Problem 31 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{6-4}{9-2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2}{7}\).

Step 1: Simplify the Numerator

The numerator of the fraction is given as \(6 - 4\). Simplify it by subtracting 4 from 6. \[6 - 4 = 2\]

Step 2: Simplify the Denominator

The denominator of the fraction is given as \(9 - 2\). Simplify it by subtracting 2 from 9. \[9 - 2 = 7\]

Step 3: Form the Simplified Fraction

Now that we have simplified the numerator and denominator, form the fraction with these values. The simplified expression is \[\frac{2}{7}\]

Step 4: Conclusion

The fraction \(\frac{6-4}{9-2}\) simplifies to \(\frac{2}{7}\). This is already in its simplest form.

Key Concepts

Numerator and Denominator

When dealing with fractions, it's essential to understand what the numerator and denominator represent. A fraction is a mathematical expression representing the division of one quantity by another. The number above the division line, known as the numerator, tells us how many parts we have or are considering. The number below the line, known as the denominator, specifies how many equal parts the whole is divided into.

For example, in the fraction \( \frac{6-4}{9-2} \), before we simplify, the expression consists of a numerator "6 - 4" and a denominator "9 - 2". Simplifying these expressions gives us a clearer version of the fraction. Understanding these parts helps us in calculations, such as addition, subtraction, or simplifying fractions.

Subtraction in Fractions

Subtraction in fractions is a crucial concept as it sets the stage for simplifying complex expressions. When subtracting within a fraction, you perform the operation separately on the numerator and the denominator.

In the exercise \( \frac{6-4}{9-2} \), subtraction is carried out in both parts. First, calculate the numerator: \( 6 - 4 = 2 \). Then, calculate the denominator: \( 9 - 2 = 7 \). These steps are necessary to reduce the fraction before attempting any further operations.

The key is performing each subtraction correctly to avoid errors in the resultant fraction. If the order is reversed or calculations are inaccurate, the entire outcome may be incorrectly simplified.

Fraction Simplification Steps

Once subtraction within the numerator and the denominator has been performed, the next step is to simplify the entire fraction by combining these results. This involves checking if the resulting fraction can be reduced further.

In our case, the fraction \( \frac{2}{7} \) cannot be simplified further because the greatest common divisor (GCD) of 2 and 7 is 1, indicating it's already in its simplest form.

• Step 1: Simplify the numerator and denominator by performing any arithmetic operations like addition, subtraction, etc.
• Step 2: After getting the simplified values, re-write the fraction.
• Step 3: Check if the fraction can be reduced further by finding the GCD of the numerator and denominator.

Simplification is complete when no further division to smaller terms is possible. Understanding these steps ensures clarity in solving and simplifying other fractional problems effectively.