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SAMPLE LESSON 3
Joseph Buczek - Instructor  
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                          LESSON 3 - Fractions

This lesson describes the principles underlying the use of fractions in algebra

LESSON 3 - What Is A Fraction?
A Fraction is a mathematical expression that is a picture of, or representation of some part of of a whole quantity. Sometimes the representation of a part of a whole quantity is called a ratio of two quantities. Both fractions and ratios are expressed as one number, representing the part of the whole, called the numerator, over another number, representing the whole, or total, called the denominator.

Working With Fractions

There are whole numbers and there are parts of whole numbers. A part of a whole number is called a fraction of the whole number. Parts of whole numbers, or fractions, can be confusing to work with. One of the primary reasons for this is that a fraction can involve different forms, with different numbers, that represent the same fraction.

The word fraction is a term that one or more of equal portions of a whole quantity. The words equal portions signify that a fraction represents a whole item cut into equal portions and one or more of the equal portions taken out of the item.

One of the easiest ways to visualize the idea behind a fraction is to consider a pizza pie. Let us think of the pizza being cut into four pieces (slices). Next, consider the pizza with one slice removed, as illustrated below.




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The one slice removed is one-fourth (1/4) of the pizza, because there were originally four slices and each slice represents one-fourth of the total pizza. With one slice removed there are three-fourths remaining in the original pizza.

Next, consider two slices removed from the pizza, as illustrated below.


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The two slices removed represent two-fourths (2/4) of the pizza. With two slices removed there are two-fourths (2/4) remaining in the original pizza.

Next, consider another pizza cut into eight pieces. Consider this pizza with one slice removed, as illustrated below.



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The one slice removed from this pizza is one-eighth (1/8) of the pizza, because there were originally eight slices and each slice represents one-eight of the total pizza. With one slice removed, there are seven eights (7/8) remaining from the original pizza.

Finally, consider 6 slices removed from this pizza, as illustrated below.



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The six slices of pizza removed from the whole pizza that was originally cut into eight pieces represents six-eighths of the total pizza. With six slices removed, there are two eights(2/8) of the pizza remaining from the original pizza.

DEFINITIONS OF THE NUMERATOR AND THE DENOMINATOR

The first lesson to be learned from the above is that an objects, such as the pizzas, above, can be cut into different numbers of pieces, or slices. In the first example, the pizza was cut into 4-slices. In the second example, the pizza was cut into 8-slices. It is conceivable that a pizza could be cut into 5, 10 or even 26 pieces.

Because a pizza can be cut into different numbers of slices and, the number of slices taken out has to be considered in terms of the original number of slices that the pizza was cut into, the number of slices that the pizza is cut into has to be reckoned as a fundamental reference when any fraction of this pie is calculated. The number of slices that a pizza is originally cut into serves as a basis for the class of the calculations to be performed with fractions regarding the pizza. The number of slices that a pizza is originally cut into is a characteristic that is held in common with any calculations performed regarding fractions.

The number of slices that a pizza is cut into determines a class of one kind of unit in a system of numbers. Anytime you are using a particular class of number unit out of a system numbers you are using a particular denomination. The word denomination signifies a name for a thing out of a class of things. When a pizza is cut into four slices, the denomination is four, because you could have cut the pizza into any number of slices (say, up to 24, before the pieces got too small to handle). With your choice of 4 slices, the denomination is 4. When you cut the pizza into 8 slices, the denomination is 8.

Another example of this would be you paying $80.00 that you owe to someone. You could pay in eight $10.00 bills. In this case the denomination that you paid in would be ten dollar bills. If you paid in sixteen $5.00 bills the denomination that you paid in would be five dollar bills. If you paid in all $1.00 bills, the denomination that you paid in would be one-dollar bills.

Because the number of equal slices that a whole pizza is divided will be used in a fractional notation, this value is called the denominator. The denominator is the name for the number that is below the line in a fraction. The number of slices removed from a pizza is the numerator, the numerator is the count of the number of slices removed from the pizza. The numerator is the number above the line in a fraction.

Although the above description of the numerator and denominator of a fraction was in terms of slices of a pizza, it should be understood that the concept of the fraction holds for anything that can be divided into parts. A dollar can be divided into 100 pennies. The distance from New York to California is about 3,000 miles, etc. Therefore, fractions will be useful for measuring anything that can be divided into parts.

ADDING AND SUBTRACTING FRACTIONS

If the sizes of the pieces into which two 12" pizzas are cut are equal, it makes sense that you can add the pieces removed (because they are the same size). But, suppose one 12" pizza is cut into six slices and a second 12" pizza is cut into eight slices. Would you trade two of your slices from the first pizza for two slices from the second pizza? Of course not! The slices from the first pizza are larger than the slices from the second pizza. You can't add two slices from the first pizza to three slices from the second pizza to get five slices of pizza. The slices are not the same size! Let us consider this addition mathematically.

