Chapter 1. Arithmetic Review
1.1. Objectives*
This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. If you would like a quick review of arithmetic before attempting the study of algebra, this chapter is recommended reading. If you feel your arithmetic skills are pretty good, then move on to Basic Properties of Real Numbers (Section 2.1). However you feel, do not hesitate to use this chapter as a quick reference of arithmetic techniques.
The other chapters include Practice Sets paired with Sample Sets with sufficient space for the student to work out the problems. In addition, these chapters include a Summary of Key Concepts, Exercise Supplements, and Proficiency Exams.
1.2. Factors, Products, and Exponents *
Overview
Factors
Exponential Notation
Factors
Let’s begin our review of arithmetic by recalling the meaning of multiplication for whole numbers (the counting numbers and zero).
Multiplication is a description of repeated addition.
In the addition 7 + 7 + 7 + 7 the number 7 is repeated as an addend* 4 times. Therefore, we say we have four times seven and describe it by writing 4 · 7 The raised dot between the numbers 4 and 7 indicates multiplication. The dot directs us to multiply the two numbers that it separates. In algebra, the dot is preferred over the symbol × to denote multiplication because the letter x is often used to represent a number. Thus, 4 · 7 = 7 + 7 + 7 + 7
In a multiplication, the numbers being multiplied are called factors. The result of a multiplication is called the product. For example, in the multiplication 4 · 7 = 28 the numbers 4 and 7 are factors, and the number 28 is the product. We say that 4 and 7 are factors of 28. (They are not the only factors of 28. Can you think of others?)Now we know that
This indicates that a first number is a factor of a second number if the first number divides into the second number with no remainder. For example, since 4 · 7 = 28 both 4 and 7 are factors of 28 since both 4 and 7 divide into 28 with no remainder.
Exponential Notation
Quite often, a particular number will be repeated as a factor in a multiplication. For example, in the multiplication 7 · 7 · 7 · 7 the number 7 is repeated as a factor 4 times. We describe this by writing 74 . Thus, 7 · 7 · 7 · 7 = 74 The repeated factor is the lower number (the base), and the number recording how many times the factor is repeated is the higher number (the superscript). The superscript number is called an exponent.
An exponent is a number that records how many times the number to which it is attached occurs as a factor in a multiplication.
Sample Set A
For Examples 1, 2, and 3, express each product using exponents.
Example 1.1.
3 · 3 · 3 · 3 · 3 · 3. Since 3 occurs as a factor 6 times, 3 · 3 · 3 · 3 · 3 · 3 = 36
Example 1.2.
8 · 8. Since 8 occurs as a factor 2 times, 8 · 8 = 82
Example 1.3.
5 · 5 · 5 · 9 · 9. Since 5 occurs as a factor 3 times, we have 53 . Since 9 occurs as a factor 2 times, we have 92 . We should see the following replacements.
Then we have 5 · 5 · 5 · 9 · 9 = 53 · 92
Example 1.4.
Expand 35 . The base is 3 so it is the repeated factor. The exponent is 5 and it records the number of times the base 3 is repeated. Thus, 3 is to be repeated as a factor 5 times. 35 = 3 · 3 · 3 · 3 · 3
Example 1.5.
Expand 62 · 10 4 . The notation 62 · 10 4 records the following two facts: 6 is to be repeated as a factor 2 times and 10 is to be repeated as a factor 4 times. Thus, 62 · 10 4 = 6 · 6 · 10 · 10 · 10 · 10
Exercises
For the following problems, express each product using exponents.
Exercise 1.2.2.
12 · 12 · 12 · 12 · 12
Exercise 1.2.4.
1 · 1
Exercise 1.2.6.
8 · 8 · 8 · 15 · 15 · 15 · 15
Exercise 1.2.8.
3 · 3 · 10 · 10 · 10
Exercise 1.2.9. (Go to Solution)
Suppose that the letters x and y are each used to represent numbers. Use exponents to express the following product. x · x · x · y · y
Exercise 1.2.10.
Suppose that the letters x and y are each used to represent numbers. Use exponents to express the following product. x · x · x · x · x · y · y · y
For the following problems, expand each product (do not compute the actual value).
Exercise 1.2.12.
43
Exercise 1.2.14.
96
Exercise 1.2.16.
27 · 34
Exercise 1.2.18.
x 6 · y 2
For the following problems, specify all the whole number factors of each number. For example, the complete set of whole number factors of 6 is 1, 2, 3, 6.
Exercise 1.2.20.
14
Exercise 1.2.22.
30
Exercise 1.2.24.
45
Exercise 1.2.26.
17
Exercise 1.2.28.
