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Test Tips

An actual ACT Mathematics Test contains 60 questions to be answered in 60 minutes.

• Read each question carefully to make sure you understand the type of answer required.
• If you choose to use a calculator, be sure it is permitted, is working on test day, and has reliable batteries.
• Use your calculator wisely.
• Solve the problem.
• Locate your solution among the answer choices.
• Make sure you answer the question asked.
• Make sure your answer is reasonable.
• Check your work.

Calculator Tips

• Review the latest information on permitted and prohibited calculators.
• You are not required to use a calculator. All the problems can be solved without a calculator.
• If you regularly use a calculator in your mathematics work, use one you're familiar with when you take the mathematics test. Using a more powerful, but unfamiliar, calculator is not likely to give you an advantage over using the kind you normally use.

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DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.

Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test.

You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator.

Note: Unless otherwise stated, all of the following should be assumed.

1. Illustrative figures are NOT necessarily drawn to scale.
2. Geometric figures lie in a plane.
3. The word line indicates a straight line.
4. The word average indicates arithmetic mean.

Utilitarian ethics is a normative ethical system that is primarily concerned with the consequences of ethical decisions; therefore it can be described as a , which are essentially the same thing, both having a notion that the consequence of the act is the most important determinant of the act being moral or not. Teleological reasoning takes into consideration that the ethical decision is dependent upon the consequences (“ends”) of the actions. In teleological reasoning, a person will do the right thing if the consequences of his or her actions are good. Additionally, if an action by a person was an act that was “not good,” but the consequences turned out to be “good,” under some theories of teleological reasoning, the act may be deemed a good ethical act. This is also referred to as “consequentialist moral reasoning,” where we locate morality in the consequences of our actions.

As a result of the consequentialist nature of utilitarianism, the means to get to the ethical decision (“end”) are secondary; the end result is that which must be considered before determining the morality of the decision.

Jeremy Bentham (1748-1832) developed the principles of utility by defining it as a measure of maximizing pleasure while minimizing pain. Bentham wrote that everyone prefers pleasure over pain. It is with this belief that utilitarian moral principles are founded (Sandel, 2010). In developing the theory of utilitarianism, Bentham may have meant pleasure as in “happiness” and pain as in “sadness”; however, Bentham’s rendering of utilitarianism sounded hedonistic, as if sensuality was the measure Bentham associated with pleasure (Hinman, 2013).

John Stuart Mill reconsidered the principles of utilitarianism and suggested that pleasure should not merely refer to sensual pleasure but also to mental pleasure, such as music, literature, and friendship. Mill sought to make intellectual pleasures preferable to sensual ones.

Hinman (2013) suggests there are four principle differences between pleasure and happiness:

1. Happiness is related to the mind, whereas pleasure is related to the body (for example sexual pleasure, eating, drinking)
2. Pleasure is of shorter duration than happiness. Happiness is long-term, focusing on the satisfaction of living well, or achieving life goals.
3. Happiness may encompass pleasure and pain.
4. There is an evaluative element in happiness versus pleasure.

There are two formulations of utilitarianism: and . Act utilitarianism concerns the consequences of the first instance, where the utility of that act is all that is regarded.

The second formulation of utilitarianism, rule utilitarianism, concerns the consequences of the majority of people following a certain rule that is immoral, which would be negative. With rule utilitarianism, to determine the ethics of an act, the questions to ask are “What would happen if there was a universal rule that condones this action?” and “Would such a rule promote the consequences that would best serve a moral society?” Rule utilitarianism operates as a check and balance for utilitarian principles, assuring that decisions that may be utilitarian in principle are qualified with the notion of universality, asking “what would the result be if everyone followed a rule that allowed this act?”

The best way to illustrate consequentialist theory is through an implausible story proposed by Michael Sandel (2010).

Imagine you are the driver of a trolley car train and are speeding along. As you are heading to the work yard, you realize the brakes don’t work. Ahead you see five workers on the track. They are busy jack hammering and do not see you approach. You as the driver have the ability to determine where the train goes by switching the tracks to another track. However that track has one worker, who is also oblivious to your approach. By physically switching the tracks, you will save five, but your actions will kill the lone worker.

The moral dilemma is such that we are required to determine what the consequences or the end result should be. The questions you need to ask are:

What action would you take?

Are the end results, or the consequences of your actions, important?

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What action would a Rule Utilitarian take?

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What action would an Act Utilitarian take?

Q. How can utilitarian theory assist law enforcement in moral dilemmas?

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Law enforcement officers possess a great deal of discretion that must be exercised by all officers of every rank, regardless of their experience. When exercising this discretion, officers will be confronted on a daily basis with issues that are complex, and may not be covered in the agency’s policy and most certainly would not have been covered in their formal education or police academy or other training. Law enforcement officers also are required to make exigent decisions, without the ability to consult with senior officers or policy and procedures. In some instances, when confronted with decisions, officers may want to rely on utilitarianism to make an ethical decision that is defensible when scrutinized in the future. For example, an officer tasked with policing a large pro-marijuana protest group may observe a person within the group selling marijuana. Legally, the officer has the duty to charge that person with trafficking in a controlled substance under the Controlled Drug and Substance Act, a serious indictable offence. However, from a utilitarian position, the officer may elect not to arrest and charge the suspect for two reasons:

1. The act of not arresting would make more pro-marijuana group happier compared to the number of people would be unhappy with that decision. We can reasonably say that society at large is becoming more relaxed about marijuana use, and the movement to legalize marijuana is strong and getting stronger. Perhaps the officer would recognize this, and make his or her decision accordingly. If the drug being trafficked was crack cocaine, then the officer would likely adjust the decision. (If the drug was a more lethal drug, that could cause death, the officer would be compelled by duty to arrest the suspect in order to prevent harm.)
2. If the arrest is made for trafficking , the consequence would likely be a serious violent confrontation with the large pro-marijuana group. The arrest by the police would not make the majority of these individuals happy. As a result, while arresting the trafficker may be the duty of the officer, the officer may come to the conclusion that the consequences of making an arrest are likely to be negative. Therefore by using discretion, the officer is utilizing utilitarian principles in his or her decision making.

