Comparing Fractions: Techniques and Examples

When dealing with fractions, it is often necessary to determine which fraction is greater or whether they are equivalent. This guide will explore various methods to compare fractions, providing detailed examples to aid in understanding.

Method 1: Using Equivalence to Find a Common Denominator

One common method to compare fractions is to find a common denominator and then compare the numerators.

Example:

Consider the fractions 3}{5} and 7}{10}. To compare these fractions, we can multiply the numerator and denominator of 3}{5} by 2 to find an equivalent fraction with a common denominator.

3}{5} (2/2) 3}{5} 6}{10}

Now, we can compare and .

Since 7 is greater than 6, is greater than , and therefore is greater than .

Method 2: Cross Multiplication

Another useful technique for comparing fractions is cross multiplication. This method involves multiplying the numerator of one fraction by the denominator of the other and comparing the results.

Example:

Let's compare the fractions 10/4 and 7/2.

First, we can simplify 10/4 to 5/2.

Now, we compare 5/2 and 7/2.

Using cross multiplication:

5 × 2 10

7 × 2 14

Since 14 is greater than 10, 7/2 is greater than 5/2.

Therefore, 7/2 is greater than 10/4.

Method 3: Decimal Conversion

A straightforward method to compare fractions is to convert them to decimal form. This method can be particularly useful when the denominator makes finding a common denominator complex.

Example:

Let's compare the fractions 7/10 and 3/5.

First, convert each fraction to a decimal:

7/10 0.7

3/5 0.6 (Note: 3/5 6/10 0.6)

Since 0.7 is greater than 0.6, 7/10 is greater than 3/5.

Method 4: Finding a Common Denominator

A straightforward approach is to multiply the numerator and denominator of one fraction to achieve a common denominator with the other fraction.

Example:

Consider the fractions 7/10 and 6/10 (which is equivalent to 3/5).

Since the denominators are already the same, we can directly compare the numerators.

Since 7 is greater than 6, 7/10 is greater than 6/10, and therefore 7/10 is greater than 3/5.

Method 5: Common LCM (Least Common Multiple)

For fractions with different denominators, finding the Least Common Multiple (LCM) can simplify the process of comparing them.

Example:

Let's compare the fractions 2/3 and 7/9.

The LCM of 3 and 9 is 9.

Convert each fraction to have a common denominator of 9:

2/3 6/9

7/9 7/9

Since 7 is greater than 6, 7/9 is greater than 6/9, and therefore 7/9 is greater than 2/3.

The difference between the two fractions is 1/9.

Short Tricks for Quick Comparison

Here is a quick trick to solve questions like comparing 2/3 and 7/9.

Steps:

Set the fractions equal to each other: 2/3 7/9 Cross-multiply: 2/3 7/9 2 × 9 3 × 7 18 21 Compare the results:

Since 21 is greater than 18, the fraction with the larger numerator, 7/9, is greater.

Therefore, 7/9 is greater than 2/3.

By understanding and practicing these methods, you can quickly and accurately compare fractions in various contexts.