2
Solutions to Exercises
1.3. Prime Factorization *
Overview
Prime And Composite Numbers
The Fundamental Principle Of Arithmetic
The Prime Factorization Of A Whole Number
Prime And Composite Numbers
Notice that the only factors of 7 are 1 and 7 itself, and that the only factors of 23 are 1 and 23 itself.
A whole number greater than 1 whose only whole number factors are itself and 1 is called a prime number.
The first seven prime numbers are2, 3, 5, 7, 11, 13, and 17The number 1 is not considered to be a prime number, and the number 2 is the first and only even prime number.Many numbers have factors other than themselves and 1. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28 (since each of these whole numbers and only these whole numbers divide into 28 without a remainder).
A whole number that is composed of factors other than itself and 1 is called a composite number. Composite numbers are not prime numbers.Some composite numbers are 4, 6, 8, 10, 12, and 15.
The Fundamental Principle Of Arithmetic
Prime numbers are very important in the study of mathematics. We will use them soon in our study of fractions. We will now, however, be introduced to an important mathematical principle.
Except for the order of the factors, every whole number, other than 1, can be factored in one and only one way as a product of prime numbers.
When a number is factored so that all its factors are prime numbers, the factorization is called the prime factorization of the number.
Sample Set A
Example 1.6.
Find the prime factorization of 10. 10 = 2 · 5 Both 2 and 5 are prime numbers. Thus, 2 · 5 is the prime factorization of 10.
Example 1.7.
Find the prime factorization of 60.
The numbers 2, 3, and 5 are all primes. Thus,
22
·
3
·
5
is the prime factorization of 60.
Example 1.8.
Find the prime factorization of 11.11 is a prime number. Prime factorization applies only to composite numbers.
The Prime Factorization Of A Whole Number
The following method provides a way of finding the prime factorization of a whole number. The examples that follow will use the method and make it more clear.
Divide the number repeatedly by the smallest prime number that will divide into the number without a remainder.
When the prime number used in step 1 no longer divides into the given number without a remainder, repeat the process with the next largest prime number.
Continue this process until the quotient is 1.
The prime factorization of the given number is the product of all these prime divisors.
Sample Set B
Example 1.9.
Find the prime factorization of 60.Since 60 is an even number, it is divisible by 2. We will repeatedly divide by 2 until we no longer can (when we start getting a remainder). We shall divide in the following way. 
The prime factorization of 60 is the product of all these divisors.
Example 1.10.
Find the prime factorization of 441.Since 441 is an odd number, it is not divisible by 2. We’ll try 3, the next largest prime. 
The prime factorization of 441 is the product of all the divisors.
Exercises
For the following problems, determine which whole numbers are prime and which are composite.
Exercise 1.3.2.
25
Exercise 1.3.4.
2
Exercise 1.3.6.
5
Exercise 1.3.8.
9
Exercise 1.3.10.
34
Exercise 1.3.12.
63
Exercise 1.3.14.
339
For the following problems, find the prime factorization of each whole number. Use exponents on repeated factors.
Exercise 1.3.16.
26
Exercise 1.3.18.
54
Exercise 1.3.20.
56
Exercise 1.3.22.
480
Exercise 1.3.24.
2025
Solutions to Exercises
1.4. The Least Common Multiple*
Overview
Multiples
Common Multiples
The Least Common Multiple (LCM)
Finding The Least Common Multiple
Multiples
When a whole number is multiplied by other whole numbers, with the exception of Multiples zero, the resulting products are called multiples of the given whole number.
| Multiples of 2 | Multiples of 3 | Multiples of 8 | Multiples of 10 | |||
| 2 · 1 = 2 | 3 · 1 = 3 | 8 · 1 = 8 | 10 · 1 = 10 | |||
| 2 · 2 = 4 | 3 · 2 = 6 | 8 · 2 = 16 | 10 · 2 = 20 | |||
| 2 · 3 = 6 | 3 · 3 = 9 | 8 · 3 = 24 | 10 · 3 = 30 | |||
| 2 · 4 = 8 | 3 · 4 = 12 | 8 · 4 = 32 | 10 · 4 = 40 | |||
| 2 · 5 = 10 | 3 · 5 = 15 | 8 · 5 = 40 | 10 · 5 = 50 | |||
| … | … | … | … |
Common Multiples
There will be times when we are given two or more whole numbers and we will need to know if there are any multiples that are common to each of them. If there are, we will need to know what they are. For example, some of the multiples that are common to 2 and 3 are 6, 12, and 18.
Sample Set A
Example 1.11.
We can visualize common multiples using the number line. 
Notice that the common multiples can be divided by both whole numbers.
