Writing a Fraction in Simplest Form

In the preceding section, we showed that and are equivalent fractions. Note that we could have written in its simplest form by dividing its numerator and denominator by 4.

To Simplify (Reduce) a Fraction

such as divide its numerator and denominator by the same number.

neither b nor n is 0

A fraction is said to be in simplest form (or reduced to lowest terms) when the only common factor of its numerator and its denominator is 1.

EXAMPLE 9

Express in simplest form.

Solution

To reduce this fraction, we can divide both its numerator and its denominator by 3.

To be sure that we have not made an error, let's check whether the cross products are equal:

3 · 5 = 15 and 1 · 15 = 15.

To reduce a fraction to lowest terms, we divide the numerator and denominator by all the factors that they have in common. To find these common factors, it is often helpful to express both the numerator and denominator as the product of prime factors. We can then divide out or cancel all common factors.

EXAMPLE 10

Write in lowest terms.

Solution

Express the numerator and denominator as the product of primes.

Cancel the common factors, noting that 1 remains.

Multiply the remaining factors.

EXAMPLE 11

Suppose that your annual income is $39,000. If you pay $9,000 for rent and $3,000 for food per year, rent and food account for what fraction of your income? Simplify your answer.

Solution

You must first find the total part of the income that you pay for rent and food per year.

The total part is $12,000 and the whole is $39,000, so the fraction is . You can simplify this fraction in the following way.

Note that canceling a 0 is the same as dividing by 10.

Therefore of your income goes for rent and food.