CliffsNotes CBEST - BTPS TESTING Ph.D., Jerry Bobrow Ph.D. & 8 more 2021

Specific strategies with examples
Introduction to the mathematics section

A number of different approaches can be helpful in solving mathematics problems. To be successful on the exam, read the question entirely, try not to overthink the question, write down (or circle) key words, translate the words into a numeric question, and always check your work.

Note: This section provides specific strategies to help you approach math problems. It is not a content review of math topics. Although explanations are provided below, you may need to refer to chapters 4—7 to review specific math topics related to the examples. For example, Example 3 is an algebra problem. If you need further algebra instruction to answer the problem, refer to Chapter 5, “Algebra Review,” for further clarification.

Now, let’s take a look at specific test-taking strategies to prepare for the CBEST.

1. Use reasoning

2. Write down key words

3. Pull out information

4. Work from the answer choices

5. Plug in simple numbers

6. Approximate

7. Make comparisons

8. Show procedural steps

9. Analyze statistical data

10. Find the “possibilities” in probability questions

11. Mark or draw diagrams

12. Interpret graphs

Use Reasoning

It’s important to remember that the CBEST math section focuses on your ability to reason quantitatively using word problems. Sometimes word problems can be confusing because you are required to take words and translate them into a math problem, but using reasoning skills will help you solve the problems. Pay attention to the words used, their meaning, and how they are connected. See Chapter 7 for additional information about word problems.

Examples:

1. If the price of apples is changed from two dozen for $5 to three dozen for $6, how many more apples can be purchased for $30 now than could be purchased before?

A. 36

B. 48

C. 72

D. 144

E. 180

Reasoning can help you solve this question. Let’s start by looking at the first part of the problem. If the price of apples is $5 for two dozen, then $30 will allow you to purchase six groups of two dozen apples, or 6 × 2 × 12 = 144 apples.

The second part of the problem shows that when the price of apples is changed to three dozen for $6, then $30 will allow you to purchase five groups of three dozen apples, or 5 × 3 × 12 = 180 apples.

Now let’s put these two together and look at the question again. “How many more apples can be purchased for $30 now than could be purchased before?” Therefore, 180 — 144 = 36 more apples can be purchased now. The correct answer is choice A.

TEST TIP: In the next example, if you immediately recognize the method or proper formula to solve the question, use reasoning skills to solve the problem.

2. Which of the following numbers is between and ?

A. 0.45

B. 0.35

C. 0.29

D. 0.22

E. 0.20

First circle (or write down) to help you stay focused on the question. A quick peek at the answer choices tips you off that you should work in decimals (not fractions). If you know that is 0.25 and is 0.333..., you have insight into the problem and should be able to solve it. Because 0.29 is the only answer choice between 0.25 and 0.333..., it is reasonable to assume that 0.29 is the correct answer. The correct answer is choice C.

3. Robert works at a clothing store and folds shirts on Monday and Tuesday. On Monday, he folds 22 more shirts than he folds on Tuesday. If Robert folds a total of 88 shirts in 2 days, how many shirts were folded on Tuesday?

A. 22

B. 33

C. 66

D. 99

E. 110

First, circle (or write down) how many shirts were folded on Tuesday. Using reasoning, you can immediately eliminate choices D and E because they are greater than 88 shirts (the total for both days). Choice C is also an unreasonable answer because if Robert folds 66 shirts on Tuesday, and 22 more than the 66 on Monday, that is 88 shirts on Monday alone, which equals the total for the 2 days. The only possible answer is choice A or B.

To solve the problem, let x be the number of shirts Robert folds on Tuesday; then x + 22 is the number of shirts he folds on Monday. So the equation looks like this:

Solving gives you:

2x + 22 = 88

Subtract 22 from each side:

Dividing by 2 leaves:

Is choice B, 33, reasonable? Well, if Robert folds 33 shirts on Tuesday, and 22 more than 33 on Monday, he folded 55 shirts on Monday. So the total of Monday and Tuesday is 33 + 55 = 88. Yes, it is reasonable. It checks. The correct answer is choice B.

Write Down Key Words

Writing down key words is an effective test-taking technique, especially for math word problems. Pulling out and writing down information from the word problem can often give you additional insight and help you focus on precisely what you are being asked to calculate.

