Introduction: Division of Whole Numbers

Division undoes multiplication. Division and multiplication are often called opposite operations.

Learning: Division of Whole Numbers

Dividend, Divisor, and Quotient

Division undoes multiplication. Division and multiplication are often called opposite operations. The quotient of two numbers is the result of dividing one number by the other. The number being divided is the dividend. The number the dividend is divided by is the divisor. In the equation 14 ÷ 2 = 7, read "fourteen divided by two equals seven," 14 is the dividend, 2 is the divisor, and 7 is the quotient.

Quotient

The result of dividing one number by another.
Dividend

The number being divided in a division problem.
Divisor

The number dividing the dividend in a division problem.
Division Symbols

The most common symbol used to represent division is ÷. The forward slash (/) or a horizontal bar also denotes division. Write the quotient of 12 and 3 as 12 ÷ 3 or frac(12,3) or 12/3. In either form, the first or top number is the dividend and the second or bottom number is the divisor.
Dividing by Single Digit, No Remainder

Rectangular arrays can be used to find the quotient of two whole numbers.

Use a rectangular array to find the quotient of 18 and 6. Create rows of 6 boxes. Shade rows until there are 18 boxes in the array. Then observe how many rows of 6 boxes are in 18 squares.

The rectangular array in Figure 1 contains 3 rows of 6 shaded boxes for a total of 18. Therefore, the quotient of 18 and 6 is 3 or 18 ÷ 6 = 3. This division equation is related to the multiplication facts 3 × 6 = 18 and 6 × 3 = 18. Notice, because there are 6 columns of 3 shaded boxes, it is also true that 18 ÷ 3 = 6.
Division with Remainder

Picturing the division of a whole number by a nonfactor results in an array with leftover boxes. The number of these leftover boxes is known as the remainder of the division problem.
Remainder

The quantity that remains when one number does not divide another evenly.
Figure 2 shows the quotient and remainder of 20 ÷ 7 in an array of rows of 7 squares. The 2 complete rows of 7 shaded squares represent the quotient, 2. The incomplete row containing 6 shaded squares is the remainder, 6. The solution to 20 ÷ 7 is 2 with remainder 6.
Never Divide by 0

The quotient of 0 and any nonzero number is 0. Using mathematical symbols, this translates to frac(0,a)=0. However, when 0 is the divisor, the quotient is undefined (does not exist).

Dividing by Larger Whole Number, with Remainder

Arrays become cumbersome when dividing larger whole numbers. The division of multidigit whole numbers requires repeated use of these products.

Use the long division algorithm to divide multidigit numbers. The dividend goes inside the long-division symbol, , while the divisor is written to the left of the symbol.

Notation of Remainders

When the final difference is 0, the division problem has no remainder. When the final difference is greater than 0, this number is the remainder. Denote the remainder by writing R and the remainder next to the quotient.

To illustrate, consider Figure 3.

For the problem on the left of Figure 3, the divisor, 4, is greater than the first dividend digit, 3, so proceed to using the first two dividend digits, 31. The quotient of 31 and 4 is 7. Write it over 1, the second digit. Write the product, 28, under the first two digits, 31. Continue, to find 319 ÷ 4 = 79 R3.

FIGURE 3 Examples of Long Division For the problem on the right of Figure 3 the second application of step 3 gives a difference of 0, and the next dividend digit is 2. Since 6 is greater than 2, write 0 as the next quotient digit, and write the next dividend digit, 4, next to the 2. Then the quotient of 24 and 6 is 4, written next to the 0, and the final difference is 8424 ÷ 6 = 1404. There is no remainder
To divide by a multidigit number, the steps are essentially the same.

The rest of the steps are the same.

To find the quotient of 457 and 21, write 457 under the long-division symbol and write 21 to the left of the long-division symbol.

Divide the first two digits of 457, 45, by 21. Write the quotient, 2, above the second digit, 5. Multiply this result by the divisor, 21, and write the product beneath the first two digits of 457. Subtract.
The divisor 21 goes into 45 two times with a remainder of 3.