The Least Common Multiple (LCM)
Notice that in our number line visualization of common multiples (above) the first common multiple is also the smallest, or least common multiple, abbreviated by LCM.
The least common multiple, LCM, of two or more whole numbers is the smallest whole number that each of the given numbers will divide into without a remainder.
Finding The Least Common Multiple
To find the LCM of two or more numbers,
Write the prime factorization of each number, using exponents on repeated factors.
Write each base that appears in each of the prime factorizations.
To each base, attach the largest exponent that appears on it in the prime factorizations.
The LCM is the product of the numbers found in step 3.
Sample Set B
Find the LCM of the following number.
Example 1.12.
9 and 12
The bases that appear in the prime factorizations are 2 and 3.
The largest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 2 and 2 (or 22 from 12, and 32 from 9).
The LCM is the product of these numbers. LCM = 22 · 32 = 4 · 9 = 36
Thus, 36 is the smallest number that both 9 and 12 divide into without remainders.
Example 1.13.
90 and 630
The bases that appear in the prime factorizations are 2, 3, 5, and 7.
The largest exponents that appear on 2, 3, 5, and 7 are, respectively, 1, 2, 1, and 1.
The LCM is the product of these numbers. LCM = 2 · 32 · 5 · 7 = 2 · 9 · 5 · 7 = 630
Thus, 630 is the smallest number that both 90 and 630 divide into with no remainders.
Example 1.14.
33, 110, and 484
The bases that appear in the prime factorizations are 2, 3, 5, and 11.
The largest exponents that appear on 2, 3, 5, and 11 are, respectively, 2, 1, 1, and 2.
The LCM is the product of these numbers.
Thus, 7260 is the smallest number that 33, 110, and 484 divide into without remainders.
Exercises
For the following problems, find the least common multiple of given numbers.
Exercise 1.4.2.
8, 10
Exercise 1.4.4.
9, 18
Exercise 1.4.6.
7, 9
Exercise 1.4.8.
24, 36
Exercise 1.4.10.
20, 24
Exercise 1.4.12.
24, 54
Exercise 1.4.14.
36, 48
Exercise 1.4.16.
7, 11, 33
Exercise 1.4.18.
4, 5, 21
Exercise 1.4.20.
15, 25, 40
Exercise 1.4.22.
12, 16, 24
Exercise 1.4.24.
6, 9, 12, 18
Solutions to Exercises
1.5. Equivalent Fractions *
Overview
Equivalent Fractions
Reducing Fractions To Lowest Terms
Raising Fractions To Higher Terms
Equivalent Fractions
Fractions that have the same value are called equivalent fractions.
For example, 
and 
represent the same part of a whole quantity and are therefore equivalent. Several more collections of equivalent fractions are listed below.
Example 1.15.
Example 1.16.
Example 1.17.
Reducing Fractions To Lowest Terms
It is often useful to convert one fraction to an equivalent fraction that has reduced values in the numerator and denominator. When a fraction is converted to an equivalent fraction that has the smallest numerator and denominator in the collection of equivalent fractions, it is said to be reduced to lowest terms. The conversion process is called reducing a fraction.
We can reduce a fraction to lowest terms by
Expressing the numerator and denominator as a product of prime numbers. (Find the prime factorization of the numerator and denominator. See Section (Section 1.3) for this technique.)
Divide the numerator and denominator by all common factors. (This technique is commonly called “cancelling.”)
Sample Set A
Reduce each fraction to lowest terms.
Example 1.18.
Example 1.19.
Example 1.20.
Example 1.21.
Raising a Fraction to Higher Terms
Equally important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator. The higher, equivalent fraction is constructed by multiplying the original fraction by 1.Notice that 
and 
are equivalent, that is 
Also, 
This observation helps us suggest the following method for raising a fraction to higher terms.
A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same nonzero number.
For example, 
can be raised to 
by multiplying both the numerator and denominator by 8, that is, multiplying by 1 in the form 
How did we know to choose 8 as the proper factor? Since we wish to convert 4 to 32 by multiplying it by some number, we know that 4 must be a factor of 32. This means that 4 divides into 32. In fact, 32÷4 = 8. We divided the original denominator into the new, specified denominator to obtain the proper factor for the multiplication.
Sample Set B
Determine the missing numerator or denominator.
Example 1.22.
Example 1.23.
Exercises
For the following problems, reduce, if possible, each fraction lowest terms.
Exercise 1.5.2.
Exercise 1.5.4.
Exercise 1.5.6.
Exercise 1.5.8.
Exercise 1.5.10.
Exercise 1.5.12.
Exercise 1.5.14.
