COMPASS Exam - Bob Miller's Math Prep

Miller.

             CHAPTER 1: The Beginnings

Our great adventure begins with our basic terms. It is very important to understand what a question asks as well as how to answer it. The word numbers has many meanings, as we start to see here.

NUMBERS

Whole numbers: 0, 1, 2, 3, 4, …

Integers: 0, ±1, ±2, ±3, ±4, …, where ±3 stands for both +3 and −3.

Positive integers are integers that are greater than 0. In symbols, x > 0, x an integer.

Negative integers are integers that are less than 0. In symbols, x < 0, x an integer.

Even integers: 0, ±2, ±4, ±6, …

Odd integers: ±1, ±3, ±5, ±7, …

Inequalities

For any numbers represented by a, b, c, or d on the number line:

We say c > d (c is greater than d) if c is to the right of d on the number line.

We say d < c (d is less than c) if d is to the left of c on the number line.

c > d is equivalent to d < c.

a ≤ b means a < b or a = b; likewise, a ≥ b means a > b or a = b.

Example 1: 4 ≤ 7 is true because 4 < 7; 9 ≤ 9 is true because 9 = 9; but 7 ≤ 2 is false because 7 > 2.

Example 2: Find all integers between −4 and 5.

Solution: {−3, −2, −1, 0, 1, 2, 3, 4}.

Notice that the word between does not include the endpoints.

Example 3: Graph all the multiples of five between 20 and 40 inclusive.

Solution:

Notice that inclusive means to include the endpoints.

Odd and Even Numbers

Here are some facts about odd and even integers that you should know.

• The sum of two even integers is even.

• The sum of two odd integers is even.

• The sum of an even integer and an odd integer is odd.

• The product of two even integers is even.

• The product of two odd integers is odd.

• The product of an even integer and an odd integer is even.

• If n is even, n is even. If n² is even and n is an integer, then n is even.

• If n is odd, n² is odd. If n² is odd and n is an integer, then n is odd.

OPERATIONS ON NUMBERS

Product is the answer in multiplication, quotient is the answer in division, sum is the answer in addition, and difference is the answer in subtraction.

Because 3 × 4 = 12, 3 and 4 are said to be factors or divisors of 12, and 12 is both a multiple of 3 and a multiple of 4.

A prime is a positive integer with exactly two distinct factors, itself and 1. The number 1 is not a prime because only 1 × 1 = 1. It might be a good idea to memorize the first eight primes:

2, 3, 5, 7, 11, 13, 17, and 19

The number 4 has more than two factors: 1, 2, and 4. Numbers with more than two factors are called composites. The number 28 is a perfect number because if we add the factors less than 28, they add to 28.

Example 4: Write all the factors of 28.

Solution:     1, 2, 4, 7, 14, and 28.

Example 5: Write 28 as the product of prime factors.

Solution:     28 = 2 × 2 × 7.

Example 6: Find all the primes between 70 and 80.

Solution:     71, 73, 79. How do we find this easily? First, because 2 is the only even prime, we have to check only the odd numbers. Next, we have to know the divisibility rules:

• A number is divisible by 2 if it ends in an even number. We don’t need this here because then it can’t be prime.

• A number is divisible by 3 (or 9) if the sum of the digits is divisible by 3 (or 9). For example, 456 is divisible by 3 because the sum of the digits is 15, which is divisible by 3 (it’s not divisible by 9, but that’s okay).

• A number is divisible by 4 if the number named by the last two digits is divisible by 4. For example, 3936 is divisible by 4 because 36 is divisible by 4.

• A number is divisible by 5 if the last digit is 0 or 5.

• The rule for 6 is a combination of the rules for 2 and 3.

• It is easier to divide by 7 than to learn the rule for 7.

• A number is divisible by 8 if the number named by the last three digits is divisible by 8.

• A number is divisible by 10 if it ends in a zero, as you know.