                                            2/6 + 3/8 = 5/? (? = 6?, 8?)

We are not able to perform the addition in this fraction because the denominators are different. In order to perform any addition the denominators must be the same!

There is a trick that can be used to find a mathematical answer to the above addition. The solution will not be in terms of a slice size of 1/6 or 1/8, but, will be a mathematical determination of the size that fits among the 1/6 and the 1/8 sizes. The method for determining this other size of slice is called the common denominator and, as you know, is found by multiplying the two denominators together. The denominators are 6 and 8. Multiplying 6 times 8 gives us 48. This is our new denominator. But, since we multiplied the first denominator by 8, we must multiply the numerator in that fraction by 8, so,

                                          (2/6)( 8/8) = (2)(8)/(6)(8) =16/48

The denominator in the 3/8 fraction has been multiplied by 6 to make it equal to 48. In order to keep this fraction the same value, it's numerator must be multiplied by 6, then,

                                            3/8 = (3/8)(6/6) = (18/48)

Adding the fractions with the common denominators of 48 results in

                                             16/48 + 18/48 = 34/48

The solution to the original problem, then, is

                                                     2/6 + 3/8 = 34 /48

The reason that we had to use a common denominator was that we tried to add different sizes of pizza slices. All we could do was find a common sized slice that would give us a mathematical solution to our problem, but, not one that fit either of our original size slices.

The above indicates that fractions can be added or subtracted if the denominators of the fractions involved are the same value. Let us solve a few problems involving fractions.

PROBLEM 1: Add 3/16 and 12/16

SOLUTION: Since the denominators in this problem are the same value, all we have to do to add these fractions is to write the common denominator on the other side of an equal sign, add the two numerators, and put the sum of these numerators over the common denominator, as

                                       3/16 + 12/16 = (3 + 12) /16 = 15/16

PROBLEM 2: Subtract 144/128 -188/128

SOLUTION: The denominators are the same so place the common denominator on the right side of an equal sign place attached to the problem and subtract the numerators, as,

                          144/128 - 188/128 = (144 - 188)/128 = -44/128

More than two fractions can be added or subtracted in one operation as the following problem shows.

PROBLEM 3: Combine the following fractions into a single fraction.

                                       9/37 + 55/37 - 74/37 + 17/37

SOLUTION: Since the denominators are the same, we may combine all the numerators over the common denominator, 37, as,

                 9/37 + 55/37 - 74/37 + 17/37 = (9 + 55 - 74 + 17)/37 = 7/37

A fraction such as 21 over 6 is called an improper fraction because the numerators has a larger value than the denominator. In terms of the pizza example this number would mean taking 21 slices of pizza from a pie that was only cut into 8 slices. The meaning, of course, is that from several pies of six slices each we took out 21 slices. The number of pies, and the fraction of a pies slices removed may readily be determined by dividing the denominator into the numerators, as in,

                                                        21/6 = 3.5

The fraction 21/6, then signifies, that out of pizzas cut into 6 slices each, three and one-half pies would be needed to make 21 slices of pizza.

A number such as 12(2/7) is called a mixed number since a whole number, 12, is combined with a fraction 2/7. This number signifies 12 whole units of an item and 2/7 of that item which means out of one unit of the item divided into 7 parts, 2 of the parts are removed and combined with the 12 whole units of that item. Since the division of the whole unit is 7, the whole number 12 2/7 may be converted to a fraction by multiplying the whole number, 12, by 7 and placing this product over 7, as,

                                             12(2/7) = (12)(2)/7 = 24/7

In terms of algebraic symbols the complete concept of the addition of fractions can be expressed as

                                               a(b/c) = (ab)/c}

MULTIPLYING FRACTIONS

The rule for multiplying two fractions is: Multiply the numerator of the first fraction by the numerator of the second fraction, and, multiply the denominator of the first fraction by the denominator of the second fraction. The following example illustrates the procedure for multiplying two fractions.


PROBLEM 4: Multiply (2/3)(4/3)

SOLUTION: These fractions are multiplied together by first, multiplying the numerators together, and, then, multiplying the denominators together. The products obtained form the numerator and denominator for the product fraction, as,

                                       (2/3)(4/3) = (2)(4)/(3)(3) = 8/9

Algebraically, multiplications of fractions can be expressed as

                                      (a/b)(c/d) = (ac)/(bd)

DIVISION OF FRACTIONS

Dividing two fractions means solving a problem like

                                               (2/3)/(6/8)

Two fractions are divided by using the following rule:

Two fractions are divided by inverting the fraction in the denominator, and, multiplying it with the fraction in the numerator.

The following example shows the procedure for dividing the above two fractions.

                       (2/3)/(6/8) = (2/3)(8/6) = (2)(8)/(3)(6) = 16/8

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