Exercise 1.5.16.
Exercise 1.5.18.
Exercise 1.5.20.
Exercise 1.5.22.
Exercise 1.5.24.
For the following problems, determine the missing numerator or denominator.
Exercise 1.5.26.
Exercise 1.5.28.
Exercise 1.5.30.
Exercise 1.5.32.
Exercise 1.5.34.
Solutions to Exercises
1.6. Operations with Fractions *
Overview
Multiplication of Fractions
Division of Fractions
Addition and Subtraction of Fractions
Multiplication of Fractions
To multiply two fractions, multiply the numerators together and multiply the denominators together. Reduce to lowest terms if possible.
Example 1.24.
For example, multiply 
Notice that we since had to reduce, we nearly started over again with the original two fractions. If we factor first, then cancel, then multiply, we will save time and energy and still obtain the correct product.
Sample Set A
Perform the following multiplications.
Example 1.25.
Example 1.26.
Division of Fractions
Two numbers whose product is 1 are reciprocals of each other. For example, since 
and 
are reciprocals of each other. Some other pairs of reciprocals are listed below.
Reciprocals are used in division of fractions.
To divide a first fraction by a second fraction, multiply the first fraction by the reciprocal of the second fraction. Reduce if possible.
This method is sometimes called the “invert and multiply” method.
Sample Set B
Perform the following divisions.
Example 1.27.
Example 1.28.
Example 1.29.
Addition and Subtraction of Fractions
To add (or subtract) two or more fractions that have the same denominators, add (or subtract) the numerators and place the resulting sum over the common denominator. Reduce if possible.
CAUTIONAdd or subtract only the numerators. Do not add or subtract the denominators!
Sample Set C
Find the following sums.
Example 1.30.
Example 1.31.
Fractions can only be added or subtracted conveniently if they have like denominators.
To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as the denominator the least common multiple of the original denominators.
The least common multiple of the original denominators is commonly referred to as the least common denominator (LCD). See Section (Section 1.4) for the technique of finding the least common multiple of several numbers.
Sample Set D
Find each sum or difference.
Example 1.32.
Example 1.33.
Exercises
For the following problems, perform each indicated operation.
Exercise 1.6.2.
Exercise 1.6.4.
Exercise 1.6.6.
Exercise 1.6.8.
Exercise 1.6.10.
Exercise 1.6.12.
Exercise 1.6.14.
Exercise 1.6.16.
Exercise 1.6.18.
Exercise 1.6.20.
Exercise 1.6.22.
Exercise 1.6.24.
Exercise 1.6.26.
Exercise 1.6.28.
Exercise 1.6.30.
Exercise 1.6.32.
Exercise 1.6.34.
Exercise 1.6.36.
Exercise 1.6.38.
Exercise 1.6.40.
Solutions to Exercises
1.7. Decimal Fractions*
Overview
Decimal Fractions
Adding and Subtracting Decimal Fractions
Multiplying Decimal Fractions
Dividing Decimal Fractions
Converting Decimal Fractions to Fractions
Converting Fractions to Decimal Fractions
Decimal Fractions
Fractions are one way we can represent parts of whole numbers. Decimal fractions are another way of representing parts of whole numbers.
A decimal fraction is a fraction in which the denominator is a power of 10.
A decimal fraction uses a decimal point to separate whole parts and fractional parts. Whole parts are written to the left of the decimal point and fractional parts are written to the right of the decimal point. Just as each digit in a whole number has a particular value, so do the digits in decimal positions. 
Sample Set A
The following numbers are decimal fractions.
Example 1.34.
Example 1.35.
Adding and Subtracting Decimal Fractions
To add or subtract decimal fractions,
Align the numbers vertically so that the decimal points line up under each other and corresponding decimal positions are in the same column. Add zeros if necessary.
Add or subtract the numbers as if they were whole numbers.
Place a decimal point in the resulting sum or difference directly under the other decimal points.
Sample Set B
Find each sum or difference.
Example 1.36.
Example 1.37.
Example 1.38.
Multiplying Decimal Fractions
To multiply decimals,
Multiply tbe numbers as if they were whole numbers.
Find the sum of the number of decimal places in the factors.
The number of decimal places in the product is the sum found in step 2.
Sample Set C
Find the following products.
Example 1.39.
6.5 × 4.3
6.5 × 4.3 = 27.95
Example 1.40.
23.4 × 1.96
23.4 × 1.96 = 45.864
Dividing Decimal Fractions
To divide a decimal by a nonzero decimal,
Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor’s last digit.
Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
Set the decimal point in the quotient by placing a decimal point directly above the decimal point in the dividend.
Divide as usual.