Bring down the 7 in 457 and write it next to the difference 3. Divide the newly created number, 37, by the divisor, 21. Write the quotient digit above the 7 in 457. Multiply this result by 21 and write the product beneath 37. Subtract. The divisor 21 goes into 37 one time with a remainder of 16.

There are no additional numbers to bring down from the dividend. Since 16 is smaller than the divisor, 16 is the remainder of the division problem. The quotient of 457 and 21 is 21 R16.Another way of writing remainders considers them as a fraction of the divisor. There are 16 parts of 21 left over in the above problem. 457÷21=21+frac(16,21) . Looking back at Figure 2, 20÷7=2+frac(6,7) . There are six-sevenths of a group of 7 boxes that cannot be counted as a group of 7.
Examples: Division of Whole Numbers

Example 1

Use long division to find the quotient of 78 and 3.

Solution:

Write the dividend under the long-division symbol and the divisor to the left of the long-division symbol.

Divide the first digit of the dividend by the divisor. Write the quotient above the first digit of the dividend. Multiply this quotient by the divisor and write the product beneath the first digit of the dividend. Subtract.

The divisor 3 goes into 7 two times with remainder 1.

Write the next digit of the dividend next to the 1. Divide the newly created number, 18, by the divisor. Write the quotient digit above the second digit of the dividend. Multiply this result by the divisor and write the product beneath the 18. Subtract.

The final difference is 0, so this division problem has no remainder. The quotient of 78 and 3 is 26.Example 2

Use long division to find the quotient of 291 and 18.
Solution:

Write the dividend under the long-division symbol and the divisor to the left of the long-division symbol.

Divide the first two digits of the dividend by the divisor. Write the quotient above the second digit of the dividend. Multiply this quotient by the divisor and write the product beneath the first two digits of the dividend. Subtract.The divisor 18 goes into 29 only 1 time with a remainder of 11.

Write the next digit of the dividend, 1, next to the difference 11. Divide the newly created number, 111, by the divisor. Write the quotient digit, 6, above the third digit of the dividend. Multiply this result by the divisor and write the product beneath 111. Subtract.The divisor 18 goes into 111 6 times with a remainder of 3. There are no additional numbers to bring down from the dividend. Because 3 is smaller than the divisor, 3 is the remainder of the division problem. The quotient of 291 and 18 is 16 R3.
Examples: Division of Whole Numbers

Example 1

Use long division to find the quotient of 78 and 3.

Summary: Division of Whole Numbers

Division undoes multiplication. Division and multiplication are often called opposite operations. The quotient of two numbers is the result of dividing one number by the other. The number being divided is the dividend. The number the dividend is divided by is the divisor. The quantity that remains when one number does not divide another evenly is known as the remainder of the division problem.

Summary: Division of Whole Numbers

Division undoes multiplication. Division and multiplication are often called opposite operations. The quotient of two numbers is the result of dividing one number by the other. The number being divided is the dividend. The number the dividend is divided by is the divisor. The quantity that remains when one number does not divide another evenly is known as the remainder of the division problem.

Introduction: An Introduction to Fractions

Fractions occur throughout professional and family life, in everything from recipes to finances to measures of efficiency and productivity. We are going to examine the format of fractions; how to add, subtract, multiply, and divide them; and common applications of fractions. An important concept in manipulating fractions is determining common denominators, and we will cover that topic as well.

Ajay is an IT consultant. His client, ClipperQuest, is a growing Internet company that markets discounted goods and services, helping to connect businesses with new customers. ClipperQuest is reviewing their computing needs and has asked Ajay to recommend the number and kinds of equipment they will need to purchase over the next several years.
Ajay has the following data to guide him: ClipperQuest would like to spread its computing purchases evenly out over 3 years rather than make all their purchases at once. Currently, ClipperQuest employs 250 people. Next year, they expect to increase their staff by frac(1,5)to keep up with business growth. The following year they expect to increase staff by an additional frac(1,10).

