsius
- Answer
-
86° F
Exercise \(\PageIndex{397}\)
–5° Celsius
Exercise \(\PageIndex{398}\)
–12° Celsius
- Answer
-
10.4° F
Exercise \(\PageIndex{399}\)
24° Celsius
Chapter Practice Test
Exercise \(\PageIndex{1}\)
Write as a whole number using digits: two hundred five thousand, six hundred seventeen.
- Answer
-
205,617
Exercise \(\PageIndex{2}\)
Find the prime factorization of 504.
Exercise \(\PageIndex{3}\)
Find the Least Common Multiple of 18 and 24.
- Answer
-
72
Exercise \(\PageIndex{4}\)
Combine like terms: 5n+8+2n−1.
In the following exercises, evaluate.
Exercise \(\PageIndex{5}\)
\(-|x|\) when \(x=-2\)
- Answer
-
−2
Exercise \(\PageIndex{6}\)
11−a when a=−3
Exercise \(\PageIndex{7}\)
Translate to an algebraic expression and simplify: twenty less than negative 7.
- Answer
-
−7−20;−27
Exercise \(\PageIndex{8}\)
Monique has a balance of −$18 in her checking account. She deposits $152 to the account. What is the new balance?
Exercise \(\PageIndex{9}\)
Round 677.1348 to the nearest hundredth.
- Answer
-
677.13
Exercise \(\PageIndex{10}\)
Convert \(\frac{4}{5}\) to a decimal.
Exercise \(\PageIndex{11}\)
Convert 1.85 to a percent.
- Answer
-
185%
Exercise \(\PageIndex{12}\)
Locate \(\frac{2}{3},-1.5,\) and \(\frac{9}{4}\) on a number line.
In the following exercises, simplify each expression.
Exercise \(\PageIndex{13}\)
\(4+10(3+9)-5^{2}\)
- Answer
-
99
Exercise \(\PageIndex{14}\)
−85+42
Exercise \(\PageIndex{15}\)
−19−25
- Answer
-
−44
Exercise \(\PageIndex{16}\)
\((-2)^{4}\)
Exercise \(\PageIndex{17}\)
\(-5(-9) \div 15\)
- Answer
-
3
Exercise \(\PageIndex{18}\)
\(\frac{3}{8} \cdot \frac{11}{12}\)
Exercise \(\PageIndex{19}\)
\(\frac{4}{5} \div \frac{9}{20}\)
- Answer
-
\(\frac{16}{9}\)
Exercise \(\PageIndex{20}\)
\(\frac{12+3 \cdot 5}{15-6}\)
Exercise \(\PageIndex{21}\)
\(\frac{m}{7}+\frac{10}{7}\)
- Answer
-
\(\frac{m+10}{7}\)
Exercise \(\PageIndex{22}\)
\(\frac{7}{12}-\frac{3}{8}\)
Exercise \(\PageIndex{23}\)
\(-5.8+(-4.7)\)
- Answer
-
−10.5
Exercise \(\PageIndex{24}\)
100−64.25
Exercise \(\PageIndex{25}\)
(0.07)(31.95)
- Answer
-
2.2365
Exercise \(\PageIndex{26}\)
\(9 \div 0.05\)
Exercise \(\PageIndex{27}\)
\(-14\left(\frac{5}{7} p\right)\)
- Answer
-
−10p
Exercise \(\PageIndex{28}\)
(u+8)−9
Exercise \(\PageIndex{29}\)
6x+(−4y)+9x+8y
- Answer
-
15x+4y
Exercise \(\PageIndex{30}\)
\(\frac{0}{23}\)
Exercise \(\PageIndex{31}\)
\(\frac{75}{0}\)
- Answer
-
undefined
Exercise \(\PageIndex{32}\)
−2(13q−5)
Exercise \(\PageIndex{33}\)
A movie lasted 1\(\frac{2}{3}\) hours. How many minutes did it last? ( 1 hour \(=60\) minutes)
- Answer
-
100 minutes
Exercise \(\PageIndex{34}\)
Mike’s SUV is 5 feet 11 inches tall. He wants to put a rooftop cargo bag on the the SUV. The cargo bag is 1 foot 6 inches tall. What will the total height be of the SUV with the cargo bag on the roof? (1 foot = 12 inches)
Exercise \(\PageIndex{35}\)
Jennifer ran 2.8 miles. Convert this length to kilometers. (1 mile = 1.61 kilometers)
- Answer
-
4.508 km


3.1: Introduction
So far our discussion has been limited to service businesses where companies sell expertise, knowledge, experiences, or the use of something to customers. In a service business, customers do not purchase or take ownership of a physical product.