Sample Set D
Find the following quotients.
Example 1.41.
32.66÷7.1
Example 1.42.
Check by multiplying 2.1 and 0.513. This will show that we have obtained the correct result.
Example 1.43.
12÷0.00032
Converting Decimal Fractions to Fractions
We can convert a decimal fraction to a fraction by reading it and then writing the phrase we have just read. As we read the decimal fraction, we note the place value farthest to the right. We may have to reduce the fraction.
Sample Set E
Convert each decimal fraction to a fraction.
Example 1.44.
Example 1.45.
Converting Fractions to Decimal Fractions
Sample Set F
Convert the following fractions to decimals. If the division is nonterminating, round to 2 decimal places.
Example 1.46.
Example 1.47.
Example 1.48.
Example 1.49.
Example 1.50.
This is a complex decimal. The “6” is in the hundredths position. The number 
is read as “sixteen and one-fourth hundredths.” 
Now, convert 
to a decimal. 
Exercises
For the following problems, perform each indicated operation.
Exercise 1.7.2.
15.015 − 6.527
Exercise 1.7.4.
156.33 − 24.095
Exercise 1.7.6.
44.98 + 22.8 − 12.76
Exercise 1.7.8.
1.11 + 12.1212 − 13.131313
Exercise 1.7.10.
2.97 · 3.15
Exercise 1.7.12.
5.009 · 2.106
Exercise 1.7.14.
100 · 12.008
Exercise 1.7.16.
10,000 · 12.008
Exercise 1.7.18.
51.811 ÷ 1.97
Exercise 1.7.20.
0.129516 ÷ 1004
For the following problems, convert each decimal fraction to a fraction.
Exercise 1.7.22.
0.115
Exercise 1.7.24.
48.1162
For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places.
Exercise 1.7.26.
Exercise 1.7.28.
15 ÷ 22
Exercise 1.7.30.
Solutions to Exercises
1.8. Percent *
Overview
The Meaning of Percent
Converting A Fraction To A Percent
Converting A Decimal To A Percent
Converting A Percent To A Decimal
The Meaning of Percent
The word percent comes from the Latin word “per centum,” “per” meaning “for each,” and “centum” meaning “hundred.”
Percent means “for each hundred” or “for every hundred.” The symbol % is used to represent the word percent.
Thus, 
Converting A Fraction To A Percent
We can see how a fraction can be converted to a percent by analyzing the method that 
is converted to a percent. In order to convert 
to a percent, we need to introduce 
(since percent means for each hundred).
Example 1.51.
To convert a fraction to a percent, multiply the fraction by 1 in the form 
, then replace 
with the % symbol.
Sample Set A
Convert each fraction to a percent.
Example 1.52.
Example 1.53.
Example 1.54.
Converting A Decimal To A Percent
We can see how a decimal is converted to a percent by analyzing the method that 0.75 is converted to a percent. We need to introduce 
To convert a fraction to a percent, multiply the decimal by 1 in the form 
, then replace 
with the % symbol. This amounts to moving the decimal point 2 places to the right.
Sample Set B
Convert each decimal to a percent.
Example 1.55.
Notice that the decimal point in the original number has been moved to the right 2 places.
Example 1.56.
Notice that the decimal point in the original number has been moved to the right 2 places.
Example 1.57.
Notice that the decimal point in the original number has been moved to the right 2 places.
Converting A Percent To A Decimal
We can see how a percent is converted to a decimal by analyzing the method that 12% is converted to a decimal. We need to introduce 
To convert a percent to a decimal, replace the % symbol with 
then divide the number by 100. This amounts to moving the decimal point 2 places to the left.
Sample Set C
Convert each percent to a decimal.
Example 1.58.
Notice that the decimal point in the original number has been moved to the left 2 places.
Example 1.59.
Notice that the decimal point in the original number has been moved to the left 2 places.
Example 1.60.
Notice that the decimal point in the original number has been moved to the left 2 places.
Exercises
For the following problems, convert each fraction to a percent.
Exercise 1.8.2.
Exercise 1.8.4.
Exercise 1.8.6.
Exercise 1.8.8.
Exercise 1.8.10.
Exercise 1.8.12.
8
For the following problems, convert each decimal to a percent.
Exercise 1.8.14.
0.42
Exercise 1.8.16.
0.1298
Exercise 1.8.18.
5.875
Exercise 1.8.20.
21.26
Exercise 1.8.22.
12
For the following problems, convert each percent to a decimal.
Exercise 1.8.24.
76%
Exercise 1.8.26.
67.2%
Exercise 1.8.28.
3.00156%
Exercise 1.8.30.
0.00034%