Roughly frac(1,4)of ClipperQuest’s employees are “external,” working primarily from home or while traveling on business. These employees must have a portable laptop; the rest can use desktop computers.

Every in-house employee at ClipperQuest needs access to a personal computer to perform at least some of his or her job tasks. However, management estimates that frac(1,3)of employees use the computer fewer than 2 hours per day. Management would like these minimal users to be able to share a computer between 2 or more people. ClipperQuest would like to have no more than 4 employees sharing a printer in order to avoid logjams.
QUESTIONS TO CONSIDER:

1. ClipperQuest would like to spread its computing purchases evenly out over 3 years rather than make all their purchases at once." How can this statement be expressed in terms of fractions?
2. If they increase staff as expected, how many employees will ClipperQuest have next year and the year after that?
3. How many laptops does Clipper Quest currently need? How many printers?
4. How many in-house employees use the computer fewer than 2 hours per day? For these employees, how many computers are needed if 2 share a computer? What if 3 employees share a computer?
Learning: An Introduction to Fractions

FRACTIONS AND NOTATION

There are three different types of fractions: proper fractions, improper fractions, and mixed numbers. All of these types of fractions contain numbers and a fraction bar. The numerator of a fraction is the number above the fraction bar; the denominator is the number below the fraction bar. In the fraction frac(5,3), 5 is the numerator and 3 is the denominator.

Numerator

The number above the fraction bar.

Denominator

The number below the fraction bar.
The denominator denotes the number of equal pieces in a whole, or unit. Here, the 3 denotes three equal parts of each unit. The numerator of a fraction counts the number of equal pieces. Here, each unit is divided into thirds, and there are 5 of them. The fraction is read "five thirds."
Proper Fractions

In a proper fraction, the numerator is less than the denominator. A few examples of proper fractions are frac(1,3), frac(5,6), and frac(7,9).
Proper fraction

A fraction whose numerator is less than its denominator.

Improper Fractions

In an improper fraction, the numerator is greater than the denominator. A few examples of improper fractions are frac(5,3), frac(19,12), and frac(23,10).
Improper fraction

A fraction whose numerator is greater than its denominator.
Mixed Numbers

A mixed number is one way of writing an improper fraction. Recall that a fraction bar is one way of indicating division. To change an improper fraction into a mixed number, first find the quotient of the numerator divided by the denominator without a remainder; that is the integer part of the mixed number. The remainder is the numerator of the proper fraction with the original denominator.
Mixed number

The sum of an integer and a fraction.
For example, for the improper fraction frac(23,10), 10 goes into 23 twice, with a remainder of 3. The 2 is equivalent tofrac(20,10) and frac(3,10) remain from the original improper fraction. So, frac(23,10)=2+frac(3,10). Write the improper fraction frac(23,10) as 2frac(3,10), and read it as "two and three-tenths." Here the "and" represents addition. Notice that, while "2" and "frac(3,10)" are written next to each other, they are added, not multiplied.

Least Common Multiple and Greatest Common Factor

There is an expression "you can't add apples and oranges." While it is true you can count "pieces of fruit," the sum cannot be labeled either "apples" or "oranges." Fractions represent kinds of pieces (thirds, halves, fifths, and so on), and only the same kinds of pieces can be added (or subtracted). That is, to add fractions, they must have the same denominators (sometimes referred to as common denominators).Multiples

A multiple of a given number has the given number as a factor. For example, the set of multiples of 4 includes 4, 8, 12, 16, 20, 24, and so on. While the product of denominators is a multiple of each denominator, it is not necessarily the most efficient multiple to use to rewrite the fractions.Multiple

For any given number, any number that has the given number as a factor.

Finding the Least Common Multiple

To add fractions with denominators 6 and 8, first consider the sets of multiples of each denominator. The set of multiples of 6 is 6, 12, 18, 24, 30, 36, 42, 48..., and the set of multiples of 8 is 8, 16, 24, 32, 40, 48, ....

