## Simplifying Fractions

Three concepts help explain the process of simplifying fractions:

1. Multiplying a quantity by 1 has no effect

2. A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)

$\dfrac{17a^2b}{17a^2b}~~=~~1~~~~a\ne 0,~~b\ne 0$

3. A product of two fractions can be rewritten as a fraction of two products (and vice versa)

$\dfrac{a}{b} \cdot \dfrac{c}{d}~~=~~\dfrac{ac}{bd}\\*~\\*~\\*~\dfrac{ac}{bd}~~=~~\dfrac{a}{b} \cdot \dfrac{c}{d}$

To simplify a fraction:

• Rewrite both numerator and denominator as products of factors (if they are not already factored)
• Examine both numerator and denominator to see if they share any factors
• If they do share factors, use concept (3) above to move the shared factors into a separate fraction
• That separate fraction should now have a numerator that is exactly the same as its denominator, which by concept (2) above means that it must equal 1, therefore by concept (1) above we can drop it from the expression

Consider the following fraction… can it be simplified?

## Negative Differences

Algebra is a set of rules that allow us to change the appearance of an expression without changing the quantitative relationship that it represents. Sometimes the changes in appearance are greater than expected, causing us to doubt whether two expressions really do represent the same quantitative relationship.  The ways in which negative differences can be rewritten seem to surprise people until they become accustomed to them.

Consider a difference that is being subtracted:

$b-(a-3)$

If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign that is in front of them first.  Leaving the parentheses in place while distributing the negative produces:

## Negative Fractions

Question: Where should I put the negative sign when I am writing a fraction like negative two thirds?

Answer: As long as you write only one negative sign, it does not matter where you put it.

Two ideas are useful to keep in mind during the explanation that follows:
– Subtraction is the same thing as the addition of a negative.
– The negative of a number can be created by multiplying the number by negative one.

These principles apply to fractions as well, so:

$-\dfrac{3}{5}\\*~\\*~\\*=(-1)(\dfrac{3}{5})\\*~\\*~\\*=(\dfrac{-1}{~1})(\dfrac{3}{5})\\*~\\*~\\*=\dfrac{-3}{~5}$

Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to adding the same fraction, but with a negative numerator. But this is not the only option…

## Algebra Intro 11: Dividing Fractions, Equivalent Fractions

Once a person is comfortable with multiplying fractions, dividing one fraction by another becomes fairly straightforward.

### Dividing Fractions

An alternative to division by any number (not just a fraction) is “multiplying by the reciprocal”. Dividing by two has the same effect as multiplying by one half. Multiplying by the reciprocal of a number will always produce the exact same result as dividing by the original number.

Using this approach, any division problem can be Continue reading Algebra Intro 11: Dividing Fractions, Equivalent Fractions

## Equivalence Deserves More Attention

Most students taking courses in Algebra or higher seem quite comfortable with the idea of “equivalent fractions”: improper or unsimplified fractions all of which evaluate to the same decimal value. An example would be

$\dfrac{2}{3}=\dfrac{4}{6}=\dfrac{12}{18}=\dfrac{60}{90}=0.\overline{666}$

To create such fractions, multiply whatever fraction you wish to start with by 1 (the multiplicative identity) in the form of a fraction whose numerator and denominator are the same:

$\dfrac{2}{3}\cdot \dfrac{2}{2}=\dfrac{4}{6}=0.\overline{666}$

$\dfrac{2}{3}\cdot \dfrac{6}{6}=\dfrac{12}{18}=0.\overline{666}$

The key concepts here are that
a) an infinite number of equivalent fractions can easily be created, and
b) while all these equivalent fractions sure look different, they all represent the same decimal value or simplified fraction.

Turning to algebra, the very similar concept of “equivalent equations” is helpful in explaining how to solve algebra problems. I like to define “algebra” as: a set of rules for changing the appearance of an expression without changing the quantitative relationship that it defines.  This is exactly what was being done with the fractions above.

Continue reading Equivalence Deserves More Attention