· Computer-administered test-takers: You will be given scratch paper or a writing board to write down important key words. Remember to write down just one or two key words; any more than that will waste valuable testing time.

· Paper-based test-takers: Circling and/or underlining key words in questions is an effective test-taking technique because it focuses your attention on what is being asked in the problem. Keep in mind that you are allowed to mark and write on your test booklet.

Examples:

4. If 3 yards of fabric cost $12.96, what is the price per foot?

A. $1.44

B. $4.00

C. $4.32

D. $5.32

E. $12.96

The key word here is foot. It is important to immediately make a note of the word “foot” to help you stay focused on what is being asked in the problem. Dividing $12.96 by 3 will tell you only the price per yard. By quickly glancing over the question and not writing down key words, you may have selected choice C. This is because $12.96 divided by 3 equals $4.32. However, choice C is incorrect because you need to divide by 3 again (since there are 3 feet per yard) to find the cost per foot. Choice B is incorrect because it’s an approximation of $4.32, and choices D and E are both incorrect. To solve the problem:

$12.96 ÷ 3 yards = $4.32 per “yard”

$4.32 ÷ 3 feet = $1.44 per “foot”

The correct answer is choice A.

5. Together a bat and glove cost $110. If the bat costs $40 more than the glove, what is the cost of the bat?

A. $35.00

B. $40.00

C. $70.00

D. $75.00

E. $110.00

Circling or writing down the key words can help you stay focused on what is being asked. You should have immediately eliminated choice E because $110 is the total cost of the bat and glove. The key words here are cost of the bat, so mark or write down these words. Let x = cost of the glove. Then x + 40 = cost of the bat (cost is $40 more than the glove), eliminating choices A and B. Together they cost $110.

Solving this question algebraically:

The equation shows you that the cost of the glove is $35. Therefore, since x = $35, then the cost of the bat is x + 40 = $75. Always answer the question that is being asked. The correct answer is choice D.

6. Dan is 18 years old. He works for his father for of the year, and he works for his brother for the rest of the year. What is the ratio of the time per year that Dan spends working for his brother to the time he spends working for his father?

A.

B.

C.

D.

E.

The key words rest of the year point to the answer:

Remember, the way you write the ratio is important to finding the correct solution. Find the ratio of to .

Note that you don’t need Dan’s age to solve the problem. The correct answer is choice B.

Pull Out Information

Pulling information out of the word problem can make the problem more workable and give you additional insight into the problem. Pull out the given facts and identify which of those facts will help you solve the problem. Notice in the examples that follow, not all facts will always be needed!

Examples:

7. Jason is 10 years older than his sister. If Jason were 25 years of age in 2018, in what year could he have been born?

A. 1983

B. 1993

C. 1994

D. 1998

E. 2003

The key words here are in what year and could he have been born. Thus, the solution is 2018 — 25 = 1993. Notice that you should pull out the information 25 years of age and in 2018. Jason’s age in comparison to his sister’s age is not relevant, and therefore, should not be pulled out. The correct answer is choice B.

8. An employee’s annual salary was increased by $15,000. If her new annual salary now equals $90,000, what is the percent increase?

A. 15%

B.

C. 20%

D. 22%

E. 24%

Mark or write down what you are looking for—in this case, percent increase.

Now pull out information. If the employee’s salary was increased by $15,000 to $90,000, then the starting salary was $90,000 — $15,000 = $75,000. Therefore, . The correct answer is choice C.

TEST TIP: When pulling out information, write the numbers and/or letters to put them into a helpful format and eliminate some of the wording. The next two examples present ratio questions. Practice this type of question as often as possible so that you’re comfortable with solving ratio questions on the CBEST.

9. If a mixture is alcohol by volume and water by volume, what is the ratio of the volume of alcohol to the volume of water in this mixture?

A.

B.

C.

D.

E.

The first bit of information to pull out is what you’re looking for: ratio of the volume of alcohol to the volume of water. You should have circled or written this information.

Rewrite the question as A (alcohol): W (water) and then insert this into a ratio: .