• A number is divisible by 11 if the difference between the sum of the even-place digits (2nd, 4th, 6th, etc.) and the sum of the odd-place digits (1st, 3rd, 5th, etc.) is a multiple of 11. For example, for the number 928,193,926: the sum of the odd-place digits (9, 8, 9, 9, and 6) is 41; the sum of the even-place digits (2, 1, 3, and 2) is 8; and 41 −8 is 33, which is divisible by 11. So 928,193,926 is divisible by 11.

That was a long digression!!!!! Let’s get back to Example 6.

We have to check only 71, 73, 75, 77, and 79. The number 75 is not a prime because it ends in a 5. The number 77 is not a prime because it is divisible by 7. To see if the other three are prime, for any number less than 100 you have to divide by the primes 2, 3, 5, and 7 only. You will quickly find that 71, 73, and 79 are primes.

Rules for Operations on Numbers

() are called parentheses (singular: parenthesis); [ ] are called brackets; {} are called braces.

Rules for adding signed numbers

1. If all the signs are the same, add the numbers and use that sign.

2. If two signs are different, subtract them, and use the sign of the larger numeral.

Example 7:  a. 3 + 7 + 2 + 4 = +16

b. −3 − 5 − 7 − 9 = −24

c. 5 − 9 + 11 − 14 = 16 − 23 = −7

d. 2 − 6 + 11 − 1 = 13 − 7 = +6

Rules for multiplying and dividing signed numbers

Look at the minus signs only.

1. Odd number of minus signs—the answer is minus.

2. Even number of minus signs—the answer is plus.

Example 8:

Solution:   Five minus signs, so the answer is minus, namely −8.

Rule for subtracting signed numbers

The sign (−) means subtract. Change the problem to an addition problem.

Example 9:  a. (−6) − (−4) = (−6) + (+4) = −2

b. (−6) − (+2) = (−6) + (−2) = −8, because it is now an adding problem.

Order of Operations

In doing a problem such as 4 + 5 × 6, the order of operations tells us whether to multiply or add first:

1. If given letters, substitute in parentheses the value of each letter.

2. Do operations in parentheses, inside ones first, and then the tops and bottoms of fractions.

3. Do exponents next. (Chapter 3 discusses exponents in more detail.)

4. Do multiplications and divisions, left to right, as they occur.

5. The last step is adding and subtracting. Left to right is usually the safest way.

Example 10: 4 + 5 × 6 =

Solution:       4 + 30 = 34

Example 11: (4 + 5)6 =

Solution:       (9)(6) = 54

Example 12: 1000 ÷ 2 × 4 =

Solution:       (500)(4) = 2000

Example 13: 1000 ÷ (2 × 4) =

Solution:       1000 ÷ 8 = 125

Example 14: 4[3 + 2(5 − 1)] =

Solution:       4[3 + 2(4)] = 4[3 + 8] = 4(11) = 44

Example 15:

Solution:

Example 16: If x = −3 and y = −4, find the value of:

a. 7 − 5x − x²

b. xy² − (xy)²

c.

Solutions:     a. 7 − 5x − x² = 7 − 5(−3) − (−3)² = 7 + 15 − 9 = 13

b. xy² − (xy)² = (−3)(−4)² − ((−3)(−4))² = (−3)(16) − (12)²

                        = −48 − 144 = −192

c.

Before we get to the exercises, let’s talk about ways to describe a group of numbers (data).

DESCRIBING DATA

Four of the measures that describe data are necessary to know. The first three are measures of central tendency; the fourth, the range, measures the span of the data.

Mean: Add up the numbers and divide by how many numbers you have added up.

Median: Middle number. Put the numbers in numeric order and see which one is in the middle. If there are two middle numbers, which happens with an even number of data points, take their average.

Mode: Most common numbers—those that appear the most times. A set with two modes is called bimodal. There can actually be any number of modes.

Range: Highest number minus the lowest number.

Example 17: Find the mean, median, mode, and range for 5, 6, 9, 11, 12, 12, and 14.

