Classify and Convert Rational Numbers: Learn It 2

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than [latex]1[/latex], in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

simplified fraction

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

For example,

  • [latex]\Large\frac{2}{3}[/latex] is simplified because there are no common factors of [latex]2[/latex] and [latex]3[/latex].
  • [latex]\Large\frac{10}{15}[/latex] is not simplified because [latex]5[/latex] is a common factor of [latex]10[/latex] and [latex]15[/latex].

The process of simplifying a fraction is often called reducing the fraction. We can use the Equivalent Fractions Property in reverse to simplify fractions.

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]

How to: Simplify/Reduce 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.

Note: To simplify a negative fraction, we use the same process as above. Remember to keep the negative sign

After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator.

Let’s simplify the fraction we saw earlier.

Simplify: [latex]\Large\frac{10}{15}[/latex]


Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.

To identify the prime factors of a number using the factor tree method, start by dividing the given number into two factors. Continue breaking down each factor into smaller factors until you’re left with only prime numbers. These prime numbers, found at the “leaves” of your factor tree, are the prime factors of the original number. Remember a prime number is a whole number greater than [latex]1[/latex] that can only be divided by [latex]1[/latex] and itself without a remainder.

Simplify: [latex]\Large\frac{210}{385}[/latex]


We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.

Simplify: [latex]\Large\frac{5xy}{15x}[/latex]