Next, pull out the volumes of each: . Now, you can figure out the answer by inspection or substitution using . Invert the bottom fraction and multiply to get . The ratio of the volume of alcohol to the volume of water is 3 to 4. The correct answer is choice C.

10. If the ratio of boys to girls in a drama class is 2 to 1, which of the following is a possible number of students in the class?

A. 10

B. 16

C. 19

D. 25

E. 30

First, write down the words possible number of students. Pulling out this information gives you the following: b:g = 2:1 (the ratio of boys to girls).

Because the ratio of boys to girls is 2:1, the possible total number of students in the class must be a multiple of 2 + 1 (boys plus girls), or 3. The multiples of 3 are 3, 6, 9, 12, 15, and so on. Using reasoning, only 30 is a multiple of 3. The correct answer is choice E.

Work from the Answer Choices

If you don’t immediately recognize a method or formula, or if using the method or formula would take a great deal of time, try working backward from the answer choices. Because the answer choices are usually given in ascending or descending order, always start by plugging in choice C first. Then you know whether to go up or down with your next try. (In some cases, you may want to plug in the simplest answer first.)

Examples:

11. If , what is the value of x?

A. —2

B. —1

C. 0

D. 1

E. 2

You should first underline or write down value of x. If you’ve forgotten how to solve this kind of equation, work backward by plugging in answer choices. Start with choice C and plug in 0:

Using reasoning, you can see that 0 is too small. Let’s try choice D, a larger number. Plugging in 1 gives you:

This answer is true. Working from the answer choices is sometimes a valuable technique. The correct answer is choice D.

12. Find the counting number that is less than 15, and when divided by 3 has a remainder of 1, but when divided by 4 has a remainder of 2.

A. 5

B. 8

C. 10

D. 12

E. 13

By working from the answer choices, you can eliminate wrong answer choices. For example, you can eliminate choices B and D immediately because they are divisible by 4, leaving no remainder. Choices A and E can also be eliminated because they each leave a remainder of 1 when divided by 4. Check choice C:10 leaves a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4. The correct answer is choice C.

13. What value for x makes the following expression a true statement?

x + x + 20 is greater than 80

A. 15

B. 22

C. 25

D. 27

E. 35

When working from the answer choices, start in the middle with choice C because the answer choices are in ascending or descending order. When using this strategy, you should be able to quickly recognize that the correct answer is 25 (choice C), less than 25 (choices A and B), or greater than 25 (choices D and E). Check choice C:

Choice C, 25 + 25 + 20 = 70. 70 is not greater than 80, therefore, choice C is incorrect.

In this case, the correct answer must be greater than 25, so you can eliminate choices A, B, and C. Now check choices D and E:

Choice D, 27 + 27 + 20 = 74. 74 is not greater than 80, therefore, choice D is incorrect.

Choice E, 35 + 35 + 20 = 90. 90 is greater than 80, therefore, the correct answer is choice E.

Plug In Simple Numbers

Substituting numbers for variables can often help you understand a problem. Remember to plug in simple, small numbers (e.g., 1, 2, 3, 4, or 5) because you have to do the work.

Examples:

14. If x is a positive integer in the equation 2x = y, then what must y be?

A. A positive even integer

B. A negative even integer

C. Zero

D. A positive odd integer

E. A negative odd integer

At first glance, this problem appears quite complex. But let’s plug in some simple numbers and see what happens. For example, first plug in 1 (the simplest positive integer) for x.

Now try 2:

Try it again. No matter what positive integer is plugged in for x, y is always positive and even. Therefore, the correct answer is choice A.

TEST TIP: Some problems may deal with percent or percent change. If you don’t see a simple method for working the problem, try plugging in the values of 10 or 100 and see what you get.

15. If 40% of the students in a class have brown eyes and 20% of those with brown eyes have brown hair, then what percent of the original total number have brown hair and brown eyes?

A. 4%

B. 8%

C. 16%

D. 20%

E. 32%

First, make note of percent of the original total number, brown hair, and brown eyes. In this problem, if you don’t identify a simple method, start with 100 students in the class. Because 40% of the students have brown eyes, then 40 students have brown eyes. Now, the problem says that 20% of those students with brown eyes have brown hair. So take 20% of 40, which gives you:

0.20 × 40 = 8

Because the question asks what percent of the original total number have brown eyes and brown hair, and because you started with 100 students, the answer is 8 out of 100, or 8%. The correct answer is choice B.

