Pre-Algebra DeMYSTiFieD, Second Edition

By Allan G. Bluman

Preempt your anxiety about PRE-ALGEBRA!

Ready to learn math fundamentals but can't seem to get your brain to function? No problem! Add Pre-Algebra Demystified, Second Edition, to the equation and you'll solve your dilemma in no time.

Written in a step-by-step format, this practical guide begins by covering whole numbers, integers, fractions, decimals, and percents. You'll move on to expressions, equations, measurement, and graphing. Operations with monomials and polynomials are also discussed. Detailed examples, concise explanations, and worked problems make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.

It's a no-brainer! You'll learn:

• Addition, subtraction, multiplication, and division of whole numbers, integers, fractions, decimals, and algebraic expressions
• Techniques for solving equations and problems
• Measures of length, weight, capacity, and time
• Methods for plotting points and graphing lines

Simple enough for a beginner, but challenging enough for an advanced student, Pre-Algebra Demystified, Second Edition, helps you master this essential mathematics subject. It's also the perfect way to review the topic if all you need is a quick refresh.

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Dec 6, 2010
• ISBN: 9780071742511

Book preview

chapter 1

Whole Numbers

Numbers make up the foundation of mathematics. The first numbers people used were the natural or counting numbers, consisting of 1, 2, 3,. … When 0 is added to the set of natural numbers, the set is called the whole numbers. This chapter explains the basic operations of addition, subtraction, multiplication, and division of these numbers.

CHAPTER OBJECTIVES

In this chapter, you will learn how to

• read whole numbers

• round whole numbers

• Add, subtract, multiply, and divide whole numbers

• Solve word problems using whole numbers

Naming Numbers

Our number system is called the Hindu-Arabic system or decimal system. It consists of 10 symbols or digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, which are used to make our numbers. Each digit in a number has a place value. The place value names are shown in Fig. 1-1.

FIGURE 1-1

In larger numbers each group of three numbers (called a period) is separated by a comma. The names at the top of the columns in Fig. 1-1 are called period names.

To name a number, start at the left and go to the right, read each group of three numbers separately using the period name at the comma. The word ones is not used when naming numbers.

EXAMPLE _____ _____ _____

Name 62,432,709.

SOLUTION _____ _____ _____

Sixty-two million, four hundred thirty-two thousand, seven hundred nine.

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EXAMPLE _____ _____ _____

Name 560,711.

SOLUTION _____ _____ _____

Five hundred sixty thousand, seven hundred eleven.

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EXAMPLE _____ _____ _____

Name 87,001,000,012.

SOLUTION _____ _____ _____

Eighty-seven billion, one million, twelve.

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MATH NOTE

Other period names after trillions in increasing order are quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, and so fourth.

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Practice

Name each number:

1. 515

2. 27,932

3. 1,607,003

4. 63,902,400,531

Answers

1. Five hundred fifteen

2. Twenty-seven thousand, nine hundred thirty-two

3. One million, six hundred seven thousand, three

4. Sixty-three billion, nine hundred two million, four hundred thousand, five hundred thirty-one

Rounding Numbers

Many times it is not necessary to use an exact number. In this case, an approximate number can be used. Approximations can be obtained by rounding numbers. All numbers can be rounded to specific place values.

To round a number to a specific place value, first locate that place value digit in the number. If the digit to the right of the specific place value digit is 0, 1, 2, 3, or 4, the place value digit remains the same. If the digit to the right of the specific place value digit is 5, 6, 7, 8, or 9, add one to the specific place value digit. In either case all digits to the right of the specific place value digit are changed to zeros.

EXAMPLE _____ _____ _____

Round 52,183 to the nearest hundred.

SOLUTION _____ _____ _____

We are rounding to the hundreds place which is the digit 1. Since the digit to the right of the 1 is 8, add 1 to 1 to get 2. Change all digits to the right to zeros. Hence 52,183 rounded to the nearest hundred is 52,200.

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EXAMPLE _____ _____ _____

Round 53,462 to the nearest thousand.

SOLUTION _____ _____ _____

We are rounding to the thousands place, which is the digit 3. Since the digit to the right of the 3 is 4, the 3 stays the same. Change all digits to the right of 3 to zeros. Hence 53,462 rounded to the nearest thousand is 53,000.

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EXAMPLE _____ _____ _____

Round 1,498,352 to the nearest ten thousand.

SOLUTION _____ _____ _____

We are rounding to the ten thousands place, which is the digit 9. Since the digit to the right of the 9 is 8, the 9 becomes a 10. Then we write the 0 and add the 1 to the next digit to the left. The 4 then becomes a 5. Hence the answer is 1,500,000.

