# Comparing Fractions Using Three Different Methods

Undoubtedly you know how to compare two integers, if they have two different signs, the positive number is greater, if they have the same sign we have two cases:

If both numbers are positive, then the farthest from zero is greater If both numbers are negative, then the farthest from zero is the smaller

But how do we compare two mixed numbers? For example, if you ate three-quarters of your pizza, and your brother ate two-thirds of his pizza, which one ate the most? This is what we will try to answer in this lesson, comparing two fractions in three different ways.

To compare two fractions, we can use three different methods:

• Decimal Method

• The Same Denominator Method

• The Same Numerator Method

## Decimal Method

This method is based on dividing the numerator by the denominator and then comparing the two whole parts. If they are equal we compare the tenths place, if they are equal we compare the hundredths place, if they are equal we compare the thousandths place and so on.

#### Example1:

Compare the frations: and

#### Solution:

Converting each fraction to a decimal:

The two whole parts are equal which is 0. Move to the next.

The digit in tenth place of is 7, and the digit in tenth place of is 6

So,

#### Example2:

Compare the frations: and

#### Solution:

Converting each fraction to a decimal:

The two whole parts are equal which is 1. Move to the next.

The digit in tenth place of is 8, and the digit in tenth place of is 8. Move to the next.

The digit in hundredths place of is 7, and the digit in tenth place of is 0.

So,

## The Same Denominator Method

When we have two fractions with like denominators, it's easy to compare them, The greatest fraction is the fraction with the greatest numerator.

### Note:

Let and represent two fractions.

If , so

When we have two fractions with unlike denominators, we make them the same (check out this lesson)

#### Example3:

Compare the frations: and

#### Solution:

The two fractions have the same denominator, we compare the numerators. It's clear that .

Thus,

#### Example4:

Compare the frations: and

#### Solution:

The two fractions withy unlike denominators, we need to make the the same.

The Greatest Common Factor: .

The equivalent fractions:

And since , therefore

Hence,

## The Same Numerator Method

When we have two fractions with like numerators, it's easy to compare them. But pay attention, the greatest fraction is the fraction with the smallest denominator. Because as we increase the denominator, the fraction decreases

#### Example5:

Compare the frations: and

#### Solution:

The two fractions withy like numerators, we need to compare the denominators. It's clear that . Thus,

#### Example6:

Compare the frations: and

#### Solution:

The two fractions withy like numerators, we need to compare the denominators. It's clear that . Thus,

## Comparing Mixed Fractions

To compare two mixed fractions, we need to convert them to improper fractions and apply the rules of comparing two fractions. (check out this lesson)

#### Example7:

Compare the mixed frations: and

#### Solution:

Firs, we need to convert the mixed fractions to improper fractions using this method:

### Note:

Let is a mixed fraction. To convert the mixed fraction to improper fraction we use the following formula:

The improper fractions obtained are unlike denominators, we need to make them the same.

The Greatest Common Factor: .

The equivalent fractions:

And since , therefore

Hence,