16. A corporation triples its annual bonus to 50 of its employees. What percent of the employees’ new bonus is the increase?

A. 50%

B.

C. 100%

D. 200%

E. 300%

Use $100 for the normal bonus. If the annual bonus was normally $100, tripled it is now $300. Therefore, the increase, $200, must be placed over the new bonus of $300:

The correct answer is choice B.

TEST TIP: Be sure to make logical substitutions. Use the appropriate type of number—a positive number, a negative number, zero, and so on—to get the full picture.

Approximate

Some questions require accurate computations. For others, estimation or approximation (or rounding off) may be all you need to arrive at the correct answer. It may be useful to look at the answer choices and see how close together or far apart they are. This will guide you in determining how close your approximation needs to be to choose the correct answer.

Examples:

17. Tia is trying to round the number 4,547 to the nearest hundred. If she gets the right answer, she receives extra credit. What answer must Tia get to receive extra credit?

A. 5,000

B. 4,600

C. 4,550

D. 4,540

E. 4,500

To round to the nearest hundred, you must check to see whether the number in the tens place is 5 or more. If it is less than 5, which it is in this case, let the hundreds place stand as is and put zeros in the tens and ones place, respectively—4,500. The correct answer is choice E.

18. What is the value of to the nearest tenth?

A. 49.1

B. 17.7

C. 4.9

D. 4.63

E. 0.5

First, underline or write down what you are looking for: nearest tenth. Before starting any computations, glance at the answers to see how far apart they are. Notice that choices A and E are not reasonable and that the only close choices are C and D. However, D is not a possible choice because it is to the hundredth place, not tenth. Making some quick approximations—0.889 ≈ 1 and 9.97 ≈ 10—leaves the problem in this form:

The closest answer is 4.9. The correct answer is choice C.

19. If 2,100 employees work in a factory but only 21% work the night shift, approximately how many people work the other shifts?

A. 400

B. 800

C. 1,100

D. 1,600

E. 2,000

First, underline or make note of approximately how many people and other shifts. Remember, you’re looking for other shifts. Notice that the answer choices are spread out.

Now, approximate 21% as 20%. Next, approximate 2,100 people as 2,000 people. So, 20% of 2,000 is 0.20 × 2,000 = 400.

But be careful; 400 is the number of those who work only on the night shift, and you want the number of people who work the other shifts. So subtract 400 from 2,000, leaving 1,600. Another method is to approximate, then subtract 20% from 100%, to get 80%, which is the percent of other workers. Multiply 80% times 2,000 and you get 1,600. The correct answer is choice D.

Make Comparisons

At times, questions will require you to compare the sizes of several decimals or of several fractions. If decimals are being compared, make sure that the numbers being compared have the same number of digits. (Remember that zeros to the far right of the decimal point can be inserted or eliminated without changing the value of the number.)

Examples:

20. Put these numbers in order from smallest to largest: .

A.

B.

C.

D.

E.

First, make a mental note (or write down) smallest to largest, then rewrite 0.6 as 0.60 so all of the decimals now have the same number of digits:

Now treat these as though the decimal point were not there (this can be done only when all of the numbers have the same number of digits to the right of the decimal). Put in numeric order:

Note: A quick method is to look for the largest place value. In this case it would be the tenths place. Since 1 is the smallest number, we can rule out choices A, B, D, and E.

The correct answer is choice C.

21. Which of the following answer choices places the fractions in order from smallest to largest?

A.

B.

C.

D.

E.

Find the common denominator: . Therefore, the order becomes . Or you can use decimal equivalents:

The order becomes . The correct answer is choice C.

22. Which of the following is equal to of 0.02%?

A. 0.4

B. 0.04

C. 0.004

D. 0.0004

E. 0.00004

Simplifying this problem first means changing to 0.2. Next, change 0.02% to 0.0002 (that is, 0.02 × 0.01 = 0.0002).

