Problem 5 - Algebra 1: Concepts and Skills

Question

Subtract. Simplify your answer. $$ \frac{8}{3 r}-\frac{1}{3 r} $$

Step-by-Step Solution

Answer

The answer to the problem is \( \frac{7}{3 r} \)

Step 1: Identify Common Denominators

The problem contains two fractions, \( \frac{8}{3 r} \) and \( \frac{1}{3 r} \). It is important to note that these fractions have the same denominator, namely \( 3 r \). When subtracting fractions with the same denominator, you retain the denominator and subtract the numerators.

Step 2: Perform the Subtraction

The subtraction operation therefore simplifies to \( \frac{8 - 1}{3 r} \), which equals \( \frac{7}{3 r} \)

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It's a powerful tool for solving problems and expressing general mathematical relationships. In algebra, we often have to manipulate equations to find unknown values. Algebra helps us in everyday situations, like calculating the cost of multiple items or figuring out how long a trip will take. Understanding algebra is crucial because:

• It allows us to solve real-world problems by creating models with equations.
• We can perform operations on unknowns and solve for them.
• It forms the basis for higher mathematics like calculus and linear algebra.
• It enhances logical thinking and problem-solving skills.

Algebra is not just about finding the right answer; it's about understanding the relationships between different variables and how they can be manipulated to find a solution.

Common Denominator

When dealing with fractions, finding a common denominator is an essential step to perform addition or subtraction. A common denominator is a shared multiple of the denominators of two or more fractions, which allows them to be combined easily.In the given exercise, both fractions \( \frac{8}{3r} \) and \( \frac{1}{3r} \) already have a common denominator of \( 3r \). This simplifies the process because:

• You don't have to find a new denominator; you just use the shared one to make calculations easier.
• It allows the fractions to be directly subtracted or added by operating on the numerators alone.
• The common denominator helps in keeping the equation balanced and consistent.

Remember, a common denominator is especially useful when dealing with more complex fractions and ratios, as it simplifies calculations and ensures accuracy in solutions.

Numerator Subtraction

Numerator subtraction is the key step when subtracting two fractions that share the same denominator. It involves subtracting the numerators while keeping the common denominator unchanged. In our exercise, we had:\[ \frac{8}{3r} - \frac{1}{3r} \]Since they share the same denominator \( 3r \), we only need to work with the numerators. We subtract as follows:\[ 8 - 1 = 7 \]This result gives us the numerator of the final fraction. So, the solution is:\[ \frac{7}{3r} \]Key points to remember when subtracting numerators:

• Always ensure the fractions have a common denominator before performing subtraction.
• Focus on subtracting the numerators while keeping the denominator the same.
• After subtraction, simplify the fraction if possible.

By understanding numerator subtraction, you ensure accurate results when working with fractions, leading to precise and simplified answers.