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The principle of equivalent fractions
FRACTIONS IN ALGEBRA are often called rational expressions. (See Topic 18 of Precalculus.) We begin with the principle of equivalent fractions, which appears as follows:
"Both the numerator and denominator may be multiplied
Both x and y have been multiplied by the factor a.
That means that, in a calculation, we could replace either one with the other.
It is the same rule as in arithmetic:
on multiplying both 2 and 3 by 5.
Problem 1. Write the missing numerator.
To see the answer, pass your mouse over the colored area.
The denominator has been multiplied by 3; therefore the numerator will also be multiplied by 3.
Problem 2. Write the missing numerator.
The denominator has been multiplied by x; therefore the numerator will also be multiplied by x.
Problem 3. Write the missing numerator.
The denominator has been multiplied by 8x2; therefore the numerator will also be multiplied by 8x2.
The student should expect that the original denominator on the left will be a factor of the new denominator on the right. It must be a factor because to produce the new denominator, the original denominator was multiplied.
Problem 4. Write the missing numerator.
("The denominator has been multiplied by _____. Therefore the numerator will also be multiplied by ____.")
Since 1 has been multiplied by b, then so will a.
The numerator ab however is simply the product of a times b. It is a kind of cross-multiplying, and the student should not have to write the denominator 1.
You do algebra with your eyes.
Problem 5. Write the missing numerator.
Part f) is The Difference of Two Squares.
There will be more problems of this type at the 2nd Level.
Reducing to lowest terms
The numerator and denominator of a fraction are called its terms. Since we may multiply both terms, then, symmetrically, we may divide both terms.
"Both the numerator and denominator may be divided
When we do that, we say that we have reduced the fraction to its lowest terms.
Again, this is the same as in arithmetic.
5 is a common factor of the numerator and denominator. Therefore we may divide each of them by 5.
One often hears that we have "canceled" the 5's. But that can be very dangerous expression, as the following examples will show.
Answer. This can not be reduced. We cannot "cancel" the 5's, because 5 is not a factor of either the numerator or the denominator. In both of them, 5 is a term.
We cannot cancel terms.
The word term does double duty in algebra. We speak of the terms of a sum and also the terms of a fraction, which are the numerator and denominator. A fraction is in its lowest terms when the terms -- the numerator and denominator -- have no common factor.
Answer. When the numerator or denominator is made up of a sum, then if every term has a common factor, we may divide every term by it.
In this example, every term in both the numerator and denominator has a factor 3. Therefore, upon dividing every term by 3, we can write immediately:
There is no more reducing. The numerator and denominator no longer have a common factor.
We could show the common factor explicitly, by writing
But to actually write that is not required.
This example illustrates the following:
To divide a sum -- 3a + 6b + 9c -- by a number,
Answer. Not possible. The numerator and denominator have no common factor. 3 is not a common factor, because 3 is not a factor of 8. 2 is not a common factor, because 2 is not a factor of 3. And a is not a common factor. That fraction is in its lowest terms.
although algebraically correct, is not called reducing. To reduce a fraction means to keep it as one fraction. Otherwise, every fraction could be "reduced."
Again, to divide a sum, every term must have a common factor, as in Example 4.
Answer. 2 is a factor of every term in both the numerator and denominator. Therefore we may divide every term by 2.
There is no more dividing. We cannot "cancel" the 4's because 4 is not a factor of the denominator. 4 is not a factor of 5.
Problem 6. Reduce to lowest terms.
on dividing both the numerator and denominator by x.
Note that we must write 1 in the numerator, for x = 1 · x.
It is not the style in algebra to write 1 as a denominator.
We can view x − 3 as a factor of the numerator, because
x − 3 = (x − 3)·1
Again, we must write 1 in the numerator.
Problem 7. Reduce.
Answer. Not possible. The numerator and denominator have no common factor.
Answer. In its present form, there is no reducing -- because there are no factors. But we can make factors:
(x −3) is now seen to be a common factor. We can divide by it. And when we do, the numerator and denominator no longer have a common factor. The end.
Answer. The only common factor is x2. And we could display it by factoring both the numerator and denominator:
The fraction is now in its lowest terms. No common factors.
Problem 8. Reduce.
upon dividing every term by their common factor, 3.
Problem 9. Make factors and reduce.
Problem 10. Reduce to lowest terms -- if possible.
Next Lesson: Multiplying and dividing algebraic fractions
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