ACT Math For Dummies (2011)
Part II. Building Your Pre-Algebra and Elementary Algebra Skills
In this part. . .
In Part II, I discuss pre-algebra and elementary algebra. These topics include basic math, percent problems, factors and multiples, statistics and probability, algebraic expressions, and basic algebraic equations. The final chapter in the part allows you to practice problems like those you may encounter on the ACT.
Chapter 4. Starting with the Basics: Pre-Algebra
In This Chapter
Beefing up your arithmetic skills
Working with factors and multiples
Handling percents, ratios, and proportions
Understanding powers and square roots
Answering questions in statistics and probability
Pre-algebra focuses on basic arithmetic skills, which are the basis for all of math. The good news is that you probably began learning arithmetic during your first few years in school, so nothing here will be new. Even so, beefing up your skills can help to improve your ACT score.
In this chapter, I start off with a review of the four basic operations — addition, subtraction, multiplication, and division. I focus on helping you answer ACT questions involving number sequences. I also review inequalities (such as <, >, ≥, and ≤) and absolute value. Next, you get some practice working with factors and multiples and a variety of questions that focus on percents, including percent increase and decrease. Finally, I show you how to use ratios and proportions to your advantage and provide info on statistics and probability.
Getting Back to Basics: Answering Basic Arithmetic Questions
Arithmetic is the basis for all of math, so you need a good grounding in it to do well on the math portion of the ACT. Fortunately, every question in this book requires a bit of arithmetic, so you get lots of practice. In this section, I start off with the basics of arithmetic, including the four basic operations, inequalities, and absolute value.
Reviewing the four basic operations
The four basic arithmetic operations, as you probably know, are addition, subtraction, multiplication, and division. You’ve been working with these operations for a long time, and you’ll be using them a lot more on the ACT.
Many of the simpler word problems toward the beginning of the ACT give you scenarios that require you to apply basic arithmetic. If you read this type of question carefully, you should be able to work it out.
Arithmetic word problems aren’t difficult, but they sometimes can be a little confusing. Use the room provided in your test booklet for scribbling notes as you work on them. Sometimes it even helps to draw a picture, as I discuss in Chapter 3.
Carla, a secretary in a law firm, is paid on a monthly basis. Each month, she works exactly 22 days and is paid $3,850. In May, Carla needed to take a day off. The firm deducted one day’s pay from her paycheck and hired a temporary secretary named Jerome, paying him $135 for the day. How much did the firm pay out in total to both Carla and Jerome that month?
To answer this question, you first need to find out how much Carla is paid per day. To do this, divide her monthly salary ($3,850) by the number of days she works each month (22):
3,850 ÷ 22 = 175
As you can see, Carla is paid $175 dollars a day. When she took the day off, the firm paid Jerome $135, which is $40 less (175 – 135 = 40). Thus, the firm paid out $40 less to Carla and Jerome than it usually pays to Carla. Subtract that $40 from Carla’s normal salary to find your answer:
3,850 – 40 = 3,810
Therefore, the firm paid $3,810, so the correct answer is Choice (A).
Jacqui works in a supermarket stocking shelves. She likes to time herself while she works and has found that it takes 10 seconds to open a box and 2 seconds to stock each item. About how long will it take her to open and stock a complete order of 24 boxes, each of which contains 36 cans of tuna?
(F) Between 1,200 and 1,400 seconds
(G) Between 1,400 and 1,600 seconds
(H) Between 1,600 and 1,800 seconds
(J) Between 1,800 and 2,000 seconds
(K) Between 2,000 and 2,200 seconds
The order contains 24 × 36 = 864 cans. Each can takes 2 seconds to stock, which in total takes 864 × 2 = 1,728 seconds. Additionally, opening all 24 boxes will take 24 × 10 = 240 seconds. Thus, the total order will take 1,728 + 240 = 1,968 seconds. So the correct answer is Choice (J).
Following along with number sequences
A number sequence is a set of numbers that’s in a specific order based on a rule. The rule that governs a sequence is usually an operation performed on each number to produce the next number in the sequence.
