# CHAPTER 7: THE WORLD OF FRACTIONS

SECTION 7.1: FRACTIONS AND RECIPROCALS

Clara Latimer Bacon August 23, 1866 - April 14, 1948

Clara Latimer Bacon was born in Hillsgrove, McDonough County, Illinois of a pioneer New England family. She was graduated from Hedding College, Abingdon, Illinois in 1886. After a year of teaching she entered Wellesley College. In 1890 she received her B.A. degree from Wellesley College, then taught secondary school in Kentucky for one year and in Illinois for five years. In 1897, at the invitation of Dr. Goucher, she began teaching at the Women's College of Baltimore (now Goucher College) as an instructor of mathematics. She arrived in Baltimore with her sister Agnes, their mother, and servant, Ida Lindsay, who took care of Clara for the rest of her life. During her time at Goucher she continued her graduate studies at the University of Chicago during the summer quarters from 1901-1904, earning a master's degree from the University of Chicago in 1904 with a thesis on " The determination and investigation of the real chords of two conics which intersect fewer than four real points." In October 1907 she began graduate work at Johns Hopkins University in mathematics, education and philosophy. A fellowship from the Baltimore Association for Promotion of University Education of Women allowed her to spend the 1910-1911 academic year at the university. In 1911 she became the first woman to receive a Ph.D. in mathematics from Johns Hopkins University. Her dissertation was on "The Cartesian oval and the elliptic functions p and σ," later published in the American Journal of Mathematics, Vol. 35, No. 3. (July, 1913), pp. 261-280.

You are all familiar with fractions like 10/2, 30/6, and 5/2. In a fraction there is a "numerator" or "dividend" and a "denominator" or "divisor." In , 10/2, 10 is the numerator or dividend and 2 is the denominator or divisor.

The result of division is called the "quotient."  In 10/5 = 2: 2, the answer, is called the quotient.

If you are given a fraction and switch the position of the numerator and the denominator the resulting fraction is called the "reciprocal" of the original fraction. The signs of reciprocals are always the same.

ex. 1) 3/7 and 7/3 are reciprocals.

ex. 2) -4/1 and 1/-4 are reciprocals.

ex. 3) 5 and 1/5 are reciprocals.

ex. 4) -1/6 and -6 are reciprocals.

SECTION 7.2: COMPLEX FRACTIONS

When a fraction exists in a numerator or the denominator of a fraction, this type of fraction is called a "complex fraction."

Here are a number of examples of complex fractions.

 ex. 1) (3/5) / (2/7) 3/5 is the numerator and 2/7 is the denominator ex. 2) (3/5) / 9 3/5 is the numerator and 9 is the denominator ex. 3) 2 / (5/7) 2 is the numerator and 5/7 is the denominator

SECTION 7.3: DIVISION BY 0

You were taught a long time ago how to check if your quotient was correct when you did division and the quotient had a remainder of 0. For example, you know 10/2 = 5 is correct. WHY?

 DIVISION CHECK RULE quotient = numerator/denominator If numerator = quotient x denominator

I will do a few examples to clarify the Division Check Rule.

 ex 1) Is 5 = 20/4? Yes; 20 = 5 x 4 ex 2) Is 25 = 50/2? Yes; 50 = 25 x 2 ex 3) Is 7 = 18/3 No; 18 ≠ 7 x 3 ex 4) Is 0 = 0/13 ? Yes; 0 = 0 x 13

The above is a very important property when we want to know if we are dividing properly. If you know your multiplication tables this will not be problematic. But guess what? There is an 'odd case" that we will consider where the above property runs into difficulties.

Now what if we divide by 0? Let's see what happens:

If we divide 27 by 0, i.e. 27/0, what do you think our answer would be?

Let's take a guess at our answer. How about 0?

Let's check using the Division Check Rule: If 27/0 = 0 then 0 x 0 should equal 27, which it does not. So 0 can't be the answer!

Oddly, when we divide by 0, we cannot determine if our answer is correct using the Division Check Rule! Therefore mathematicians have agreed on the following.

 DIVISION BY 0 RULE Division by zero is called "UNDEFINED." When you divide by zero a "meaningful" answer cannot be found.

The Division By 0 Rule may not satisfy your mathematical curiosity but that is all we have for now to deal with the problematic situation of a dividing any number by 0.

SECTION 7.4: FACTORS AND TERMS - THE KEYS TO UNDERSTANDING CANCELING IN FRACTIONS

A "term" is any math expression that is added or subtracted. A "factor" is any math expression that is multiplied. These two definitions are two of the most important concepts in algebra! Below are some examples:

 ex. 1) 5 + 6 + 3 + 8 5, 6, 3, and 8 are all terms ex. 2) -3 - 5 - 6 -7 3, 5, 6, and 7 are all term ex. 3) (5)(4)(3) 5, 4, and 3 are factors ex. 4) (-4)(-5)(-8) -4, -5, and -8 are factors.

ex. Name the terms and factors of (6)(4) + 3 + (6)(5) + 7

As you know, parentheses mean multiplication. In example 5 above, we have "factors that are parts of terms." Being able to see when "factors are parts of terms" is extremely important.

ex. Name the terms and factors of

Where are the factors and where are the terms in the above example? Can you see the trees through the forest? And the forest through the trees? I will examine this example below.

As you know, parentheses pressed next to each other, ( )( ) , with no operation between them mean multiplication. In the example above we have "terms that are parts of factors" and "factors that are parts of terms." Being able to see these distinctions is very important as you will see in the next section.