Merchandising businesses sell products. A merchandising business buys finished and packaged manufactured products, marks them up, and sells them to customers. A merchandiser , therefore, may be either the buyer or the seller in a given transaction, depending upon whether product is being purchased (and added to the stock of inventory), or sold (and removed from the stock of inventory.)
A vendor is a company or individual that a merchandiser purchases goods from. A customer is a company or individual that a merchandiser sells goods to.
Inventory consists of items that are purchased for resale. Note that inventory is different from supplies. Supplies are items that are purchased to be used in the operation of the business, not to be sold to customers. For example, a merchandiser may have Windex glass cleaner on hand. It is considered inventory if it will be resold to customers and is considered a supply if it is used in running the business to keep the check-out counters clean. Similarly, inventory is also different from fixed assets such as equipment . For example, a merchandiser may have desktop computers on hand. They are considered inventory if they are to be resold to customers, such as in the case of Best Buy , Dell , or Apple . They would be classified as equipment if the merchandiser is using the computers to run its own business operations.
Sales is the new revenue account used to record income from selling products. This account replaces Fees Earned , the revenue account used for a service business.
The following are common sequences of events for merchandising businesses. When you are the buyer , you will (1) purchase product on account; (2) return product; and (3) pay for the product. When you are the seller , you will (1) sell product on account and reduce the inventory balance; (2) accept returns and increase the inventory balance; and (3) receive payment for sales.
Most merchandising businesses use a perpetual inventory system . It is the process of keeping a current running total of inventory, both in number of
units on hand and its dollar value, at all times. When product is purchased for resale, inventory immediately increases. When product is sold, the total value of the inventory on hand is immediately reduced.
This accounts summary table lists the new accounts used by merchandising businesses that use the perpetual inventory system for timing the recording of its changes in inventory value.
| ACCOUNT TYPE | ACCOUNTS | TO INCREASE | TO DECREASE | NORMAL BALANCE | FINANCIAL STATEMENT | CLOSE OUT? |
| Asset |
Merchandise Inventory Account that keeps track of Items in stock for resale to customers |
debit | credit | debit |
Balance
|
NO |
| Contra Asset |
Estimated Inventory Returns Account that keeps track of the cost of the amount inventory that customers are expected to return |
credit | debit | credit | Balance Sheet | NO |
| Revenue |
Sales Account that keeps track of the dollar amount of purchases made by customers |
credit | debit | credit | Income Statement | YES |
| Contra Revenue |
Sales Returns Account that keeps track of the dollar amount of merchandise actually returned by customers Allowance for Sales Returns Account that keeps track of the dollar amount of merchandise esti- mated to be returned by customers Sales Discounts Account that keeps track of the dollar amount of discounts taken by customers under the gross method of recording sales Sales Discounts Not Taken Account that keeps track of the dollar amount of discounts not taken by customers under the net method of recording sales |
debit | credit | debit | Income Statement | YES |
| Expense |
Cost of Merchandise Sold Account that keeps track of what a company paid for the inventory it has sold to customers Delivery Expense Account that keeps track of the transportation charges that a seller has absorbed as an expense |
debit | credit | debit | Income Statement | YES |
When BUYING, the only new account above you may use is Merchandise Inventory. When SELLING, you may use any of the seven new accounts.


5.3: Passive Transport
By the end of this section, you will be able to do the following:
- Explain why and how passive transport occurs
- Understand the osmosis and diffusion processes
- Define tonicity and its relevance to passive transport
Plasma membranes must allow certain substances to enter and leave a cell, and prevent some harmful materials from entering and some essential materials from leaving. In other words, plasma membranes are selectively permeable (semipermeable)—they allow some substances to pass through, but not others. If they were to lose this selectivity, the cell would no longer be able to sustain itself, and it would be destroyed. Some cells require larger amounts of specific substances. They must have a way of obtaining these materials from extracellular fluids. This may happen passively, as certain materials move back and forth, or the cell may have special mechanisms that facilitate transport. Some materials are so important to a cell that it spends some of its energy, hydrolyzing adenosine triphosphate (ATP), to obtain these materials. Red blood cells use some of their energy doing just that. Most cells spend the majority of their energy to maintain an imbalance of sodium and potassium ions between the cell's interior and exterior, as well as on protein synthesis.