To answer most ACT questions involving a sequence, you need to discover the rule and then apply it. Begin by looking for a way to apply a basic operation to one number in the sequence in order to change it to the next number. Then, see if this rule works for the rest of the sequence.
Tip: When a sequence is increasing, look for ways to add or multiply (or both) one number to produce the next number.
In the number sequence 3, 10, 31, ___, 283, what number should be placed in the blank?
The rule to change 3 to 10 using only addition would be “add 7,” because 3 + 7 = 10. But this rule doesn’t change 10 to 31, because 10 + 7 = 17. Using only multiplication, the rule would be “multiply by ,” because 3 × = 10. Again, however, this rule doesn’t change 10 to 31, because 10 × = .
A rule involving multiplication and addition is “multiply by 3 and then add 1.” This does the trick, because 10 × 3 + 1 = 31. So here’s how you find the next number in the sequence: 31 × 3 + 1 = 94. Therefore, the correct answer is Choice (D).
In some cases, two different rules work to produce alternating numbers, as this next example illustrates.
If the first five numbers in a sequence are 5, 7, 14, 16, 32, what is the tenth number?
The two alternating rules producing this sequence are “add 2” and “multiply by 2.” Use these rules to continue the sequence:
5, 7, 14, 16, 32, 34, 68, 70, 140, 142, . . .
Thus, the correct answer is Choice (G).
Checking out inequalities
You probably know that an equation is a math statement that tells you two values are equivalent. In contrast, an inequality tells you that two values aren’t equivalent. The four inequalities you’ll see on the ACT are
Greater than (>)
Less than (<)
Greater than or equal to (≥)
Less than or equal to (≤)
Note that > and < both tell you that two values are not equal, and ≥ and ≤ tell you that they may be equal.
Sometimes, reversing an entire inequality can help you make better sense of it without changing its meaning. For example, the inequality 2 > x may not be immediately clear, but its equivalent x < 2 is easier to understand: x is less than 2, so you can picture values that x may be (1, 0, –1, and so forth).
An ACT question involving an inequality may ask you to plot the solution set — the set of all possible solutions — on a number line. A black dot on the number line tells you that the value is included — that is, the sign is either≥ or ≤. An open circle tells you that the value is not included — that is, the sign is either > or <.
Which of the following is the solution set for the inequality –2 ≤ x < 3
The inequality tells you that the solution set for x is between the two values, so you can rule out Choices (A) and (B). The value –2 is included, because the first inequality is ≤. So there should be a black dot at –2; thus, you can rule out Choice (C). The value 3 is not included, because the second inequality is <. As a result, there should be an open circle at 3; thus, you can rule out Choice (E). Therefore, the correct answer is Choice (D).
Evaluating a number’s absolute value
The absolute value of a number is its value when you drop the minus sign. If the number doesn’t have a minus sign to begin with, its value stays the same. For example:
|3| = 3 |–4| = 4 |0| = 0
Absolute value isn’t difficult, but you can avoid confusion by taking ACT questions with absolute value in two steps: First, evaluate and remove the absolute value bars, and then complete the problem.
When working with absolute value, |–4| – |–6| =
Begin by evaluating |–4| and |–6| separately to remove the absolute value bars: |–4| = 4 and |–6| = 6. Thus, you can rewrite the equation:
|–4| – |–6| = 4 – 6
Now the problem becomes simple: 4 – 6 = –2, so the correct answer is Choice (D).
In some cases, you may need to apply absolute value to a number line.
Which of the following number lines expresses the set of all possible values of x for the inequality x ≥ |–2|?
To begin, evaluate the absolute value: |–2| = 2. So you can rewrite the inequality as follows:
x ≥ 2
As you can see, the solution set includes 2 and all values greater than 2, so the correct answer is Choice (F).
