Problem 24 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$ \frac{10(r-6)}{10 r} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{10(r-6)}{10 r} \) is \( \frac{(r-6)}{r} \).

Step 1: Look For Common Factors

Here, you're given the fraction \( \frac{10(r-6)}{10 r} \). In this case, you can identify 10 as a common factor in the numerator and the denominator.

Step 2: Cancel Out Common Factors

Once the common factor has been identified, the next step would be to cancel out the common factor from both the numerator and the denominator. Hence, the equation will be \( \frac{(r-6)}{r} \).

Step 3: Check For Simplification

Look carefully at the result from Step 2, \( \frac{(r-6)}{r} \). This fraction cannot be simplified any further since 'r-6' in the numerator and 'r' in the denominator have no common factors. Thus, \( \frac{(r-6)}{r} \) is the simplified form of the given fraction.

Key Concepts

Common Factors in Algebra

Understanding common factors in algebra is key to simplifying algebraic expressions effectively. When we speak of common factors in the context of algebra, we refer to numerical or algebraic terms that appear both in the numerator and the denominator of a fraction.

For instance, in the given example, the expression \( \frac{10(r-6)}{10r} \) contains the number 10 in both parts of the fraction, making it a common factor. Additionally, if there were variables that appeared in both the numerator and the denominator to the same power, they too would be considered common factors.

Identifying common factors is the first crucial step in simplification because it sets up the problem for further reduction, making the expression more manageable and easier to work with.

Canceling Common Factors

Once common factors have been identified, canceling them is the next logical progression. Canceling common factors means to divide both the numerator and the denominator by the same non-zero number or algebraic expression. Doing so simplifies the fraction by eliminating redundancy.

As seen in the exercise, after identifying 10 as a common factor, it is removed from both the numerator and the denominator: \( \frac{10(r-6)}{10 r} \Rightarrow \frac{(r-6)}{r} \) This process is akin to dividing both parts of the fraction by 10, thus simplifying the expression substantially.

It's crucial to note, however, that only factors can be canceled; individual terms, especially those added or subtracted within parentheses, cannot be canceled in pieces. Misunderstanding this can lead to incorrect simplification.

Algebraic Fraction Simplification

After canceling common factors, it's important to ensure that the resulting algebraic fraction is in its simplest form. Algebraic fraction simplification involves reducing the expression to the lowest terms where no further common factors exist between the numerator and the denominator.

In our example, after canceling, the simplified form is \( \frac{(r-6)}{r} \). Here, you can't simplify the fraction further as 'r-6' and 'r' have no further common algebraic factors. Remember that simplifying algebraic fractions also means checking if the numerator or denominator can be factored further, which could then lead to more simplification.

Always re-examine the final form for any hidden common factors and reduce them if possible. This attention to detail ensures clarity and precision in algebra, which is essential to mastering the subject.