The most direct forms of membrane transport are passive. Passive transport is a naturally occurring phenomenon and does not require the cell to exert any of its energy to accomplish the movement. In passive transport, substances move from an area of higher concentration to an area of lower concentration. A physical space in which there is a single substance concentration range has a concentration gradient .
Selective Permeability
Plasma membranes are asymmetric: the membrane's interior is not identical to its exterior. There is a considerable difference between the array of phospholipids and proteins between the two leaflets that form a membrane. On the membrane's interior, some proteins serve to anchor the membrane to cytoskeleton's fibers. There are peripheral proteins on the membrane's exterior that bind extracellular matrix elements. Carbohydrates, attached to lipids or proteins, are also on the plasma membrane's exterior surface. These carbohydrate complexes help the cell bind required substances in the extracellular fluid. This adds considerably to plasma membrane's selective nature (Figure \(\PageIndex{1}\)).
Recall that plasma membranes are amphiphilic: They have hydrophilic and hydrophobic regions. This characteristic helps move some materials through the membrane and hinders the movement of others. Non-polar and lipid-soluble material with a low molecular weight can easily slip through the membrane's hydrophobic lipid core. Substances such as the fat-soluble vitamins A, D, E, and K readily pass through the plasma membranes in the digestive tract and other tissues. Fat-soluble drugs and hormones also gain easy entry into cells and readily transport themselves into the body’s tissues and organs. Oxygen and carbon dioxide molecules have no charge and pass through membranes by simple diffusion.
Polar substances present problems for the membrane. While some polar molecules connect easily with the cell's outside, they cannot readily pass through the plasma membrane's lipid core. Additionally, while small ions could easily slip through the spaces in the membrane's mosaic, their charge prevents them from doing so. Ions such as sodium, potassium, calcium, and chloride must have special means of penetrating plasma membranes. Simple sugars and amino acids also need the help of various transmembrane proteins (channels) to transport themselves across plasma membranes.
Diffusion
Diffusion is a passive process of transport. A single substance moves from a high concentration to a low concentration area until the concentration is equal across a space. You are familiar with diffusion of substances through the air. For example, think about someone opening a bottle of ammonia in a room filled with people. The ammonia gas is at its highest concentration in the bottle. Its lowest concentration is at the room's edges. The ammonia vapor will diffuse, or spread away, from the bottle, and gradually, increasingly more people will smell the ammonia as it spreads. Materials move within the cell’s cytosol by diffusion, and certain materials move through the plasma membrane by diffusion (Figure \(\PageIndex{2}\)). Diffusion expends no energy. On the contrary, concentration gradients are a form of potential energy, which dissipates as the gradient is eliminated.
Each separate substance in a medium, such as the extracellular fluid, has its own concentration gradient, independent of other materials' concentration gradients. In addition, each substance will diffuse according to that gradient. Within a system, there will be different diffusion rates of various substances in the medium.
Factors That Affect Diffusion
Molecules move constantly in a random manner, at a rate that depends on their mass, their environment, and the amount of thermal energy they possess, which in turn is a function of temperature. This movement accounts for molecule diffusion through whatever medium in which they are localized. A substance moves into any space available to it until it evenly distributes itself throughout. After a substance has diffused completely through a space, removing its concentration gradient, molecules will still move around in the space, but there will be no net movement of the number of molecules from one area to another. We call this lack of a concentration gradient in which the substance has no net movement dynamic equilibrium. While diffusion will go forward in the presence of a substance's concentration gradient, several factors affect the diffusion rate.
- Extent of the concentration gradient: The greater the difference in concentration, the more rapid the diffusion. The closer the distribution of the material gets to equilibrium, the slower the diffusion rate.
- Mass of the molecules diffusing: Heavier molecules move more slowly; therefore, they diffuse more slowly. The reverse is true for lighter molecules.
- Temperature: Higher temperatures increase the energy and therefore the molecules' movement, increasing the diffusion rate. Lower temperatures decrease the molecules' energy, thus decreasing the diffusion rate.