Understanding Factors and Multiples
Two important concepts in arithmetic are factors and multiples. Both of these are related to the simple idea of divisibility: One positive integer is divisible by another if you can divide the first integer by the second without leaving a remainder. For example:
14 is divisible by 2, because 14 ÷ 2 = 7 (with no remainder)
14 is not divisible by 3, because 14 ÷ 3 = 4 r 2 (4 with a remainder of 2)
When one number is divisible by another, you can describe the relationship between them using the words factor (the smaller number) and multiple (the larger number). For example:
2 is a factor of 14
14 is a multiple of 2
In this section, I show you how to answer a variety of ACT questions that involve factors and multiples. (For even more on these two topics, check out another book by Yours Truly: Basic Math and Pre-Algebra For Dummies[Wiley].)
The factors of a positive integer are always less than the number itself, so it’s easy to list all the factors of a number. To answer an ACT question, you may need to generate a list of all factors of a number.
Which of the following integers has exactly the same number of factors as the number 21?
Of course, to figure out this problem you have to know how many factors 21 has. Every integer greater than 1 has at least two factors: 1 and the number itself. To list all the factors of a number, begin by writing down these two numbers with some space between them, like this:
Factors of 21: 1 21
All other factors must fall between 1 and 21. Start by testing 2: Clearly 21 isn’t divisible by 2, because 21 isn’t even. Next, test whether 21 is divisible by 3: Yes, because 21 ÷ 3 = 7 with no remainder. So you can add 3 to the early part of the list and 7 to the later part:
Factors of 21: 1 3 7 21
Now all other factors (if any exist) must fall between 3 and 7. A quick test of 4, 5, and 6 shows that 21 isn’t divisible by any of these numbers. So 21 has exactly four factors: 1, 3, 7, and 21.
Use the same process to list the factors of the five answer choices until you find one that has exactly four factors:
Factors of 3: 1 3
Factors of 4: 1 2 4
Factors of 5: 1 5
Factors of 6: 1 2 3 6
As you can see, 21 and 6 both have exactly four factors, so the correct answer is Choice (D).
Knowing how to list the factors of a number in this way also is important for finding the greatest common factor (GCF) of a set of numbers — that is, the highest number that’s a factor of every number in that set.
What is the greatest common factor of 24, 36, and 50?
To answer this question, list the factors of all three numbers:
Factors of 24: 1 2 3 4 6 8 12 24
Factors of 36: 1 2 3 4 6 9 12 18 36
Factors of 50: 1 2 5 10 25 50
The greatest number that appears in all three lists is 2, so the correct answer is Choice (F).
Generating lists of multiples
The positive multiples of an integer are always greater than the integer itself. For this reason, you can never list all the multiples of a number (as you can with factors; see the preceding section for details). When answering an ACT question, it’s often handy to make a quick list of the first ten or so multiples of a number. Making this list is simple. Just keep adding the number you’re working with. For example:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 . . .
When adding the fractions , , , and , what is the lowest common denominator?
The denominators are the bottom numbers in the fractions: 3, 5, 10, and 15. To find the lowest common denominator among these four numbers, generate a list of the multiples of these numbers until you find the lowest number that appears on all three lists, like so:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40
Multiples of 10: 10, 20, 30, 40, 50, 60
Multiples of 15: 15, 30, 45, 60
As you can see, 30 is the lowest common denominator for these four fractions, so the correct answer is Choice (C).
Perfecting Your Knowledge of Percents
To do well on the ACT, you need to have a solid understanding of percents. Questions may ask you to calculate a percentage directly. Alternatively, you may need to use percents to handle a word problem or work with a graph, such as a pie chart.
Before you begin reviewing percentage problems, make sure your calculator has a percent key to help you save time when calculating percents. Even with a calculator to help, however, certain percent problems can be confusing. So read on for details.
In this section, I show you a variety of ways to answer percent questions. I also give you some practice solving the much-dreaded percent increase and percent decrease problems.
Knowing basic percent conversions
Even though you can use a calculator on the ACT, you can save a lot of time by knowing a few basics. One good place to start is by memorizing the basic conversions for percents, decimals, and fractions. I list the most common conversions in Table 4-1.
Taking advantage of a quick trick for calculating some percents
Ready for a way to save a few seconds on the ACT? In this section, I let you in on a quick trick for finding certain percents. You simply have to reverse the numbers you’re working with. Check out the following example.
