Equality of rational numbers with a common denominator means that two fractions are equal if their numerators are equal when the denominators are the same.

let’s delve into the details of understanding the equality of rational numbers with a common denominator.

Here’s a step-by-step explanation:

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer.

Mathematically, a rational number can be written as: a/b where:

• a is the numerator.
• b is the denominator and b≠0.

Common Denominator

When two fractions have the same denominator, they are said to have a common denominator. For example, in the fractions 3/8 and 5/8​, the denominator is 8 in both cases.

Equality of Rational Numbers with Common Denominator

Two fractions with a common denominator are equal if and only if their numerators are equal. Here’s why this works:

1. Same Denominator Concept:
   • When fractions have the same denominator, it means they are divided into the same number of equal parts. Therefore, the denominators tell us that the size of each part is the same.
2. Numerator Comparison:
   • The numerators tell us how many of those equal parts we have. If the number of parts (numerators) is the same, then the fractions are equal. If the number of parts is different, then the fractions are not equal.

Detailed Example

Let’s take two fractions 3/7​ and 5/7​:

• Identify the Numerators and Denominator:
  • First fraction: 3/7​
    • Numerator: 3
    • Denominator: 7
  • Second fraction: 5/7​
    • Numerator: 5
    • Denominator: 7
• Check the Denominator:
  • Both fractions have the same denominator, 7. This means both fractions are divided into 7 equal parts.
• Compare the Numerators:
  • The first fraction has 3 parts out of 7.
  • The second fraction has 5 parts out of 7.

Since the numerators (3 and 5) are different, the fractions are not equal: 3/7≠5/7

Another Example

Consider the fractions 4/9 and 4/9​:

• Identify the Numerators and Denominator:
  • First fraction: 4/9​
    • Numerator: 4
    • Denominator: 9
  • Second fraction: 4/9​
    • Numerator: 4
    • Denominator: 9
• Check the Denominator:
  • Both fractions have the same denominator, 9.
• Compare the Numerators:
  • Both fractions have the numerator 4.

Since the numerators are the same, the fractions are equal: 4/9=4/9​

Visual Representation

To further illustrate, let’s use a visual model. Imagine we have two identical circles, each divided into 8 equal parts (common denominator):

1. First Circle:
   • If 3 out of the 8 parts are shaded, it represents the fraction 3/8.
2. Second Circle:
   • If 5 out of the 8 parts are shaded, it represents the fraction 5/8​.

Visually, you can see that 3/8 and 5/8​ are different because the number of shaded parts (numerators) is different, even though the total number of parts (denominator) is the same.

Conclusion

To determine if two rational numbers with a common denominator are equal, you only need to compare their numerators. If the numerators are equal, the fractions are equal. If the numerators are different, the fractions are not equal. This method simplifies comparing fractions, especially when dealing with multiple fractions with the same denominator.