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DEFINITION. A fraction is a part of a whole. If you cut a pie into 12 equal pieces and ate 5 pieces, you would have eaten 5/12 (five-twelfths) of the pie (5/12 means "5 parts out of 12 equal parts").

Consider the pie mentioned above. What fraction of the pie still remains? (Use the "pie" chart above.)

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In text, fractions are usually written in a horizontal form, such as "5/12," for ease of reading. When performing calculations, fractions are usually written in a vertical form, such as 5.

12

The top (or first) number is called the numerator. The number on bottom (or second number) is called the denominator.

In the fraction 5/12, "5" is the and

7

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MULTIPLYING AND DIVIDING FRACTIONS. It may seem strange to begin with multiplication and division of fractions rather than addition and subtraction. The multiplication and division functions are relatively simple operations, however, and multiplication must be understood before addition and subtraction of some fractions can be performed.

To multiply fractions:

(1) Multiply the numerators together;

(2) Multiply the denominators together; and

(3) Place the product of the numerators over the product of the denominators.

3 5 3 x 5 15

Solution to Frame 2-2.

5 numerator

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Let's try four more.

2 2 4 2

2

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When you divide a fraction by another fraction, invert (flip) the second fraction and multiply the fractions.

3 5 3 4 3x4 12

To solve the problem 1/2 ) 1/4 (one-half divided by one-fourth), you would invert the (choose one -- 1/2; 1/4) and multiply.

NOTE: Just like 18 ) 6 asks, "How many groups of 6 are in 18?", so

Solution to Frame 2-4.

a. 1 

b. 3

4 8

c. 14 d. 6

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Solve these problems.

a. 1/4 ) 1/2 =

b. 1/2 ) 1/6 =

c. 2/3 ) 3/7 =

Solution to Frame 2-5.

1

4

1/2 ) 1/4 =

1/2 x 4/1 = 4/2 = 2

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ADDING AND SUBTRACTING FRACTIONS. Have you ever heard someone say, "You can't add apples and oranges"? Well, there is a similar rule when working with fractions -- you can't add fractions with different denominators.

This means that, in order for two fractions to be added together, the

numbers must be the same.

a. Top

a. 2 

b. 6

4 2

c. 14 d. 9

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If two fractions have the same denominator (called a "common denominator"), you add the fractions by simply adding the numerators together and putting the sum over the common denominator. YOU DO NOT ADD THE DENOMINATORS TOGETHER. For example:

8 8 8 8

More than two fractions can be added together at one time as long as they all have the same denominator. Complete the following exercise.

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When you add two fractions with the same denominator together, you add the numerators and put the sum over the common denominator. Likewise, when you subtract two fractions with the same denominator, you subtract the numerators and put the difference over the common denominator.

For example: 4 _ 1 = 4-1 = 3

8 8 8 8

Solve the following subtraction problems:

a. 9 _ 5 =

12 12

b. 17 _ 5 =

31 31

c. 5 _ 2 =

2+1+5+4 = 12

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Adding fractions with common denominators is like adding apples and

8 8 8

is much like saying "3 apples plus 1 apple equals 4 apples," with "apples" being "eighths."

8 4

a. 9 - 5 = 4

12 12

b. 17 - 5 = 12

31 31

c. 5 - 2 = 3

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Have you ever had a friend named James? Some people may call him "James," some may call him "Jim," some may call him "Jimmy," and his little sister may even call him "Bo," but he is the same person regardless of what you call him. Fractions also have many different "names" or forms, and you can change the fraction's name when you need to.

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The four "pies" shown below are the same size, but have been sliced differently. The same amount of pie has been removed (the shaded area), but the number of slices removed are different. The amount of the shaded area is the same in all four cases, but the name of the fraction is different in each case. Name the shaded areas.

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So, if you're "adding apples and oranges," see if you can change the "apple" name of the fraction to the "orange" name (or vice versa).

8 4

Can you change the name of the second fraction (the fraction with the smaller denominator) so that it will have the same denominator as the other fraction? (Refer back to Frame 2-12.)

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If two fractions have different denominators, see if the larger denominator is a multiple of the smaller. That is, can the smaller denominator be multiplied by a whole number and the product be the larger denominator? If so, then the larger denominator can be the common denominator.