If 80% of 50 is x, what is x percent of 25?
Of course, you can do 80% of 50 on your calculator, but an easier way in this case is to reverse the two numbers:
80% of 50 = 50% of 80
This trick always works. In this case, you can see that 50% of 80 equals half of 80, which is 40. So x = 40. Now you have to find 40% of 25, so use the trick again:
40% of 25 = 25% of 40
Again, you probably can see that 25% of 40 equals one-quarter of 40, which is 10. Therefore, the correct answer is Choice (A).
Calculating straightforward percent problems
Even if you’re really good at working with fractions, decimals, and percents, you can save yourself a lot of trouble by using your calculator to work out hairy calculations.
First of all, be sure that your calculator has a percent key. Sure, technically you can calculate percents by converting them to decimals — and as a math guy, it’s my duty to recommend that you know how to do this! But on the ACT, you don’t need to waste the time and risk making an error. So take full advantage of your calculator. The following example shows you how.
When you calculate 62.5% of 450, how many decimal places does the resulting number have?
You can answer this question by answering 0.625 × 450. But when rushing on the test, if you convert the decimal incorrectly, you’ll get the wrong answer. For this problem, your calculator is your best friend:
62.5% × 450 = 281.25
The resulting number has two decimal places, so the correct answer is Choice (C).
Untangling complex percent problems
Using your calculator to find a result is straightforward when you know the number you’re calculating with and the percentage you’re taking. For example, you may need to determine what 50% of 10 is. With or without a calculator, you can find out pretty quickly that the answer to this question is 5. But for some other ACT questions, you’re given information that’s more difficult to untangle.
When answering this type of question, the trick is to translate it into an equation. Substitute either 0.01 or for the percent sign, and let x (or any variable you like) stand for the number you’re trying to find. Then use algebra to solve for x (flip to Chapter 5 for a review of elementary algebra).
In this section, I provide two examples as practice for the problems you may see on the ACT.
What percent of 600 is 270?
A good way to solve this type of percent problem is to translate the words into an equation. (I discuss this technique further in Chapter 5.) Here’s how the example problem would look:
(x%)(600) = 270
Substitute 0.01 for the percent sign in this equation and solve for x:
x(0.01)(600) = 270
6x = 270
x = 45
Thus, 45% of 600 is 270, so the correct answer is Choice (C).
If 40% of a is 30, then a is what percent of 50?
To answer this question, you first need to find the value of a, so make an equation from the values in the first part of the question. Here’s how it would look:
(40%)(a) = 30
40(0.01)(a) = 30
Now solve this equation for a:
Thus, a = 75. Finally, you make an equation from the values in the second part of the question. Here’s how: “75 is what percent of 50” becomes
75 = (x%)(50)
Substitute 0.01 for the percent sign in this equation and solve for x:
So the correct answer is Choice (K).
Handling the ups and downs of percent increase and decrease
Many students find questions that involve percent increase and percent decrease somewhat confusing. The first step to answering these questions correctly is identifying them when they’re presented. Some common scenarios for percent increase questions include the following:
Sales tax added to the price of an item
Tipping a server at a restaurant
Interest earned on an investment
Some typical situations for percent decrease questions are
Money lost on an investment
Discount on an item being sold
Deduction from a paycheck due to taxes
After you know whether you’re dealing with percent increase or percent decrease, here’s how you handle the calculations:
Increase: Calculate a percent increase as 100% + the percent. For example, a percent increase of 15% is equal to 100% + 15% = 115% of the original value.
Decrease: Calculate a percent decrease as 100% – the percent. For example, calculate a percent decrease of 20% as 100% – 20% = 80%.
The following two examples show you how to handle both types of questions from start to finish.
Randy bought a small guitar amplifier priced at $165 with a special coupon that gave him a 15% discount. About how much did he end up paying for the amp?
(A) Less than $100
(B) Between $100 and $120
(C) Between $120 and $140
(D) Between $140 and $160
(E) More than $160
A 15% discount is a percent decrease of 15%:
100% – 15% = 85%
Use your calculator to find this percentage of the original price of $165:
85% × $165 = $140.25
Thus, the correct answer is Choice (D).