From a rule utilitarianism perspective, the officer should consider what the consequences would be if there were a rule that everyone was allowed to smoke and sell marijuana. If the officer believes that society would be well served by this rule, then the officer should allow the sale to continue. Should the officer believe the rule would be detrimental to society, the officer should consider this as well, and at least consider making the arrest.

Like all normative theories of ethics, utilitarianism cannot address all of the ethical dilemmas we face. Sometimes using utilitarian principles may be harmful to a group of people or to an individual. Some of the major problems with utilitarian consequentialist ethics include the following:

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• Measuring happiness is difficult. Happiness is subjective and as a result is open to interpretation. Is happiness in winning a million dollars more significant than the happiness a person experiences when told by a doctor that he has a clean bill of health? Likewise, does the value of happiness increase with time, or with importance? If someone won a million dollars, would this be measured as “the most possible happy,” as the million dollars will hopefully last for a long time? Conversely, a person received a clean bill of health after a routine checkup can be regarded as more important news; however this person is likely to forget this good news within days. So when we look at the happiness that is caused by these two events, we need to ask ourselves, “what makes us the most happy?”
• Utilitarian ethics is concerned about the consequences of our actions, regardless of the action itself. However, it can be difficult to know what the consequences of our actions will be because of the variables that we do not control. For example, a police officer may believe that writing tickets at an intersection will create a safe intersection environment for everyone. However, it is difficult to determine for sure that this will be the outcome. Unintended consequences may instead occur. Suppose, for example, that while the officer is writing a ticket at the intersection, a fatal accident occurs due to the officer disrupting the traffic. In this instance, the unintended consequences could not have been predicted, especially if the officer acted in a safe manner while writing the ticket. The unintended consequences may be viewed as immoral by utilitarian standards because of the end result. People who maintain this logic are referred to as Actual Consequentialists because actual consequences are what determine if the act was right or wrong. However, some consequentialists would rightly take into consideration the fact that the fatal accident could not have possibly been foreseen, and therefore the act itself was still moral in spite of the unforeseen negative consequences. This appears to be a more logical approach to consequentialism as it incorporates a mental element (mens rea) in determining if the act was moral.
• Desired ethical consequences that actually result from our actions do not always happen immediately. If the desired consequences of our actions do not occur immediately, how long must we wait for those good consequences to develop before we can say the action was ethical? Likewise, how long are we to wait to deem the consequences as positive or negative? For example, in a correctional institute, a warden who believes that weapons are being made in an inmate job program may cancel the program. The warden may decide to cancel the program due to the inability of staff to ensure that the making of weapons does not occur. The warden’s decision is ultimately based on ethics and a desire to ensure the well-being of corrections staff and inmates. However, the inmates may view this decision as punitive as the prohibited weapons are being made by only some inmates. In analyzing the ethics of the decision by the warden, the question would have to be asked, “how much time would have to expire before we could determine this was an ethical action versus a punitive one?”
• Happiness should not be the only consequence or goal that matters in some ethical dilemmas. Some goals of the ethical decision, such as human rights, may matter more than the consequences of the action. For example, consider a detective who is investigating a series of sexual assaults has located evidence which is not admissible in court but clearly demonstrates that a suspect is guilty of the crimes. The detective realizes that the suspect is likely to recommit the crime, and therefore decides to plant false forensic evidence on the suspect to implicate him. While this action may result in positive consequences (and the greater happiness for the greater number of people), the actions are wrong and cannot be condoned. By removing the notions of justice, fairness, and basic human rights owed the suspect, the actions are immoral and unethical; they are actions that will eventually erode confidence in police. The consequences, from a utilitarian perspective, should not outweigh the notion of justice. In this way, utilitarianism can provide an excuse for those who commit wrongs for noble reasons. On the other hand, utilitarians may argue that the actions are actually not utilitarian because the long-term effects may have an opposite effect: less happiness for the greater number of people should lawful investigations not be trusted by society. This is an example of rule utilitarianism, where we can look at the benefits of having a rule that allows such actions (planning evidence) by law enforcement officers that would not promote the most happiness overall.
• When utilitarian decisions benefit the majority at the expense of the minority, the minority’s rights may not be taken into account. Utilitarian principles often run contrary to individual’s rights, and at times are the antithesis to concepts of modern justice theories. When we are tempted to make a decision that will positively impact the majority, we must also consider the negative impact on the minority. In the example above, the detective who plants false forensic evidence on a suspect may feel that the maximum happiness to the majority of people makes the action ethical. However, the investigator is not respecting the rights of the individual suspect. Much the same way, when crime reduction policies, such as sex offender registries, allegedly promote community safety, offenders’ rights are ignored due to the loss of privacy. This is especially so, given that studies indicate such registries are often ineffective and do little to protect the community (Petrunik, 2002; Vess, 2008).