- Solvent density: As the density of a solvent increases, the diffusion rate decreases. The molecules slow down because they have a more difficult time passing through the denser medium. If the medium is less dense, diffusion increases. Because cells primarily use diffusion to move materials within the cytoplasm, any increase in the cytoplasm’s density will inhibit the movement of the materials. An example of this is a person experiencing dehydration. As the body’s cells lose water, the diffusion rate decreases in the cytoplasm, and the cells’ functions deteriorate. Neurons tend to be very sensitive to this effect. Dehydration frequently leads to unconsciousness and possibly coma because of the decrease in diffusion rate within the cells.
- Solubility: As we discussed earlier, nonpolar or lipid-soluble materials pass through plasma membranes more easily than polar materials, allowing a faster diffusion rate.
- Surface area and plasma membrane thickness: Increased surface area increases the diffusion rate; whereas, a thicker membrane reduces it.
- Distance traveled: The greater the distance that a substance must travel, the slower the diffusion rate. This places an upper limitation on cell size.  Therefore, cells must either be small in size, as in the case of many prokaryotes, or be flattened, as with many single-celled eukaryotes.
A variation of diffusion is the process of filtration. In filtration, material moves according to its concentration gradient through a membrane. Sometimes pressure enhances the diffusion rate, causing the substances to filter more rapidly. This occurs in the kidney, where blood pressure forces large amounts of water and accompanying dissolved substances, or solutes , out of the blood and into the renal tubules. The diffusion rate in this instance is almost totally dependent on pressure. One of the effects of high blood pressure is the appearance of protein in the urine, which abnormally high pressure "squeezes through".
Facilitated transport
In facilitated transport , or facilitated diffusion, materials diffuse across the plasma membrane with the help of membrane proteins. A concentration gradient exists that would allow these materials to diffuse into the cell without expending cellular energy. However, these materials are polar molecule ions that the cell membrane's hydrophobic parts repel. Facilitated transport proteins shield these materials from the membrane's repulsive force, allowing them to diffuse into the cell.
The transported material first attaches to protein or glycoprotein receptors on the plasma membrane's exterior surface. This allows removal of material from the extracellular fluid that the cell needs. The substances then pass to specific integral proteins that facilitate their passage. Some of these integral proteins are collections of beta-pleated sheets that form a pore or channel through the phospholipid bilayer. Others are carrier proteins which bind with the substance and aid its diffusion through the membrane.
Channels
The integral proteins involved in facilitated transport are transport proteins , and they function as either channels for the material or carriers. In both cases, they are transmembrane proteins. Channels are specific for the transported substance. Channel proteins have hydrophilic domains exposed to the intracellular and extracellular fluids. In addition, they have a hydrophilic channel through their core that provides a hydrated opening through the membrane layers (Figure \(\PageIndex{3}\)). Passage through the channel allows polar compounds to avoid the plasma membrane's nonpolar central layer that would otherwise slow or prevent their entry into the cell. Aquaporins are channel proteins that allow water to pass through the membrane at a very high rate.
Channel proteins are either open at all times or they are “gated,” which controls the channel's opening. When a particular ion attaches to the channel protein it may control the opening, or other mechanisms or substances may be involved. In some tissues, sodium and chloride ions pass freely through open channels; whereas, in other tissues a gate must open to allow passage. An example of this occurs in the kidney, where there are both channel forms in different parts of the renal tubules. Cells involved in transmitting electrical impulses, such as nerve and muscle cells, have gated channels for sodium, potassium, and calcium in their membranes. Opening and closing these channels changes the relative concentrations on opposing sides of the membrane of these ions, resulting in facilitating electrical transmission along membranes (in the case of nerve cells) or in muscle contraction (in the case of muscle cells).
Carrier Proteins
Another type of protein embedded in the plasma membrane is a carrier protein . This aptly named protein binds a substance and, thus triggers a change of its own shape, moving the bound molecule from the cell's outside to its interior (Figure \(\PageIndex{4}\)). Depending on the gradient, the material may move in the opposite direction. Carrier proteins are typically specific for a single substance. This selectivity adds to the plasma membrane's overall selectivity. Scientists poorly understand the exact mechanism for the change of shape. Proteins can change shape when their hydrogen bonds are affected, but this may not fully explain this mechanism. Each carrier protein is specific to one substance, and there are a finite number of these proteins in any membrane. This can cause problems in transporting enough material for the cell to function properly. When all of the proteins are bound to their ligands, they are saturated and the rate of transport is at its maximum. Increasing the concentration gradient at this point will not result in an increased transport rate.