You can apply this method for finding percent increase and decrease to any of the percent problems that I discuss earlier in the chapter, in “Untangling complex percent problems.”
Keith’s portfolio is currently worth $10,200, representing a 20% increase on his original investment. How much did he originally invest?
A 20% increase is calculated as 100% + 20% = 120%, so use the following formula:
(120%)(n) = 10,200
Change the percent sign to 0.01 and solve:
120(0.01)(n) = 10,200
Therefore, the correct answer is Choice (J).
Using Ratios and Proportions to Compare Quantities
A ratio is a comparison of two quantities based on the operation of division. For example, if a school has one teacher for every eight students, you can express the teacher-to-student ratio in any of the following ways:
1:8 1 to 8
Notice that this ratio expresses the ratio of teachers to students. Thus, the 1 goes before the 8 and, in the fraction, the 1 goes on top of the 8.
When answering an ACT question that includes a ratio, a good strategy is to express the ratio as an equivalent fraction. Then you can pull out all the tools you already have for working with fractions — for example, reducing, converting to decimals, and so forth.
A company has a total of 150 employees, 25 of whom are managers. What is the ratio of managers to non-managers?
(A) 1 to 3
(B) 1 to 4
(C) 1 to 5
(D) 1 to 6
(E) 2 to 5
The company has 25 managers, so the remaining 125 employees are non-managers. Express this ratio as a fraction and then reduce it:
The ratio of managers to non-managers is 1 to 5, so the correct answer is Choice (C).
One of the most practical applications of the ratio is a proportion, which is an equation based on a ratio. For example, if you know the ratio of boys to girls, you can express this as a fraction, set it equal to another fraction that includes a variable, and then solve. The following example illustrates how this concept works.
A summer camp has a boy-to-girl ratio of 8:11. If the camp has 88 boys, what is the total number of children at the camp?
Begin by setting up the proportion as the following equation:
Before continuing, notice that the ratio specifically mentions boys first and girls second, so this order is maintained in the equation. The camp has 88 boys, so substitute this number for Boys in the equation. You don’t know how many girls there are, so use the variable g. Here’s what your equation now looks like:
To find out how many girls are at the camp, solve for g using algebra, which I discuss in greater detail in Chapter 5. First, cross-multiply to get rid of the two fractions:
88(11) = 8(g)
968 = 8g
Now divide both sides by 8:
121 = g
The camp includes 121 girls and 88 boys, so you know it has a total of 209 children; therefore, the correct answer is Choice (K).
Working with Powers and Square Roots
When you raise a number to a power (also called an exponent), you multiply that number by itself a given number of times. Powers and roots — the inverse operation to powers — are important operations to understand for the ACT. In this section, I walk you through a variety of practice problems using these skills.
Harnessing the power of exponents
When you raise a number to a power, you multiply that number by itself repeatedly. For example:
34 = 3 × 3 × 3 × 3 = 81
Raising a number to a power involves two numbers: The base is the number that’s multiplied and the exponent is the number of times that it’s multiplied. You can use any number as a base, including negative numbers and fractions. For example:
(–2)5 = (–2)(–2)(–2)(–2)(–2) = –32
A minus sign in the exponent changes the result to the reciprocal of the result — that is, the numerator (top number) and denominator (bottom number) are reversed. Or, if a number has no denominator, the numerator becomes the denominator. Consider these examples:
When you raise a number to a fractional exponent, take the numerator as a power and the denominator as a square root. The most common fractional exponent is , which equals a square root. For example:
If , how many different integers could k equal?
(E) More than 3
To begin, evaluate the two powers:
You can rewrite the inequality as follows:
Therefore, k could equal 1, 2, or 3, so the correct answer is Choice (D).
Be square: Squaring and square roots
Squares and square roots are common on the ACT. In this section, I show you how to work with them to answer basic ACT questions.