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Still Struggling

Remember to change digits to the right of the rounding place to zero when rounding. For example, when you round 4,672 to the nearest hundred, you get 4,700 not 47.

Practice

1. Round 7,831 to the nearest thousand.

2. Round 294,183 to the nearest ten thousand.

3. Round 92,308 to the nearest ten.

4. Round 682,611 to the nearest hundred thousand.

5. Round 163,793,244 to the nearest million.

Answers

1. 8,000

2. 290,000

3. 92,310

4. 700,000

5. 164,000,000

Addition of Whole Numbers

In mathematics, addition, subtraction, multiplication, and division are called operations. The numbers being added are called addends. The total is called the sum.

To add two or more numbers, first write them in a column, and then add the digits in the columns from right to left. If the sum of the digits in any column is 10 or more, write the one’s digit and carry the ten’s digit to the next column to the left and add it to the numbers in that column.

EXAMPLE _____ _____ _____

Add: 192 + 7 + 5,684 + 273.

SOLUTION _____ _____ _____

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MATH NOTE

To check addition, add from the bottom to the top.

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Practice

Add:

1. 443 + 27 + 7

2. 4,593 + 14 + 863

3. 7,324 + 625,713

4. 18 + 46,933 + 36 + 557

5. 5,641 + 300 + 65 + 77,325

Answers

1. 477

2. 5,470

3. 633,037

4. 47,544

5. 83,331

Subtraction of Whole Numbers

In subtraction, the top number is called the minuend. The number being subtracted (below the top number) is called the subtrahend. The answer in subtraction is called the remainder or difference.

To subtract two numbers, write the numbers in a vertical column and then subtract the bottom digits from the top digits. Proceed from right to left. When the bottom digit is larger than the top digit, borrow one from the digit at the top of the next column and add ten to the top digit before subtracting. When borrowing, be sure to reduce the top digit in the next column by 1.

EXAMPLE _____ _____ _____

Subtract: 19,784 − 4,213.

SOLUTION _____ _____ _____

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EXAMPLE _____ _____ _____

Subtract: 5,386 − 748.

SOLUTION _____ _____ _____

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MATH NOTE

To check subtraction, add the difference to the subtrahend to see if you get the minuend.

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Practice

Subtract:

1. 961 − 87

2. 24,271 − 6,314

3. 867,281 − 23,779

4. 73,307,641 − 863,259

5. 8,000,000 − 81,406

Answers

1. 874

2. 17,957

3. 843,502

4. 72,444,382

5. 7,918,594

Multiplication of Whole Numbers

In multiplication, the top number is called the multiplicand. The number directly below it is called the multiplier. The answer in multiplication is called the product. The numbers between the multiplier and the product are called partial products.

To multiply two numbers when the multiplier is a single digit, write the numbers in a vertical column and then multiply each digit in the multiplicand from right to left by the multiplier. If any of these products is over nine, add the tens digit to the product of numbers in the next column.

EXAMPLE _____ _____ _____

Multiply: 416 × 7.

SOLUTION _____ _____ _____

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To multiply two numbers when the multiplier contains two or more digits, arrange the numbers vertically and multiply each digit in the multiplicand by the right-most digit in the multiplier. Next multiply each digit in the multiplicand by the next digit in the multiplier and place the second partial product under the first partial product, moving one space to the left. Continue the process for each digit in the multiplier and then add the partial products to get the final product.

EXAMPLE _____ _____ _____

Multiply: 3,742 × 814.

SOLUTION _____ _____ _____

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MATH NOTE

To check the multiplication problem, multiply the multiplier by the multiplicand.

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Practice

Multiply:

1. 92 × 5

2. 651 × 87

3. 4,135 × 216

4. 61,405 × 892

5. 154,371 × 43

Answers

1. 460

2. 56,637

3. 893,160

4. 54,773,260

5. 6,637,953

Division of Whole Numbers

In division, the number under the division box is called the dividend. The number outside the division box is called the divisor. The answer in division is called the quotient. Sometimes the answer does not come out even; hence, there will be a remainder.

The process of long division consists of a series of steps. They are divide, multiply, subtract, and bring down. When dividing, it is also necessary to estimate how many times the divisor divides into the dividend. When the divisor consists of two or more digits, the estimation can be accomplished by dividing the first digit of the divisor into the first one or two digits of the dividend. The process is shown next.

EXAMPLE _____ _____ _____

Divide: 863 by 52.

SOLUTION _____ _____ _____

Step 1:

Step 2:

Step 3:

Step 4:

Repeat Step 1:

Repeat Step 2:

Repeat Step 3:

Hence, the answer is 16 remainder 31 or 16 R31. Stop when you run out of digits in the dividend to bring down.