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8 4 8 8 8 8

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2 2

The number "1" has several forms, or names.

Some are 1 , 2 , 3, and 4.

1 2 3 4

In each case, the numerator and the denominator are the same. Multiplying a fraction by one of the forms of "1" allows you to change the appearance of the fraction so that it has a different denominator. Now let's find some different names (forms) of the fraction 1/2.

(Remember, multiplying a fraction by "1" [regardless of the form of "1" you use] yields a fraction whose actual value has not changed, even if its form has changed.)

5

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2 6

The denominators are not the same, so you must find a common denominator. Since "6" is a multiple of "2" (2 x 3 = 6), you can change 1/2 to a form that has the same denominator as the other fraction.

3473

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Since 3/6 is the same as 1/2, you can substitute (switch) 3/6 for 1/2 and work the problem.

3

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Subtraction of fractions works very similar to addition. Find the common denominator, change one or both of the fractions until they have the same denominator, and subtract the numerators.

Work these problems on your own:

2 6

4

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What happens, though, when one denominator is not a multiple of the other. If you can't change the apples to oranges or oranges to apples, maybe you can change them both to grapefruit. That is, find a common denominator to which both denominators can be changed.

The common denominator will be a multiple of the _________________

6 6

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Suppose you had two fractions, one with a denominator of "3" and the other with a denominator of "4." One method of getting a common denominator is to multiply the denominators together. For a problem with two denominators ("3" and "4"), a common denominator would be "12" (3 x 4 = 12 and 4 x 3 = 12).

Finish solving the following problem:

denominator

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Solve this subtraction problem:

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Although multiplying the denominators together will always give you a common denominator, sometimes a smaller common denominator can be found. Consider the previous problem? Can you think of a common denominator for 5/6 and 3/8 that is smaller than 48? What number will both 6 and 8 divide into and the quotients be whole numbers (no remainders)?

____________

6x8 8x6 48

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You can add and subtract fractions in the same problem. Just make sure that each denominator divides evenly into the common denominator. Try this problem.

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The solution to Frame 2-24 is shown below (not sufficient space in column to right). The problem is worked two ways. The first shows the problem worked with the common denominator being the product of all of the denominators (2x3x5x7x9 = 1890). The second shows the problem being worked with a lower common denominator (2x5x7x9 = 630). Did you notice that the denominator "3" divides evenly into the denominator "9"?

2 3 5 7 9

2x3x5x7x9 3x2x5x7x9 5x2x3x7x9 7x2x3x5x9 9x2x3x5x7

1890 1890 1890 1890 1890 1890

1890 1890

NOTE: There are several ways of adding and subtracting the numerators. One way is shown above. Another (and usually better) way is to add all of the pluses (positives) together, add all of the minuses (negatives) together, and subtract as shown below.

(945+378+210) – (630+270) = 1533 – 900 = 633

Using 630 (2x5x7x9) as the common denominator

2x5x7x9 3x2x5x7x3 5x2x7x9 7x2x5x9 9x2x5x7

630 630 630 630 630 630

630 630

The converted fractions can also be added and subtracted as below:

633 or 211

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REDUCING FRACTIONS. Usually, you will want your answers "reduced." That is, you will want to use the name (form) of the fraction that has the smallest denominator possible that will still allow both the numerator and denominator to remain whole numbers. The fraction is then "reduced to its lowest form."

Below are three fractions, each in different forms. Circle the reduced form of each fraction.

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To reduce a fraction, find the largest whole number that can be divided into both the numerator and denominator evenly (no remainders). Then divide both the numerator and denominator by that number.

When reducing a fraction you must divide both the and the

by the same number.

2 3 8

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For example, the fraction 8/12 can be reduced as shown below.

12 12 ) 4 3

Sometimes you may have to divide more than once to reach the reduced form. For instance, the example can also be worked as follows:

12 12 ) 2 6

6 6 ) 2 3

Reduce the following fractions.

10 30

Solution to Frame 2-27.

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Solve the following problems. Reduce the answers.

10 10

3 6

8 3

Solution to Frame 2-28.