SECTION 7.5: EQUIVALENT FRACTIONS

 EQUIVALENT FRACTIONS If one fraction is equal to another fraction then those fractions are said to be "equivalent." ex. 1) 12/24 is equal to 1/2 hence these are equivalent fractions ex. 2) 15/20 is equal to 3/4 hence these are equivalent fractions

 CREATING EQUIVALENT FRACTIONS To construct equivalent fraction to a/b multiply "a" and "b" by the same integer other than 0. Comment: An integer is ...-3, -3, -1, 0, 1, 2, 3, ...

ex. 1) Create two equivalent fractions for 3/7

I multiplied the numerator and denominator by the integer 5.

I multiplied the numerator and denominator by the integer 10.

Therefore,

SECTION 7.6: THE RULE OF CANCELING AND REDUCING A FRACTION TO SIMPLEST FORM USING PRIME NUMBERS

A "prime number" is a number that is ONLY divisible by 1 and itself producing a remainder of 0. 2, 3, 5, 7, 11, 13 are examples of prime numbers.

- 2 is prime since it is divisible by 1 and itself. 2/1 and 2/2
- 5 is prime since it is divisible by 1 and itself. 5/1 and 5/5
- 29 is prime since it is divisible by 1 and itself: 29/1 and 29/29
- 9 is NOT prime: 9/1 and 9/9 but 9/3
- 12 is NOT prime: 12/1 and 12/9 but 12/2 and 12/3 and 12/4 and
12/6

REWRITING A NUMBER AS A PRODUCT OF PRIME NUMBERS

Many students will say that they do not know how to rewrite a number as a product of prime numbers. Here are some examples of numbers rewritten as a product of prime numbers.

 ex. 1) 12 = 2 x 2 x 3 12 is written as a product of primes ex. 2) 70 = 2 x 5 x 7 70 is written as a product of primes

Before I show you how to rewrite a number as a product of prime numbers please learn the following rules of what a number is divisible by.

- A number is divisible by 2 if the last digit is divisible by 2, which means it must be 0, 2, 4, 6, or 8.
- A number is divisible by 3 if the sum of the digits is divisible by 3.
- A number is divisible by 5 if the last digit is divisible by 5, which means it must be 0 or 5.

example) Rewrite 60 as a product of prime numbers.

Here is what I believe is the simplest way to rewrite a number as a product of primes.

The original number, in this case 60, will be equal to all the boxed prime numbers multiplied together.

 RULE OF CANCELING When reducing fractions, you can only cancel common factors of the numerator and denominator. These common factors cannot be  parts of terms.

You may be wondering why I spent so much time with factors and terms. Understanding of the distinction between terms and factors will enable you to understand canceling in fractions.

You know that 5/10 is the same a 1/2. Well, I hope you do! But why? Can you explain this mathematical truth in terms of arithmetic? Well, I think you were taught that which enabled you to cancel the 5's in the dividend (numerator) and the divisor (denominator).

 EXAMPLE OF RULE OF CANCELING Comment: The common factor, 5, in the numerator and the denominator was cancelled.

 REWRITING A NUMBER AS A PRODUCT OF PRIME NUMBERS If you rewrite a fraction's numerator and denominator as a product of prime numbers and then cancel the common factors in the numerator and denominator then you will reduce the original fraction to its "simplest form" which is also an "equivalent fraction."

ex. Reduce 210/1155 to its simplest form:

 BEWARE! DO NOT DO THE FOLLOWING! Some students try to apply the rule of canceling to a problem like this: . The student will try to do what follows: The original problem , (10 + 9)/(5 + 3) is equal to 19/8 not 5. Where did we go wrong? We canceled terms; but we can only cancel factors!

ex. Apply the rule of cancelling to the following:

ans: I cancelled the common factors, 7 and 8, in the numerator and denominator. Those common factors WERE NOT PARTS OF TERMS.

SECTION 7.7: PROPORTIONS

 WHAT IS A PROPORTION? If equivalent fractions are set equal to each other then the resulting equation is called a "proportion" ex. 1) 1/2 = 3/6 This equation is called a proportion. ex. 2) 7/9 = 28/36 This equation is called a proportion.

 EXTREMES MEANS RULE OF PROPORTIONS If a/b = c/d then ad = bc where a, b, c and d are real numbers, and "a" and "d" are called the "extremes" and "b" and "c" are called the "means." TRANSLATED: If one fraction is in fact equal to another fraction then the extremes will equal the means. ex 1) Is 3/5 = 9/15 ? Yes, because (3)(15) is equal to (9)(5). Also, 3 and 15 are the "extremes", and 9 and 5 are the "means." ex. 2) Is 3/5 = 22/10 ? No, because (3)(10) is not equal to (5)(22); therefore these two fractions are not equal.

SECTION 7.8: ADDING, SUBTRACTING, MULTIPLYING AND DIVIDING FRACTIONS

 Adding and Subtracting Fractions Common Denominators Rule 1) a/b +  c/b = (a + c)/b Rule 2 ) a/b -  c/b = (a - c)/b

ex. of Rule 1)

ex. of Rule 2)

 Adding and Subtracting Fractions Uncommon Denominators Rule 3) a/b + c/d = (ad + bc)/(bd) Rule 4) a/b -  c/d = (ad - bc)/(bd)

IMPORTANT: Using rules 3 and 4 for adding and subtracting fractions with uncommon denominators is very useful, but your result may not be a fraction in lowest terms.

ex. of Rule 3)
ex. of Rule 4)

 Multiplying Fractions Rule - Rule 5 a/b   x  c/d = (ac)/(bd)

ex. of Rule 5)

 Dividing Fractions Rule - Rule 6 a/b ÷ c/d  =  a/b x d/c   =  (ad) /(bc) Comment: Notice that c /d had to be rewritten as its reciprocal d /c .

ex. of Rule 6)

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