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? Continue reading Simplifying Fractions

Algebra Intro 8: Division

The last “arithmetic” operation introduced in school is usually division. While multiplication allows us to calculate the total needed for a group when a fixed quantity is required for each person, division allows us to determine how much each person will  get when a fixed quantity is divided equally among all in a group.   Division is the inverse of multiplication.

But how does one go about dividing?  Should we use our knowledge of Continue reading Algebra Intro 8: Division

Algebra Intro 9: Fractions, Reciprocals, and Properties of Division

Now that fractions and rational numbers have been introduced, let’s explore what they do for us a bit

Fractions And Rational Numbers

A rational number is one that can be represented by the ratio of two integers.

A fraction is a number that is written Continue reading Algebra Intro 9: Fractions, Reciprocals, and Properties of Division

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