Now that you have simplified the problem, multiply 0.2 × 0.0002, which gives 0.00004. The correct answer is choice E.

Show Procedural Steps

Some problems may not ask you to solve for a numerical answer or even an answer including variables. Rather, you may be asked to set up the equation or expression without doing any solving. A quick glance at the answer choices can help you know what is reasonably expected.

Examples:

23. 51 × 6 could be quickly mentally calculated by

A. 50 × 6 + 1

B. 51 + 51 + 51 + 51 + 51 + 51

C. (50 × 6) + (1 × 6)

D.

E. adding 51 sixes

The quickest method of calculating 51 × 6 is to first multiply 50 × 6 (resulting in 300), then multiplying 1 × 6 (resulting in 6), and adding the products together: 300 + 6 = 306. Answer choices B and E give the correct answer (306) as well, but neither is the best way to quickly calculate the answer. The correct answer is choice C.

TEST TIP: Let’s try a similar problem; however, as this next example demonstrates, it is sometimes necessary to work the problem.

24. The fastest method to solve is to

A. invert the second fraction and then multiply.

B. multiply each column across and then reduce to lowest terms.

C. find the common denominator and then multiply across.

D. divide 7 into the numerator and denominator, divide 6 into the numerator and denominator, and then multiply across.

E. reduce the first fraction to lowest terms and then multiply across.

In this problem, the way to determine the fastest procedure may be to actually work the problem as you would if you were working toward an answer. Then see whether that procedure is listed among the choices. You should then compare it to the other methods listed. Is one of the other correct methods faster than the one you used? If so, select the fastest.

These types of problems are not constructed to test your knowledge of obscure ways to solve mathematical equations. Rather, they test your knowledge of common procedures used in standard mathematical equations. Thus, the fastest way to solve this sample problem is to first divide 7 into the numerator and denominator:

Then divide 6 into the numerator and denominator:

Then multiply across:

The correct answer is choice D.

25. Which of the following equations can be used to find the perimeter, P, of a rectangle that has a length of 18 feet and a width of 15 feet?

A. P = (18)(15)

B. P = 18 + 15

C. P = 2(15)(18)

D. P = 2(15) + 18

E. P = 2(15 + 18)

The perimeter of a rectangle can be found by adding the length to the width and doubling this sum: P = 2(15 + 18). The correct answer is choice E.

Analyze Statistical Data

Basic statistical questions on the CBEST appear in a unique format compared to other math questions. A solid understanding of arithmetic concepts in Chapter 4 (fractions and proportions) will help you understand standardized test scores (e.g., stanine scores). Also see Chapter 6 for a review of basic statistics.

Examples:

Question 26 refers to the following table.

Mason’s Psychology Exams: Fall Semester

26. What is the mean score on Mason’s fall semester psychology exams?

A. 87%

B. 88%

C. 89%

D. 90%

E. 91%

To find the mean, find the sum of all of Mason’s scores.

80 + 83 + 86 + 88 + 90 + 90 + 91 + 96 = 704

Next, divide the total sum of his scores (704) by the number of scores (8) to find his average score.

704 ÷ 8 = 88

The correct answer is choice B.

27. Kelsey took a standardized math test, and her grade report showed that her score was in the 89th percentile. Which of the following is the best interpretation of the meaning of this percentile?

A. She got 89% of the problems correct.

B. Eighty-nine percent of all those taking the exam had scores below Kelsey’s score.

C. Eighty-nine percent of all those taking the exam had scores above Kelsey’s score.

D. Only 89% of those taking the exam received passing scores.

E. There was an 89% probability that her score was above the mean on the exam.

The percentile associated with a score represents how many whole percent of all scores lie below that score. Therefore, if a score is in the 89th percentile, this means that 89% of all other scores are below that score. The correct answer is choice B.

Find the “Possibilities” in Probability Questions

Some questions on the CBEST involve probability. Probability questions predict the chances that a particular outcome will occur. If you have a strong knowledge of arithmetic concepts (fractions and proportions), you should be successful on these types of questions. If you can’t remember a formal method, try some reasonable “possibilities” by setting up possible combinations. Remember to set up only what is necessary to solve the problem. See pp. 128—131 in Chapter 6 for a review of probability.