An example of this process occurs in the kidney. In one part, the kidney filters glucose, water, salts, ions, and amino acids that the body requires. This filtrate, which includes glucose, then reabsorbs in another part of the kidney. Because there are only a finite number of carrier proteins for glucose, if more glucose is present than the proteins can handle, the excess is not transported and the body excretes this through urine. In a diabetic individual, the term is “spilling glucose into the urine.” A different group of carrier proteins, glucose transport proteins, or GLUTs, are involved in transporting glucose and other hexose sugars through plasma membranes within the body.
Channel and carrier proteins transport material at different rates. Channel proteins transport much more quickly than carrier proteins. Channel proteins facilitate diffusion at a rate of tens of millions of molecules per second; whereas, carrier proteins work at a rate of a thousand to a million molecules per second.
Osmosis
Osmosis is the movement of free water molecules through a semipermeable membrane according to the water's concentration gradient across the membrane, which is inversely proportional to the solutes' concentration. While diffusion transports material across membranes and within cells, osmosis transports only water across a membrane and the membrane limits the solutes' diffusion in the water. Not surprisingly, the aquaporins that facilitate water movement play a large role in osmosis, most prominently in red blood cells and the membranes of kidney tubules.
Mechanism
Osmosis is a special case of diffusion. Water, like other substances, moves from an area of high concentration of free water molecules to one of low free water molecule concentration. An obvious question is what makes water move at all? Imagine a beaker with a semipermeable membrane separating the two sides or halves (Figure \(\PageIndex{5}\)). On both sides of the membrane the water level is the same, but there are different dissolved substance concentrations, or solute , that cannot cross the membrane (otherwise the solute crossing the membrane would balance concentrations on each side). If the solution's volume on both sides of the membrane is the same, but the solute's concentrations are different, then there are different amounts of water, the solvent, on either side of the membrane.
To illustrate this, imagine two full water glasses. One has a single teaspoon of sugar in it; whereas, the second one contains one-quarter cup of sugar. If the total volume of the solutions in both cups is the same, which cup contains more water? Because the large sugar amount in the second cup takes up much more space than the teaspoon of sugar in the first cup, the first cup has more water in it.
Returning to the beaker example, recall that it has a solute mixture on either side of the membrane. A principle of diffusion is that the molecules move around and will spread evenly throughout the medium if they can. However, only the material capable of getting through the membrane will diffuse through it. In this example, the solute cannot diffuse through the membrane, but the water can. Water has a concentration gradient in this system. Thus, water will diffuse down its concentration gradient, crossing the membrane to the side where it is less concentrated. This diffusion of water through the membrane—osmosis—will continue until the water's concentration gradient goes to zero or until the water's hydrostatic pressure balances the osmotic pressure. Osmosis proceeds constantly in living systems.
Tonicity
Tonicity describes how an extracellular solution can change a cell's volume by affecting osmosis. A solution's tonicity often directly correlates with the solution's osmolarity. Osmolarity describes the solution's total solute concentration. A solution with low osmolarity has a greater number of water molecules relative to the number of solute particles. A solution with high osmolarity has fewer water molecules with respect to solute particles. In a situation in which a membrane permeable to water, though not to the solute separates two different osmolarities, water will move from the membrane's side with lower osmolarity (and more water) to the side with higher osmolarity (and less water). This effect makes sense if you remember that the solute cannot move across the membrane, and thus the only component in the system that can move—the water—moves along its own concentration gradient. An important distinction that concerns living systems is that osmolarity measures the number of particles (which may be molecules) in a solution. Therefore, a solution that is cloudy with cells may have a lower osmolarity than a solution that is clear, if the second solution contains more dissolved molecules than there are cells.
Hypotonic Solutions
Scientists use three terms—hypotonic, isotonic, and hypertonic—to relate the cell's osmolarity to the extracellular fluid's osmolarity that contains the cells. In a hypotonic situation, the extracellular fluid has lower osmolarity than the fluid inside the cell, and water enters the cell. (In living systems, the point of reference is always the cytoplasm, so the prefix hypo - means that the extracellular fluid has a lower solute concentration, or a lower osmolarity, than the cell cytoplasm.) It also means that the extracellular fluid has a higher water concentration in the solution than does the cell. In this situation, water will follow its concentration gradient and enter the cell.