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Instead of looking for the biggest number that divides into both the numerator and denominator evenly (called the "largest common denominator"), you can divide by prime numbers. A prime number is a number that cannot be divided by any whole number other than itself and 1 without leaving a remainder. Prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on. Begin with "2." If "2" divides into both the numerator and denominator evenly (no remainder), then reduce the fraction by two. Take the new fraction and try to reduce the new numerator and denominator by "2" again. Continue until the fraction can no longer be reduced by 2. Then do the same with the next prime number ("3"). Continue until there is no whole number (other than 1) that will divide into both the numerator and the denominator evenly. For example:

72 72 ) 2 36 36 ) 2 18 18 ) 2 9 9 ) 3 3

NOTE: On 6/9, 2 will divide evenly into 6 but not into 9. Therefore, you go to the next prime number.

A variation is to break both the numerator and the denominator down into prime factors (prime numbers that yield the original number when multiplied). If the same factor appears in both the numerator and denominator, mark it out. Mark out factors one at a time. [For example, if you have "2" as a factor 3 times in the numerator but only twice in the denominator, you can only mark out two of the "2's" in the numerator.] When you are finished, multiple the remaining factors to obtain the reduced fraction. For example:

72 2 x 2 x 2 x 3 x 3 2 x 2 x 2 x 3 x 3 3

10 5

6 2

24 4

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IMPROPER FRACTIONS. Notice that the answer to the last problem in Frame 2-29 is unusual in that the fraction has a numerator that is larger than the denominator. Such a fraction is called an improper fraction.

An improper fraction is a fraction in which the numerator is equal to or larger than the denominator.

A proper fraction is a fraction in which the numerator is less than the denominator.

3/5 is a(n) .

5/3 is a(n) .

200 = 2x2x2x5x5 =

375 3x5x5x5

2x2x2x5x5 = 2x2x2=

3x5x5x5 3x5

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When you worked the problem 7/8 divided by 1/4, you came up with an improper fraction (7/2) as the answer. When your answer is an improper fraction, you will usually change it to its mixed number form (this is also referred to as "reducing"). To change an improper fraction to a mixed number, divide the numerator by the denominator and put the remainder (if any) over the denominator.

3

5 proper fraction

5

3 improper fraction

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Reduce the following improper fractions. If the remainder is zero, then the improper fraction reduces to a whole number.

3

14

3

6

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In working some problems, it may be more convenient to multiply by a fraction rather than a mixed number. In such cases, you need to know how to change a mixed number into an improper fraction.

One way of thinking about a mixed number is as a whole number plus a fraction. To change a mixed number to an improper fraction

(1) Change the whole number to an improper fraction with the same

denominator as the fraction, then

(2) Add the two fractions together.

1 3 1 x 3 3 3 3 3 3

Change the following mixed numbers to improper fractions.

Solution to Frame 2-33.

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A shortcut for changing a mixed number to an improper fraction is to:

(1) Multiply the whole number by the denominator,

(2) Add the numerator to the product, and

(3) Put the sum over the denominator.

3 3 3

Change the following mixed numbers to improper fractions using the shortcut method:

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To change a whole number to an improper fraction:

(1) Choose the desired denominator,

(2) Multiply the whole number by the denominator, and

(3) Place the product over the denominator.

For example:

5 5

Fill in the following:

2 = __

2

7 = __

5

10 = __

Solution to Frame 2-35.

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2 = 4/2

7 = 35/5

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SELF-TEST. Complete the self-test exercises below. After you have worked all the exercises, turn to the solution sheet on the following page and check your work. For each exercise answered incorrectly, reread the appropriate lesson frame(s) and rework the exercise.

1. Add and reduce:

3 3

2 8

4 3

2. Subtract and reduce:

7 7

8 2

8 3

3. Multiply and reduce:

2 5

3 4

4. Divide and reduce:

5 8

5 10

5. Change to improper fractions:

b. 7 = __

4

(a + b) (c + d) =

ac + ad + bc + bd

(2 + 1/2) (3 + 1/3) =

(2)(3) + (2)(1/3) +

(1/2)(3) + (1/2)(1/3) =

6 + 2/3 + 3/2 + 1/6 =

6 + 4/6 + 9/6 + 1/6 =

6 + 14/6 =

6 + 2 2/6 =

(6+2) + 1/3 =

CHECK:

5/2 x 10/3 =