When you square a number, you multiply it by itself. The symbol that denotes a square number is a superscript 2. For example:
42 = 4 × 4 = 16
It’s a good idea to be familiar with the square numbers up to 100. Table 4-2 can help you bone up on them.
The product of two consecutive square numbers is 900. What is the sum of these two numbers?
A quick comparison of values for the first ten square numbers shows you that the pair of numbers is 25 and 36, because 25 × 36 = 900. Their sum is 25 + 36 = 61, so the correct answer is Choice (C).
The inverse operation to squaring is taking a square root (also called a radical). When you take a square root of a number, you find a value that, when multiplied by itself, produces the number you started with. For example:
because 4 × 4 = 16
Whenever you take the square root of a positive number, two different answers are produced: one positive and the other negative. So the following also is true:
because –4 × –4 = 16
In many cases, an ACT question will specify the positive value of a square root or otherwise rule out the negative value.
Taking the square root of a negative number produces an imaginary number, which is outside the set of real numbers.
Although ACT questions may include imaginary numbers, they’re rare. Your ACT probably will have no more than one question that includes imaginary numbers. (For more on working with imaginary numbers, see Chapter 11.)
If x is divisible by 4 and , what is the value of x2?
The value of could be either 7 or –7, but in this case only the positive value satisfies the inequality, because 0 < x:
0 < x < 7
The variable x is divisible by 4, and 4 is the only number between 0 and 7 that’s divisible by 4, so x = 4. Thus, x2 = 42 = 16, so the correct answer is Choice (C).
Real-World Math: Studying Statistics
Statistics is the study of numerical information, called data, that’s observed in the real world. This data often is presented in tables and graphs, so in this section I show you how to work with these visual representations. I also show you how to calculate the mean, median, and mode (which are three types of averages) of data sets.
Presenting data in tables and graphs
Statistics deals with real-world information called data sets. Two common ways to present data sets are tables and graphs — and the ACT test writers make great use of these in their questions. So in this section, I show you how to read and answer a variety of ACT questions that involve tables and graphs.
Organizing information in tables
A table is a two-dimensional grid containing data that’s organized into horizontal rows and vertical columns. A typical ACT question that includes a table may ask you to compare or calculate data, or it may ask you to fill in a missing value in the table. Often, one table will be used for more than one question.
Use the following table, which provides the gross sales amounts among six salespeople for four months (January through April), to answer the following two questions.
What is the total gross sales for Ronaldo’s team in February?
Ronaldo’s team includes Beth, Carly, and Jefferson, whose February sales totaled $72,000 + $62,000 + $61,000 = $195,000. Therefore, the correct answer is Choice (C).
The first quarter is January through March, inclusive. If the quarterly sales quota is $180,000, which of the following salespeople failed to make his or her first-quarter quota?
You can add up the numbers for the first three months for each of the five salespeople, but there’s a quicker way. Notice that the quota of $180,000 over three months requires a monthly average of $60,000. A quick look at the table shows that Patrick made less than $60,000 for two of the three months, so start with him. Patrick’s sales for January through March were $49,000 + $59,000 + $70,000 = $178,000, so the correct answer is Choice (K).
Representing data with graphs
A graph is a visual representation of a data set. Some common types of graphs include the following:
Bar graphs, which show comparisons among numerical data
Pie charts, which show data represented as percentages of a whole
Line graphs, which show changes that occur over time
Scatterplots, which show general data trends by providing a large variety of data in two dimensions
Pictograms, which show comparisons among numerical data in discrete increments
Graphs tend to be self-explanatory — that’s why they’re so useful. If you’re clear on what the numbers in a graph mean, you’ll probably be able to answer just about any question that the ACT throws at you.
In this section, I provide questions that include a variety of graphs. Additionally, the questions in Chapter 6 give you more practice working with graphs.
The following pictogram shows the population of five neighboring counties. Which county has a population of 10,000?
Each figure represents 5,000 people, so two figures represents 10,000. As you can see, Brandt county has 10,000 people, so the correct answer is Choice (B).
The following bar graph shows the number of pots of coffee brewed by the Filterfresh Coffeehouse on each day last week. On which day did they brew exactly 17 pots of coffee?