The common multiples of 6 and 8 include 24 and 48. While 48 is the product of the numbers, the first number they share in the ordered sets is 24. The least common multiple (LCM) is 24.Least common multiple (LCM)

For any given set of numbers, the least number that is a multiple of each of the given numbers.Given numbers 3, 5, and 6, look at the sets of multiples:

The least number common to all three sets is 30; 30 is the LCM of 3, 5, and 6.

Notice that looking at the set of multiples of the greatest of the three numbers, 6, and then checking to see which of those multiples is a multiple of 3 and 5, might be a quicker way to find the LCM for these relatively small numbers.Notice, too, that a number that is a multiple of another number contains all the factors of that number. For very large numbers, considering the sets of multiples may be nearly impossible, even using a calculator. The best way to find the LCM is to factor the given numbers into products of primes, and construct the LCM as the product of all the factors needed as factors for any one of the numbers. Do not use any more factors than necessary.
This method works for numbers of any size. For the numbers 6 and 8, 6 has prime factors 2 and 3, while 8 has three factors of 2. So, the LCM needs three factors of 2 and a factor of 3; the LCM is 2 × 2 × 2 × 3 = 23 × 3 = 24.

Consider again 3, 5, and 6. Only 6 can be factored further: 6 = 2 × 3. So, for the LCM, in addition to 3, and 5, the 6 introduces only an additional factor of 2. So, the LCM is 2 × 3 × 5 = 30.
For the numbers 78, 102, and 174, prime factoring gives 78 = 2 × 3 × 13, 102 = 2 × 3 × 17, and 174 = 2 × 3 × 29. Each of the numbers has one 2 and one 3 as a factor, but 13, 17, and 29 are prime, and unique in each number. Therefore the LCM is 2 × 3 × 13 × 17 × 29 = 38,454. Using the sets of multiples to find this LCM would indeed be uselessly time consuming. Prime factoring is the method of choice.
Finding the Greatest Common Factor

Another important tool when working with fractions is the greatest common factor (GCF). The GCF is the product of all the prime factors the numbers have in common.Greatest common factor (GCF)

For a set of numbers, the greatest number that evenly divides each of the given numbers.

Consider the numbers 9 and 12. Factor each number into a product of prime numbers: 9 = 3 × 3 = 32, and 12 = 2 × 2 × 3 = 22 × 3. The greatest (and only) factor they share is 3, so 3 is the GCF of 9 and 12.

Consider the numbers 24, 48, and 60. In factored form, 24 = 2 × 2 × 2 × 3 = 23 × 3, 48 = 2 × 2 × 2 × 2 × 3 = 24 × 3, and 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5. All three numbers have 3 as a factor, and 2 as a factor twice, so the GCF is 2 × 2 × 3 = 12. It is also correct to factor the three numbers as 24 = 12 × 2, 48 = 12 × 4, and 60 = 12 × 5, and find 12 more quickly, as the remaining factors (3, 4, and 5) have no common factors. However, for more complicated sets of numbers, prime factoring is a more reliable route to the correct GCF.
It can be easy to confuse factors and multiples. The factors of a number are always less than or equal to the number. The multiples of a number are always greater than or equal to the number.

Example 1

a. What is the least common multiple of 4 and 6?

b. What is the LCM of 49 and 77?
Writing Equivalent Fractions

There are an infinite number of ways to write a numeric expression that is equivalent to a given expression. For example, 5 = 3 + 2 = 17 − 12 = 2 × 7 − 9 = ... and so on. Similarly there are an infinite number of fractions equivalent to a given fraction.

Two fractions are equivalent if the numerator and denominator of one fraction are the same multiple of the numerator and denominator of the other fraction. So, to create an equivalent fraction, multiply or divide the numerator and denominator of a given fraction by the same quantity.