Hypertonic Solutions
As for a hypertonic solution, the prefix hyper - refers to the extracellular fluid having a higher osmolarity than the cell’s cytoplasm; therefore, the fluid contains less water than the cell does. Because the cell has a relatively higher water concentration, water will leave the cell.
Isotonic Solutions
In an isotonic solution, the extracellular fluid has the same osmolarity as the cell. If the cell's osmolarity matches that of the extracellular fluid, there will be no net movement of water into or out of the cell, although water will still move in and out. Blood cells and plant cells in hypertonic, isotonic, and hypotonic solutions take on characteristic appearances (Figure \(\PageIndex{6}\)).
Visual Connection
Visual Connection
A doctor injects a patient with what the doctor thinks is an isotonic saline solution. The patient dies, and an autopsy reveals that many red blood cells have been destroyed. Do you think the solution the doctor injected was really isotonic?
Link to Learning
Link to Learning
Watch the video below for a view of the diffusion process in solutions.
Tonicity in Living Systems
In a hypotonic environment, water enters a cell, and the cell swells. In an isotonic condition, the relative solute and solvent concentrations are equal on both membrane sides. There is no net water movement; therefore, there is no change in the cell's size. In a hypertonic solution, water leaves a cell and the cell shrinks. If either the hypo- or hyper- condition goes to excess, the cell’s functions become compromised, and the cell may be destroyed.
A red blood cell will burst, or lyse, when it swells beyond the plasma membrane’s capability to expand. Remember, the membrane resembles a mosaic, with discrete spaces between the molecules comprising it. If the cell swells, and the spaces between the lipids and proteins become too large, the cell will break apart.
In contrast, when excessive water amounts leave a red blood cell, the cell shrinks, or crenates. This has the effect of concentrating the solutes left in the cell, making the cytosol denser and interfering with diffusion within the cell. The cell’s ability to function will be compromised and may also result in the cell's death.
Various living things have ways of controlling the effects of osmosis—a mechanism we call osmoregulation. Some organisms, such as plants, fungi, bacteria, and some protists, have cell walls that surround the plasma membrane and prevent cell lysis in a hypotonic solution. The plasma membrane can only expand to the cell wall's limit, so the cell will not lyse. The cytoplasm in plants is always slightly hypertonic to the cellular environment, and water will always enter a cell if water is available. This water inflow produces turgor pressure, which stiffens the plant's cell walls (Figure \(\PageIndex{7}\)). In nonwoody plants, turgor pressure supports the plant. Conversely, if you do not water the plant, the extracellular fluid will become hypertonic, causing water to leave the cell. In this condition, the cell does not shrink because the cell wall is not flexible. However, the cell membrane detaches from the wall and constricts the cytoplasm. We call this plasmolysis . Plants lose turgor pressure in this condition and wilt (Figure \(\PageIndex{8}\)).
Tonicity is a concern for all living things. For example, paramecia and amoebas, which are protists that lack cell walls, have contractile vacuoles. This vesicle collects excess water from the cell and pumps it out, keeping the cell from lysing as it takes on water from its environment (Figure \(\PageIndex{9}\)).
Many marine invertebrates have internal salt levels matched to their environments, making them isotonic with the water in which they live. Fish, however, must spend approximately five percent of their metabolic energy maintaining osmotic homeostasis. Freshwater fish live in an environment that is hypotonic to their cells. These fish actively take in salt through their gills and excrete diluted urine to rid themselves of excess water. Saltwater fish live in the reverse environment, which is hypertonic to their cells, and they secrete salt through their gills and excrete highly concentrated urine.
In vertebrates, the kidneys regulate the water amount in the body. Osmoreceptors are specialized cells in the brain that monitor solute concentration in the blood. If the solute levels increase beyond a certain range, a hormone releases that slows water loss through the kidney and dilutes the blood to safer levels. Animals also have high albumin concentrations, which the liver produces, in their blood. This protein is too large to pass easily through plasma membranes and is a major factor in controlling the osmotic pressures applied to tissues.
Contributors and Attributions
Remixed and/or curated from the following works:
Clark, M. A., Douglas, M., & Choi, J. (2018). 2.2.3 Passive Transport . In Biology 2e . OpenStax (CC BY 4.0; Access for free at https://openstax.org/books/biology-2e/pages/1-introduction ).