The coffeehouse brewed exactly 17 pots of coffee on Thursday, so the correct answer is Choice (J).
The following pie chart shows how $500,000 of funding for an organization will be allocated. Which of the following pairs of services will account for exactly $275,000 of the funding allocated?
(A) Children’s services and housing
(B) Counseling and children’s services
(C) Counseling and housing
(D) Counseling and medical services
(E) Housing and medical services
Because the pie chart divides the $500,000 by percentages, you need to figure out how much each slice of the pie is worth in dollars. Children’s services and counseling are both 20%, so each of these slices represents $100,000. Housing is 25%, so this slice represents $125,000. Medical services represents the remaining funding, so it represents $175,000. The combination of counseling and medical services represents $100,000 + $175,000 = $275,000, so the correct answer is Choice (D).
Getting a better-than-average grasp of averages
ACT questions involving statistics often require you to calculate one of three types of averages for a data set: the mean, the median, and the mode. I show you how to calculate all three types of averages in this section.
Finding the meaning of the mean
The most common type of average is the mean (also called the arithmetic mean and the mean average). When an ACT question asks for an average and doesn’t specify which type, assume that the question is asking for the mean.
To calculate the mean of a list of numbers, use the following formula:
Sasha bowls every week in a bowling league. Each week, she bowls a three-game series and then calculates the average score for her three games. This week she bowled 165 on her first game and 175 on her second. After the third game, she calculated her average score at 172. What was her score on the third game?
In this question, the mean is 172 and the number of values (games played) is 3. Two of the three values are 165 and 170, and the third value is unknown, so let x equal this value:
To find the remaining value, simplify and solve for x (for more on solving equations with algebra, see Chapter 5):
Sasha bowled 176 on her third game, so the correct answer is Choice (E).
In some cases, you may need to find the average of a set of values that includes one or more variables. The following example shows you how.
If the average (arithmetic mean) of p, 9p, 10p, and 19 is 6, what is the value of p?
In this case, the mean is 6 and you’re given all four values, so you simply plug these into the formula:
Simplify and solve for p (refer to Chapter 5 as necessary to get help with the algebra):
As you can see, the correct answer is Choice (J).
Centering on the median
Another important type of average is the median, which is the middle number in a list of numbers that are in sequential order. When a list contains an even number of values, the median is the arithmetic mean of the two middle numbers.
The Garfield Alternative School has six classrooms for students from kindergarten through fifth grade. The current enrollment for these six classes is shown in the following table. What is the median number of students in the six classes?
Place the six values in ascending order: 23, 24, 26, 30, 31, 33. The two middle values are 26 and 30, so you have to find the mean of these two numbers:
The median is 28, so the correct answer is Choice (D).
Discovering what’s so popular about the mode
A third type of average is the mode, which is the most frequently-appearing number in a list of numbers. To find the mode, you simply have to count which number occurs most in a list. Pretty easy, huh?
Geoff played golf every day last week. His scores were 85 on Sunday, 83 on Monday, 85 on Tuesday, 82 on Wednesday, 86 on Thursday, 83 on Friday, and 83 on Saturday. What were his median and mode scores for the week?
(A) Median = 83, Mode = 83
(B) Median = 83, Mode = 84
(C) Median = 83, Mode = 85
(D) Median = 84, Mode = 83
(E) Median = 84, Mode = 85
To find the median, put the seven scores in order from lowest to highest, like so:
82, 83, 83, 83, 85, 85, 86
The median score, as I explain in the previous section “Centering on the median,” is the number that appears in the middle of the list. Here in this problem it’s 83. The mode score is the one that appears most frequently in the list, so it’s also 83. Thus, the correct answer is Choice (A).
Looking at Likelihood: Probability
Probability is the mathematical likelihood that something will happen. ACT questions about probability usually involve certain common scenarios such as tossing a coin, picking a card, or choosing an item at random. I help you become familiar with probability in the following sections.
Learning to count
Calculating probability requires a systematic way to count outcomes, which are the possible ways in which a set of events can take place. For example, if you toss four coins, one possible outcome is that all four come up heads. All four of these events are independent because what happens to one coin doesn’t affect any of the others.