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EXAMPLE _____ _____ _____

Divide: 4,378 by 67.

SOLUTION _____ _____ _____

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Division can be checked by multiplying the quotient by the divisor, adding the remainder, and seeing if you get the dividend. For the previous example, multiply 65 × 67 and add 23.

Practice

Divide:

1. 2,494 ÷ 43

2. 43,967 ÷ 7

3. 40,898 ÷ 143

4. 688 ÷ 31

5. 10,568 ÷ 738

Answers

1. 58

2. 6,281

3. 286

4. 22 R6

5. 14 R236

Word Problems

In order to solve word problems follow these steps:

1. Read the problem carefully.

2. Identify what you are being asked to find.

3. Perform the correct operation or operations.

4. Check your answer or at least see if it is reasonable.

In order to know what operation to perform, it is necessary to understand the basic concept of each operation.

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MATH NOTE

Always read the word problem at least twice before starting it.

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Addition

When you are asked to find the sum or the total or how many in all, and the items are the same in the problem, you add.

EXAMPLE _____ _____ _____

Find the total calories in a breakfast consisting of a sausage and egg sandwich (450 calories), hash brown potatoes (140 calories), and low-fat milk (95 calories).

SOLUTION _____ _____ _____

Since you want to find the total number of items and the items are the same (calories), you add: 450 + 140 + 95 = 685 calories. Hence, the total number of calories in the meal is 685.

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Subtraction

When you are asked to find the difference, that is, how much more, how much less, how much larger, how much smaller, etc., and the items are the same, you subtract.

EXAMPLE _____ _____ _____

The height of the Matterhorn is 14,690 feet, and the height of Mt. McKinley is 20,320 feet. How much higher is Mt. McKinley than the Matterhorn?

SOLUTION _____ _____ _____

Since you are asked how much higher Mt. McKinley is than the Matterhorn, you subtract: 20,320 − 14,690 = 5,630. Hence, Mt. McKinley is 5,630 feet higher than the Matterhorn.

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Multiplication

When you are asked to find a total and the items are different, you multiply.

EXAMPLE _____ _____ _____

An auditorium consists of 18 rows with 24 seats in each row. How many people can the auditorium seat?

SOLUTION _____ _____ _____

Since you want a total and the items are different (rows and seats), you multiply:

The auditorium will seat 432 people.

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Division

When you are given a total and are asked to find how many items are in each part, you divide.

EXAMPLE _____ _____ _____

If 72 cameras are packed in 6 boxes, how many cameras would be placed in each box?

SOLUTION _____ _____ _____

In this case, the total is 72, and they are to be put into 6 boxes, so to find the answer, divide:

Each box would have 12 cameras in it.

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Still Struggling

When you do a word problem, always make sure that the answer you get sounds reasonable. This will help you in determining whether or not you made a mistake. For example, if you are asked to find the reduced price of an item that costs $50, and you get $75, you know this is a mistake because the reduced price of the item would be more than the original price.

Practice

Solve:

1. Find the total number of passes completed for the following players: Montana, 3,409; Lomax, 1,817; Anderson, 2,654; and White, 1,761.

2. Ieisha paid $675 for his new computer system. Included in the price was the printer at a cost of $83. How much would the system have cost without the printer?

3. Lunch for 8 people costs $112. If they decided to split the cost equally, how much would each person pay?

4. A real estate developer bought 43 acres of land at $2,750 per acre. What was the total cost of the land?

5. Mark purchased 12 roses for $36. How much was each rose for?

Answers

1. 9,641 passes completed

2. $592

3. $14

4. $118,250

5. $3

In this chapter, the basic operations of addition, subtraction, multiplication, and division of whole numbers were explained.

QUIZ

1. Name 32,321.

A. three ten thousands, two thousands, three hundreds, two tens, and one ones

B. thirty-two thousand, three hundred twenty-one

C. thirty-two million, three hundred twenty-one

D. thirty-two hundred, three hundred twenty-one

2. Name 50,000,002.

A. fifty million two

B. fifty billion two

C. fifty-two million

D. fifty-two billion

3. Round 6,314,259 to the nearest hundred thousand.

A. 6,000,000

B. 6,310,000

C. 6,314,000

D. 6,300,000

4. Round 789,961 to the nearest hundred.

A. 790,000

B. 789,600

C. 800,000

D. 789,000

5. Round 2,867 to the nearest ten.

A. 2,900

B. 2,960

C. 3,000

D. 2,870

6. Add 97 + 148 + 6 + 40.

A. 291

B. 281