Website owner: James Miller
Fractions. Rules. Reduction to lowest terms.
Def. Rational algebraic fraction. A fraction whose numerator and denominator are both polynomials.
Rules of fractions. The rules for the manipulation of algebraic fractions are the same as for fractions in arithmetic.
Rule 1. The value of a fraction is unchanged if its numerator and denominator are both multiplied or divided by the same (non-zero) quantity.
Two fractions are said to be equivalent if one can be obtained from the other by multiplying or dividing numerator and denominator by the same quantity. The two fractions in the example are equivalent fractions.
Reduction of a fraction to its lowest terms. A fraction is said to be in its lowest terms if its numerator and denominator contain no common factor other than 1. To reduce a fraction to its lowest terms:
1) factor both the numerator and denominator into their constituent prime factors
2) divide both numerator and denominator by all of their common factors.
Problem. Reduce the following fraction to lowest terms:
where we divided numerator and denominator by the common factor 3ab3(4x - 3).
The operation of dividing out common factors in the numerator and denominator is called canceling. Cancellation is sometimes indicated by a sloped line. See Fig. 1. The process of reducing a fraction to its lowest terms is called simplifying it.
Rule 2. Signs. There are three signs associated with a fraction: the sign of the numerator, the sign of the denominator, and the sign of the entire fraction. Any two of these signs can be changed without changing the value of the fraction. If any one of these signs is changed, the sign of the fraction changes. If there is no sign before a fraction, a plus sign is implied.
Rule 3. Addition or subtraction of fractions.
Case 1. Common denominator. The sum and difference of two fractions with a common denominator are given by the formulas
Case 2. Different denominators. To add or subtract two fractions with different denominators we first multiply the numerators and denominators of both fractions by such quantities as will make their denominators equal, thus changing the fractions into equivalent fractions with the same denominator. We then add or subtract them by the rule for adding or subtracting fractions with a common denominator.
The usual procedure for changing the fractions into equivalent fractions with the same denominator is as follows:
1) Decompose the denominators of both fractions into their prime factors
2) Find the least common multiple of the two denominators. This will be the common denominator to be used. It is called the Least Common Denominator (or L.C.D.).
3) Multiply the numerator and denominator of each fraction by whatever factors are needed to create the Least Common Denominator in that fraction.
where we multiplied the numerator and denominator of the first fraction by (x - 1) and the numerator and denominator of the second fraction by x to create the Least Common Denominator of x(x-1)(x-2) in each fractions.
Rule 4. Multiplication of fractions. The product of two fractions is given by the following formula:
In other words, the product of two fractions is the product of the numerators divided by the product of the denominators.
Rule 5. Division of fractions. The quotient of two fractions is given by the following formula:
In other words, to divide one fraction by another, we invert the divisor and multiply.
Complex fractions. A simple fraction contains no fraction in either its numerator or denominator. A complex fraction is a fraction containing a fraction in either it numerator or denominator or both.
Example. The following is a complex fraction:
To simplify a complex fraction:
1) Reduce the numerator and denominator to simple fractions
2) Divide the two resulting fractions
Hawks, Luby, Touton. Second-Year Algebra
Murray R. Spiegel. College Algebra
Raymond W. Brink. A First Year of College Mathematics
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?