Equivalent fractions

Two fractions in which the numerator and denominator of one fraction are the same multiple of the numerator and denominator of the other fraction.Finding Equivalent Fractions by Raising to Higher Terms

Recall that adding fractions requires fractions with the same denominator (so as to add like pieces). Usually this requires raising the fractions to higher terms. Raising a fraction to higher terms makes both the numerator and denominator of the fraction greater by multiplying them each by the same number.

Finding Equivalent Fractions by Raising to Higher Terms

Recall that adding fractions requires fractions with the same denominator (so as to add like pieces). Usually this requires raising the fractions to higher terms. Raising a fraction to higher terms makes both the numerator and denominator of the fraction greater by multiplying them each by the same number.

To rewrite two fractions as equivalent fractions with the same denominator, find the LCM of the denominators. Then multiply each fraction's numerator and denominator by whatever factors are needed to make its new denominator the LCM.

Least common denominator (LCD)

The LCM of the denominators of two or more fractions.

For example, write equivalent fractions with the same denominator for the fractions frac(5,8) and frac(1,6). Find the LCM of 8 and 6. Factoring, 8 = 2 × 2 × 2 and 6 = 2 × 3. So the LCM is 2 × 2 × 2 × 3 = 24. To write frac(5,8) as an equivalent fraction with 24 as the denominator, note that the denominator is missing the factor 3 from the LCM. So multiply the numerator, 5, and the denominator, 8, by 3 to write the equivalent fraction frac(5,8)=frac(5×3,8×3)=frac(15,24). Similarly, 6 must be multiplied by 4 to make the denominator 24. The LCD of the fractions is 24.

Writing a Fraction in Simplest Terms

It is usually easier to deal with smaller, rather than larger, numbers. For fractions, to write a fraction as a simpler equivalent fraction in lowest terms (reduced form or simplest terms), divide both the numerator and denominator by a common factor. The most efficient way to do this is to find the GCF. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 16 are 1, 2, 4, 8, and 16. To write frac(12,16) in lowest terms it is possible to divide numerator and denominator each by 2, getting frac(12 ÷ 2,16 ÷ 2)=frac(6,8) , and then, repeating the process, getting frac(6 ÷ 2,8 ÷ 2) = frac(3,4). Because 3 and 4 have no common factor—that is, they are relatively prime —this is the reduced form of the fraction.

Lowest terms (for a fraction)

The numerator and denominator of the fraction are relatively prime.

Relatively prime numbers

Numbers that have no common factors.

However, notice that it is more efficient to recognize that 4 is the greatest common factor of 12 and 16. Dividing numerator and denominator by 4 reduces frac(12,16) to frac(3,4)in one step.

Cancellation

Cancellation is a notation for showing the reduction of a fraction to lowest terms, particularly when factors are shown. It illustrates the division of the numerator and denominator by the same factor(s) by striking out the common factor(s). Looking back at frac(12,16), write frac(12,16) =frac(2 × 2 × 3,2 × 2 × 2 × 2). Then cancel (divide out) the factors common to the numerator and denominator: frac(12,16) =frac(strikeup(2) × strikeup(2) × 3,strikeup(2) × strikeup(2) × 2 × 2) and write the reduced fractionfrac(12,16) = frac(3,2 × 2) = frac(3,4). This is equivalent to frac(12,16) = frac(12 ÷ 4,16 ÷ 4) = frac(3,4). It is equally correct to writefrac(12,16) = frac(strikeup(4) × 3,strikeup(4) × 4) = frac(3,4).

Improper and Mixed Number Equivalent Fractions

There is nothing in the work of writing equivalent fractions, whether as higher or lowest terms, that requires the fraction to be a proper fraction. Improper fractions are treated in the same way as proper fractions. Mixed numbers may need to be rewritten as improper fractions, though sometimes it is sufficient to keep the whole number and write a fraction equivalent to the fractional part of the number.

