Math for the ACT 2nd Ed., Bob Miller's

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CHAPTER 1: Basic Basics

All math begins with whole numbers. Master them and you will begin to speak the language of math.

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, n² 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:

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 are 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 (1 st, 3rd, 5th, etc.) is a multiple of 11. For example, for the number 928, 193, 926: the sum of the odd digits (9, 8, 9, 9, and 6) is 41; the sum of the even digits (2, 1, 3, 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 they 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, –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 = –3and y = –4, find the value of:

a.7 – 5x – x²

b.xy² – (xy)²

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

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 used on the ACT. 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, take the average of them. This happens with an even number of data points.

Mode: Most common. Which number(s) appears the most times? A set with two modes is called bimodal. There can actually be any number of modes, including everything is a mode.

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:Mean:

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

Range: 20 – 4 = 16

Example 19:Jim received 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 exercises.

Exercise 1:If x = –5, the value of – 3 – 4x –x²is

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

Exercise 7:If m and n are odd integers, which of the following is odd?

A. mn + 3

B. m² + (n + 2)²

C. mn + m + n

D. (m + 1)(n – 2)

E. m⁴ + m³ + m² + m

Exercise 8:If m + 3 is a multiple of 4, which of these is also a multiple of 4?

A. m – 3

B. m

C. m + 4

D. m + 9

E. m + 11

Exercise 9:If p and q are primes, which one of the following can’t be a prime?

A. pq

B. p + q

C. pq + 2

D. 2pq + 1

E. p² + q²

Exercise 10:The sum of the first n positive integers is p. In terms of n and p, what is the sum of the next n positive integers?

A. np

B. n + p

C. n² + p

D. n + p²

E. 2n + 2p

Exercise 11:Let p be prime, with 20p divisible by 6; p could be

A. 3

B. 4

C. 5

D. 6

E. 7

For Exercises 12–14, use the following numbers: 8, 10, 10, 16, 16, 18

Exercise 12:The mean is

A. 8

B. 10

C. 13

D. 16

E. There are two of them.

Exercise 13:The median is

A. 8

B. 10

C. 13

D. 16

E. There are two of them.

Exercise 14: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.

Sandy’s Quiz Scores

This chart is for Exercises 15–17.

Exercise 15:The mean is

A. 96

B. 97

C. 98

D. 99

E. 100

Exercise 16:The median is

A. 96

B. 97

C. 98

D. 99

E. 100

Exercise 17: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.

Answer 4:B: The median is 100; the mode is 100; for the mean, we can use 2 and 3 instead of 20 and 30 because the ratio is the same:

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: Only C is the sum of three odd integers. All of the other answer choices are even.

Answer 8:E: If m + 3 is a multiple of 4, then m + 3 + 8 is a multiple of 4 because 8 is a multiple of 4.

Answer 9:A: By substituting proper primes, all the others might be prime.

Answer 10:C: Say n = 5; p = 1 + 2 + 3 + 4 + 5.The next five are (1 + 5) + (2 + 5) + (3 + 5) + (4 + 5) + (5 + 5) = p + n².

Answer 11:A: 3 and 6 will work, but only 3 is a prime.

Answer 12:C:

Answer 14:E: It’s bimodal; the modes are 10 and 16, each appearing twice.

Answer 15:B: The mean is the longest measure to compute:

Answer 16: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 17:E: The mode is 100 because that is the most common score; there are four of them.

CHAPTER 2: We Must Look at Arithmetic

One must master the parts as well as the whole to fully understand.

Although the ACT 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. The ACT will expect you to do this problem. Let’s try another.

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