Examples:

Questions 28—30 refer to the following scenario.

Ryan is pulling marbles out of a hat. There are 6 red marbles, 4 green marbles, and 2 yellow marbles. There are no other marbles in the hat.

28. What is the probability that Ryan will pull a green marble on his first pull?

A.

B.

C.

D.

E.

Because there are 12 total marbles and 4 green marbles, the probability is . The correct answer is choice B.

29. What is the least number of marbles that Ryan can pull to ensure that he gets at least one of each color?

A. 3

B. 4

C. 5

D. 7

E. 11

To ensure getting at least one of each color, Ryan must pull 11 marbles. He could pull 6 reds and then 4 greens, so he would have pulled 10 marbles, and still only have two different colors. But on his next pull he would have to get a yellow because only yellow marbles would be left. The correct answer is choice E.

30. If Ryan pulls a green marble first and doesn’t put it back in the hat, what is the probability that his next pull will be another green marble?

A.

B.

C.

D.

E.

When the first green marble is pulled and not put back, it leaves 3 greens, 6 reds, and 2 yellows. So the probability of pulling another green marble is . The correct answer is choice D.

Mark or Draw Diagrams

When a figure is included with the problem, mark or draw the given facts on the diagram. This helps you visualize all the facts given. If no diagram is given, drawing a diagram to meet the conditions set by the word problem can often make the problem easier for you to work. Being able to see the facts is more helpful than just reading the words.

Examples:

Question 31 refers to the following diagram.

31. In the diagram, if each small square is 2 inches in length, what is the perimeter of the shaded area?

A. 20 inches

B. 24 inches

C. 40 inches

D. 48 inches

E. 60 inches

Mark in the diagram as you start counting from the lower left-hand corner. As you go around the outside of the shaded region, your markings can help you make sure that you don’t miss a side. You should have 20 numbered sides in the perimeter. But remember, each side is 2 inches long, so multiply 20 × 2 = 40 inches.

The correct answer is choice C.

32. What is the maximum number of pieces of birthday cake of size 4" by 4" that can be cut from a cake 20" by 20"?

A. 5

B. 10

C. 16

D. 20

E. 25

Sketching the cake and marking it as follows makes this a fairly simple problem.

Notice that five pieces of cake will fit along each side; therefore, 5 × 5 = 25. Finding the total area of the cake and dividing it by the area of one of the 4 × 4 pieces also gives you the correct answer, but beware of this method because it may not work if the pieces do not fit evenly into the original area. The correct answer is choice E.

Interpret Graphs

Some math questions are based on interpreting data provided in graphs, charts, and tables. To answer questions, you must accurately read and draw conclusions about visual graphic illustrations. Your familiarity with a wide range of graphic illustrations discussed in Chapter 7 (see “Graph Interpretation,” pp. 142—152) will help you answer these types of questions. Spend a few moments studying the title, labels/categories, and numeric values before reading the question.

· Title: The title always provides an overview of the graph.

· Labels/categories: Each category provides information about the whole picture.

· Numeric values: The visual illustration of each category quickly distinguishes variations in data (greatest and lowest numerical values).

Examples:

Questions 33—34 refer to the following graph.

33. If a grade of C or better is required to take the next level mathematics course, what percent of the students qualify?

A. 16%

B. 22%

C. 54%

D. 76%

E. 83%

The number of students who received a grade of C or better is:

28 + 49 + 189 = 266

Next, you should take 266 over the “total number of students.” Since , 76% of the students qualify to take the next level mathematics course. The correct answer is choice D.

34. What is the ratio of students who received a grade of B to the total number of students who completed the course?

A.

B.

C.

D.

E.

Since 28 students withdrew from the class and did not receive a letter grade, 350 − 28 = 322 students completed the course, of which 49 earned a grade of B.

The correct answer is choice A.

Question 35 refers to the following graph.

35. According to the bar graph, Thompson’s score exceeds Leslie’s score by how many points?

A. 1,225

B. 122.50

C. 12.25

D. 1.225

E. 0.1225

Thompson’s score was 7,488, and Leslie’s score was 6,263.

Therefore, 7,488 — 6,263 = 1,225. The correct answer is choice A.