Solutions:    Mean:

Median: 11

Mode: 12

Range: 14 − 5 = 9

Example 18: Find the mean, median, mode, and range for 4, 4, 7, 10, 20, 20.

Solutions:

Median: For an even number of points, it is the mean of the middle two:

Mode: There are two: 4 and 20 (blackbirds?)

Range: 20 − 4 = 16

Example 19: Jim received grades of 83 and 92 on two tests. What grade must the third test be in order to have an average (mean) of 90?

Solution:       There are two solution methods.

Method 1: To get a 90 average on three tests, Jim needs

3(90) = 270 points. So far, he has 83 + 92 = 175 points. So Jim needs

270 − 175 = 95 points on the third test.

Method 2 (my favorite): 83 is −7 from 90. 92 is + 2 from 90.

−7 + 2 = −5 from the desired 90 average. Jim needs

90 + 5 = 95 points on the third test. (Jim needs to make up the 5-point

deficit, so add it to the average of 90.)

The second method is my choice, but it is always your choice which method you can do easier and faster.

Finally, after a long introduction, we get to some multiple-choice exercises.

Exercise 1: If x = −5, the value of − 3 − 4x − x² is

A. −48

B. − 8

C. 2

D. 13

E. 4

Exercise 2:

A. 0

B. −2

C. −4

D. −6

E. Undefined

Exercise 3: The scores on three tests were 90, 91, and 98. What does the score on the fourth test have to be in order to get exactly a 95 average (mean)?

A. 97

B. 98

C. 99

D. 100

E. Not possible

Exercise 4: On a true-false test, 20 students scored 90, and 30 students scored 100. The sum of the mean, median, and mode is

A. 300

B. 296

C. 295

D. 294

E. 275

Exercise 5: On a test, m students received a grade of x, n students received a grade of y, and p students received a grade of z. The average (mean) grade is:

A.

B.

C.

D.

E.

Exercise 6: The largest positive integer in the following list that divides evenly into 2,000,000,000,000,003 is

A. 33

B. 11

C. 10

D. 3

E. 1

For Exercises 7–9, use the following numbers: 8, 10, 10, 16, 16, 18

Exercise 7: The mean is

A. 8

B. 10

C. 13

D. 16

E. There are two of them.

Exercise 8: The median is

A. 8

B. 10

C. 13

D. 16

E. There are two of them.

Exercise 9: The mode is

A. 8

B. 10

C. 13

D. 16

E. There are two of them.

Sometimes statistics are given in frequency distribution tables, such as this one showing the grades Sandy received on 10 English quizzes.

This chart is for Exercises 10–12.

Exercise 10: The mean is

A. 96

B. 97

C. 98

D. 99

E. 100

Exercise 11: The median is

A. 96

B. 97

C. 98

D. 99

E. 100

Exercise 12: The mode is

A. 96

B. 97

C. 98

D. 99

E. 100

Let’s look at the answers.

Answer 1:   B: −3 − 4(−5) − (−5)² = −3 + 20 − 25 = −8.

Answer 2:   B: 0 − 0 − 2 = −2.

Answer 3:   E: 95(4) = 380 points; 90 + 91 + 98 = 279 points. The fourth test would have to be 380 − 279 = 101% (not possible).

Answer 4:   B: . So the sum is 100 + 100 + 96 = 296.

Answer 5:   D:

Answer 6:   E: The number is not divisible by 11 because 3 − 2 = 1, which is not a multiple of 11. It is not divisible by 3 because 3 + 2 = 5 is not divisible by 3. Because is it not divisible by 11 or 3, it is not divisible by 33. Finally, it is not divisible by 10 because it does not end in a 0. So the answer is E because all numbers are divisible by 1.

Answer 7:   C:

Answer 8:   C: .

Answer 9:   E: It’s bimodal; the modes are 10 and 16, each appearing twice.

Answer 10: B:

Answer 11: C: The median is determined by putting all of the numbers in order, so we have 100, 100, 100, 100, 98, 98, 98, 95, 95, 86. The middle terms are 98 and 98, so the median is 98.