But if you pick three letters from a bag that contains one copy of each letter from A to Z, one possible outcome is picking the letters A, M, and Z. In this case, however, the three events are dependent, because if you pick A first, you can’t also pick it second or third.
In the following sections, I show you how to count outcomes of both independent events (events that don’t affect each other) and dependent events (events that affect each other).
Counting the outcomes of independent events
When two (or more) events are independent, the outcome of one event has no effect on the outcome of the other. Tossing two coins and rolling a pair of dice are considered independent events, because what happens to one coin or die doesn’t affect what happens to the other. Counting independent events is an important first step to calculating probability, and it can also help you directly when answering some ACT questions.
If you toss ten coins into the air, in how many distinct ways can they land?
(A) Fewer than 100
(B) Between 100 and 200
(C) Between 200 and 500
(D) Between 500 and 1,000
(E) More than 1,000
In this question, how each coin lands (either heads or tails) is an event. These events don’t affect each other — that is, how one coin lands has no effect on any of the other coins — so they’re independent events. To count the number of possible outcomes for these ten events, track each coin separately as follows:
Each coin, #1 through #10, can land in one of two ways: either heads or tails. To calculate the number of possible outcomes, multiply 2 by itself 10 times. A fast way to do this on your calculator is 210:
210 = 1,024
The correct answer is Choice (E).
Counting the outcomes of dependent events
When two (or more) events are dependent, the outcome of one event affects the outcome of the other. Picking letters from a bag and socks from a drawer are dependent events: The outcome of the first event affects the outcome of the second event, because after you pick a letter or sock, you can’t pick the same one again. Identifying dependent events and counting them correctly is important for calculating probability.
Alex Ward Patton noticed that his own initials (AWP) contain no repeated letters, but the initials of his best friend, James Dean Jackson (JDJ), contain repeated letters. How many different sets of three initials have no repeating letters?
In this question, each initial is a different event. Because no initial can be repeated, each event affects the other events. For example, if a first initial is A, then the second initial can’t also be A. Thus, the number of possible first initials is 26, the number of possible second initials is 25, and the number of possible third initials is 24.
To find the number of possible outcomes, you simply multiply these three numbers together:
26 × 25 × 24 = 15,600
You can see that the correct answer is Choice (C).
Probability, the mathematical likelihood of an outcome, is always a number from 0 to 1. When the probability of an outcome is 0, the outcome is impossible; when the probability is 1, the outcome is certain.
Here’s the formula for probability:
In this formula, you replace target outcomes with the number of ways in which a specific outcome occurs. You replace total outcomes with the number of ways in which any outcome can occur.
If you toss two coins, what is the probability that at least one of them will land heads up?
To find the total number of outcomes, count the number of events. When doing so, you discover that the first coin has two and the second coin has two. Now multiply these numbers together (pretty tough, isn’t it?):
2 × 2 = 4
So you can assume 4 total outcomes, as follows:
HH HT TH TT
Of these, 3 outcomes have at least one heads, so the target outcomes is 3. Plug these numbers into the formula for probability:
The probability that at least one coin lands heads up is three-fourths, so the correct answer is Choice (E).
If a bag contains four white socks and four black socks, what is the probability of pulling three black socks at random from the bag?
To calculate the probability, you need to calculate the total outcomes and the target outcomes. Keep in mind that the question involves dependent events, because if you pull a certain sock out of the bag first, you can’t pull it out of the bag second or third.
First, count the total outcomes: You can pull eight possible socks out first, any of the remaining seven socks second, and any of the remaining six socks third. Multiply to find the total outcomes:
8 × 7 × 6 = 336
So this scenario has a total of 336 possible outcomes.
Next, count up the target outcomes: You can pull any of the four black socks out first, any of the remaining three black socks second, and either of the remaining two black socks third. Multiply to find the target outcomes:
4 × 3 × 2 = 24
Now, to get your answer, simply plug the numbers into the formula for probability:
The correct answer is Choice (H).