Problem 13 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(\frac{6 r}{11}+\frac{2 r}{11}\)

Step-by-Step Solution

Answer

The sum is \(\frac{8r}{11}\) in simplest form.

Step 1: Identify Like Terms

Both terms, \(\frac{6r}{11}\) and \(\frac{2r}{11}\), have the same denominator, which is 11. Additionally, both numerators contain the variable \(r\). This means we can directly add the numerators.

Step 2: Add the Numerators

Since the denominators are the same, we keep 11 as the common denominator and add the numerators. This gives us \(6r + 2r = 8r\).

Step 3: Write the Resulting Fraction

After adding the numerators, the sum becomes \(\frac{8r}{11}\).

Step 4: Simplify the Fraction

Check if \(\frac{8r}{11}\) can be simplified further. Since 11 is a prime number and there are no common factors with 8, the fraction is already in its simplest form.

Key Concepts

Understanding Like Terms

Like terms are expressions that share the same variable raised to the same power. Here, both terms, \(\frac{6r}{11}\) and \(\frac{2r}{11}\), contain the variable \(r\) and have identical denominators. This is essential in operations involving algebraic fractions as it allows us to combine these terms. When you identify like terms, you can add or subtract their coefficients while maintaining the variable part. In our exercise, since both terms are "like," you should focus on combining the numbers in front of the variable, also known as coefficients, which are 6 and 2.

Finding a Common Denominator

A common denominator is crucial when adding or subtracting fractions. It means equalizing the bottom numbers (denominators) of two or more fractions so that they can be directly added or subtracted. In our example, both fractions already have the common denominator of 11.

• This means we can directly combine the fractions without any further adjustments.
• It's always easier when fractions have the same denominator because it simplifies the process of addition or subtraction.

When dealing with different denominators, you would need to find a common multiple to proceed with operations.

Simplifying Fractions

After finding the sum or difference of fractions, it's always wise to check if the resulting fraction can be simplified. Simplifying fractions involves reducing them to their lowest terms, making them easier to interpret and use in further calculations.
In our original expression, after adding the numerators, you get \(\frac{8r}{11}\). To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• If the GCD is not 1, divide both the numerator and the denominator by the GCD.

In our case, since 8 and 11 have no common factors (other than 1), \(\frac{8r}{11}\) is already as simple as it gets.

Explaining Variable Coefficients

Variable coefficients refer to the numbers that are multiplied by the variables in an expression. In algebra, understanding coefficients is key to performing operations like addition and subtraction with algebraic fractions.
In the problem given, \(6r\) and \(2r\) are similar terms where 6 and 2 are the coefficients of the variable \(r\).

• Because the fractions share the same denominator, we can add the coefficients: 6 + 2 = 8.
• The result \(8r\) maintains the relationship and proportionality set by the original terms.

This approach helps simplify processes in algebra, ensuring each step is clear and understandable.