General Problem Solving: Background You’ll Need 1

  • Simplify Fractions

Simplify Fractions

There are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.

A fraction is considered simplified if there are no common factors, other than [latex]1[/latex], in the numerator and denominator. A common factor is a number that divides both the numerator and the denominator of a fraction evenly, without leaving a remainder. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing these common factors.

Simplified Fraction

A fraction is considered simplified if there are no common factors in the numerator and denominator.

The process of simplifying a fraction is often called reducing the fraction. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.

Equivalent Fractions Property

If [latex]a,b,c[/latex] are numbers where [latex]b\ne 0,c\ne 0[/latex], then

 

[latex]{\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}\text{ and }{\Large\frac{a\cdot c}{b\cdot c}}={\Large\frac{a}{b}}[/latex].
Notice that [latex]c[/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.

How To: Simplify a Fraction

  1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
  2. Simplify, using the equivalent fractions property, by removing common factors.
  3. Multiply any remaining factors.
Simplify: [latex]\Large\frac{10}{15}[/latex]

To simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.

Simplify: [latex]\Large-\frac{18}{24}[/latex]