Problem 32 - Algebra 1: Concepts and Skills

Question

Write the difference in simplest form. $$ \frac{5 c}{15}-\frac{2+c}{25 c} $$

Step-by-Step Solution

Answer

The difference in simplest form is \( \frac{25c - 3 - \frac{6}{c}}{75} \)

Step 1: Simplify Each Fraction

The first fraction can be simplified as \( \frac{5c}{15} = \frac{c}{3} \). However, the second fraction cannot be simplified.

Step 2: Find Common Denominator

The common denominator of \(3\) and \(25c\) is \(75c\). The process of equating denominators involves multiplying the first term by \((25c)/(25c)\) and the second term by \((3)/(3)\) to ensure they have the same denominator.

Step 3: Subtract the Fractions

After finding a common denominator, subtract the two fractions: \( \frac{25c^{2}}{75c}-\frac{6+3c}{75c} \)

Step 4: Simplify

Simplify the numerator by subtracting corresponding coefficients to give: \( \frac{25c^{2}-6-3c}{75c} = \frac{25c^{2}-3c-6}{75c} \).

Step 5: Reduce the Expression

The entire expression can be reduced by c, resulting in \( \frac{25c - 3 - \frac{6}{c}}{75} \)

Key Concepts

Understanding a Common Denominator

When dealing with fractions, especially in algebra, a common denominator is crucial for performing addition or subtraction. Fractions are parts of a whole, and their denominators tell us into how many parts the whole is divided. Thus, when two fractions have different denominators, we need to find a way to make them the same to effectively perform operations like addition or subtraction.

Finding a common denominator involves:

• Identifying the least common multiple (LCM) of the denominators.
• Equating the fractions by multiplying each by a form of 1 that makes each denominator the LCM.

In the exercise, the denominators are 3 and 25c. The LCM of 3 and 25c is 75c. This allows both fractions to be expressed in terms of the same "whole," making subtraction possible.

Simplifying Fractions

Simplifying fractions is about making them as simple as possible while maintaining their value. This process involves reducing both the numerator and the denominator by their greatest common divisor (GCD).

In the solution, the fraction \( \frac{5c}{15} \) was simplified to \( \frac{c}{3} \) by dividing both the numerator and denominator by 5, their GCD.

Remember:

• A simplified fraction is usually easier to work with.
• Simplification doesn't change the value of the fraction, just makes it cleaner.

In some algebraic fractions, you might find terms that can be canceled out, like variables or constants shared by both the numerator and denominator.

Algebraic Expressions in Fractions

Combining algebra with fractions can feel tricky at first because it involves both numbers and symbols. Algebraic expressions in the numerator or denominator require us to follow the same rules of arithmetic, but with the added complexity of variables.

Key steps include:

• Making sure to apply operations such as addition or subtraction carefully across the entire expression.
• Checking that all like terms are combined to simplify the expression further.

In our problem, we had to subtract two algebraic fractions with the common denominator of 75c. After the subtraction, simplifying the expression \( \frac{25c^{2} - 3c - 6}{75c} \) involved reducing terms and checking for any common factors that could further reduce the fraction. It's an insightful exercise in handling numbers and symbols simultaneously.