To write a mixed number as an improper fraction, reverse the process of writing an improper fraction as a mixed number. Recall two relationships: the whole number part is the quotient of the numerator and denominator of the improper fraction, and the denominator of the fractional part is the denominator of the improper fraction. So the original numerator is the product of the quotient and the denominator plus the numerator of the fractional part, which was the remainder of the division.

For example, write frac(23,5) as a mixed number. The quotient of 23 and 5 is 4, with a remainder of 3, so frac(23,5) = 4frac(3,5). To rewrite 4frac(3,5) as an improper fraction, multiply the whole number (quotient) 4 by the denominator 5, add 3 to it, and divide by 5. So 4frac(3,5) is equivalent to the improper fraction frac(4 × 5 + 3,5) = frac(23,5).

Example 3

Find fractions with the same denominator that are equivalent to frac(3,5) and frac(2,7).

Solution:

Both 5 and 7 are prime, so the LCM is just their product, 35. Multiply 3 and 5 in the fraction frac(3,5) by 7: frac(3 × 7,5 × 7)= frac(21,35). And multiply 2 and 7 by 5 in the fraction frac(2,7)to get frac(2 × 5,7 × 5) = frac(10,35).

Example 4

Write frac(32,72) in lowest terms.

Solution:

Factoring 32 and 72, 32 = 8 × 4 and 72 = 8 × 9. Because 4 and 9 are relatively prime, 8 is the GCF. (Alternately, factoring 32 and 72 into primes, 32 = 25 and 72 = 23 × 32, showing that the GCF is 23 = 8.) In either case, divide the numerator and the denominator by the GCF to write the fraction in lowest terms: frac(32,72) = frac(32 ÷ 8,12 ÷ 8) = frac(4,9).

Example 5

Write frac(15,12) and frac(16,15) as equivalent fractions with the same denominator.

Solution:

Factoring 12 and 15, 12 = 22 × 3 and 15 = 3 × 5. The LCM of 12 and 15 is 22 × 3 × 5 = 60. So, multiply the numerator and denominator of frac(15,12) by 5, and the numerator and denominator of frac(16,15) by 4. The equivalent fractions are frac(15,12)= frac(15 × 5,12 × 5) = frac(75,60) and frac(16,15) = frac(16 × 4,15 × 4) = frac(64,60).

Example 6

Write 8frac(5,6) and frac(10,21)as equivalent fractions with the same denominator.

Solution:

First, write 8frac(5,6) as an improper fraction. Multiplying 6 and 8 gives 42, and adding the numerator 5 gives 47 as the numerator for the denominator 6. The improper fraction is frac(47,6).

Then find the LCM of 6 and 21. Because 6 = 2 × 3 and 21 = 3 × 7, the LCM is 2 × 3 × 7 = 42.

The equivalent fractions are frac(47,6) = frac(47 × 7,6 × 7) = frac(329,42) and frac(10,21) = frac(10 × 2,21 × 2) = frac(20,42).

Summary: An Introduction to Fractions

There are three different types of fractions: proper fractions, improper fractions, and mixed numbers. The denominator denotes the number of equal pieces in a whole, or unit. Here, the 3 denotes three equal parts of each unit. The numerator of a fraction counts the number of equal pieces. In a proper fraction, the numerator is less than the denominator. In an improper fraction, the numerator is greater than the denominator.

Two fractions are equivalent if the numerator and denominator of one fraction are the same multiple of the numerator and denominator of the other fraction. So, to create an equivalent fraction, multiply or divide the numerator and denominator of a given fraction by the same quantity. It is usually easier to deal with smaller, rather than larger, numbers.

For fractions, to write a fraction as a simpler equivalent fraction in lowest terms (reduced form or simplest terms), divide both the numerator and denominator by a common factor. There is nothing in the work of writing equivalent fractions, whether as higher or lowest terms, that requires the fraction to be a proper fraction. Improper fractions are treated in the same way as proper fractions. Mixed numbers may need to be rewritten as improper fractions, though sometimes it is sufficient to keep the whole number and write a fraction equivalent to the fractional part of the number.