Answer 12: E: The mode is 100 because that is the most common score; there are four of them.

Chapter 1 Quiz

  1. List the even integers between −4 and 7.

  2. List the multiples of seven between 42 and 73 inclusive.

  3. List the primes between 100 and 110.

  4. Write all the factors of 24.

  5. Write 90 as the product of prime factors.

  6. Why is 4.56 × 10¹⁰⁰⁰ divisible by 3 and not by 9?

  7. 9 − 3³ − 1⁴ =

  8. 7 − 3(−5) − 7(2) =

  9. −1¹⁰ + (−1)¹¹ =

For questions 11 and 12, let a = −5 and d = 3; evaluate the following:

11. ad − a²

12. ad² − (ad)²

For questions 13–16, consider the set {1, 2, 2, 5, 6, 8, 10}.

13. Find the mean.

14. Find the mode.

15. Find the median.

16. Find the range.

For questions 17–20, consider the set {−9, −9, −9, 4, 4, 4}.

17. Find the mean.

18. Find the mode.

19. Find the median.

20. Find the range.

Answers to Chapter 1 Quiz

  1. −2, 0, 2, 4, and 6; between doesn’t include the end numbers. Remember, 0 is an even integer.

  2. 42, 49, 56, 63, and 70; inclusive means to include the end number.

  3. 101, 103, 107, and 109 (101 and 103, 107 and 109 are sometimes called twin primes because they differ by 2; see if you can find more of them).

  4. 1, 2, 3, 4, 6, 8, 12, and 24.

  5. 2 × 3 × 3 × 5, or 2 × 3² × 5.

  6. 4 + 5 + 6 = 15 is divisible by 3 but not by 9.

  7. 9 − 27 − 1 = −19.

  8. 7 + 15 − 14 = 8.

  9. −1 + (−1) = −2.

11. (−5)(3) − (−5)² = −15 − 25 = −40.

12. (−5)(3)² − [(−5)(3)]² =(− 5)(9) − (−15)² = −45 − 225 = −270.

14. 2.

15. 5.

16. 10 − 1 = 9.

.

18. −9 and 4 are both modes (bimodal).

20. 4 − (−9) = 13.

Let’s do some decimals, fractions, and percentages.

Answer to Bob Asks: 9 cuts.

             CHAPTER 2: Arithmetic We Must Know

Although the COMPASS does allow calculators, it is necessary to be able to do some of the work without the calculator, especially fractions. Let’s start with decimals.

DECIMALS

Rule 1: When adding or subtracting, line up the decimal points.

Example 1: Add: 3.14 + 234.7 + 86

Solution:

Example 2: Subtract: 56.7 − 8.82

Solution:

Rule 2: In multiplying numbers, count the number of decimal places and add them. In the product, this will be the number of decimal places for the decimal point.

Example 3: Multiply 45.67 by .987.

Solution:     The answer will be 45.07629. You will need to know the answer has five decimal places.

Example 4: Multiply 2.8 by .6:

Solution:     The answer is 1.68.

Example 5: What is the value of 2b² − 2.4b − 1.7 if b = .7?

Solution:     The standard way to do this type of problem is to directly substitute. 2(.7)² − 2.4(.7) − 1.7 = 2(.49) − 1.68 − 1.7 = .98 − 1.68 − 1.7 = −2.40.

You will be given answers from which to choose, so approximations may work just as well and are quicker. (.7)(.7) = .49; times 2 is .98, or approximately 1. − 2.4 times .7 = 1.68 or approximately 1.7; 1 − 1.7 = −.7; added to −1.7 is −2.4. In this case, we get the same numerical value. If we didn’t, our approximate answer would still probably be closer to the correct answer than to the other choices.

In the real world, and on this test, a good skill to save time is the ability to make reasonable approximations. This book will point out problems that can be approximated.

Rule 3: When you divide, move the decimal point in the divisor and the dividend the same number of places.

Example 6: Divide 23.1 by