Problem 63 - Fundamentals of Mathematics

Question

(Section 5.6) Find the value \(\frac{6-\frac{1}{4}}{6+\frac{1}{4}}\).

Step-by-Step Solution

Answer

The value is \(\frac{23}{25}\).

Step 1: Simplify the Numerator

First, we simplify the numerator of the expression: \(6 - \frac{1}{4}\). We convert 6 into a fraction with a denominator of 4, which is \(\frac{24}{4}\). Now it reads:\[6 - \frac{1}{4} = \frac{24}{4} - \frac{1}{4} = \frac{24-1}{4} = \frac{23}{4}.\]

Step 2: Simplify the Denominator

Next, we simplify the denominator of the expression: \(6 + \frac{1}{4}\).Convert 6 into a fraction with a denominator of 4, so it becomes \(\frac{24}{4}\). This gives us:\[6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{24+1}{4} = \frac{25}{4}.\]

Step 3: Simplify the Overall Expression

Now that we have both the numerator and denominator in fraction form, we insert them back into the original expression:\[\frac{\frac{23}{4}}{\frac{25}{4}}.\]To simplify this, multiply the numerator by the reciprocal of the denominator:\[\frac{23}{4} \times \frac{4}{25} = \frac{23 \times 4}{25 \times 4} = \frac{23}{25}.\]

Key Concepts

Numerator Simplification

Numerator simplification is the process of rewriting the top part of a fraction to make it easier to handle. In our exercise, we had to simplify the numerator of the form \(6 - \frac{1}{4}\). This involves converting the whole number 6 into a fraction with a common denominator. For this example, we convert 6 into \(\frac{24}{4}\) because 4 is the denominator of the fractional part we are subtracting. After the conversion, the expression becomes:

• Rewritten Expression: \(6 - \frac{1}{4} = \frac{24}{4} - \frac{1}{4}\)
• Simplified Numerator: \(\frac{24 - 1}{4} = \frac{23}{4}\)

Simplifying numerators helps in reducing fractions to their simplest form which makes further calculations much easier to perform.

Denominator Simplification

Just like with numerators, denominator simplification involves rewriting the bottom part of a fraction. In the given problem, we had a denominator of \(6 + \frac{1}{4}\). To simplify, we transform 6 into a fraction that matches the denominator of \(\frac{1}{4}\). Thus, we convert 6 to \(\frac{24}{4}\), making the entire expression:

• Rewritten Expression: \(6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4}\)
• Simplified Denominator: \(\frac{24 + 1}{4} = \frac{25}{4}\)

By breaking down and simplifying the denominator, we ensure consistency throughout the division process, leading to more accurate results.

Reciprocal

The concept of a reciprocal plays a crucial role when simplifying compound fractions. A reciprocal is essentially flipping a fraction so that its numerator becomes the denominator and vice versa. In this exercise, once both the numerator and the denominator were simplified to fractions, we had to divide \(\frac{23}{4}\) by \(\frac{25}{4}\). To do this division, we multiplied \(\frac{23}{4}\) by the reciprocal of \(\frac{25}{4}\), which is \(\frac{4}{25}\):

• Division becomes Multiplication: \(\frac{23}{4} \div \frac{25}{4} = \frac{23}{4} \times \frac{4}{25}\)
• Resulting Simplified Fraction: \(\frac{23 \times 4}{25 \times 4} = \frac{23}{25}\)

Understanding recprocals simplifies the division of fractions into straightforward multiplication, which is often easier to carry out.

Fraction Conversion

Fraction conversion is a essential skill, especially when dealing with mixed numbers or whole numbers combined with fractions. In the context of our problem, we needed to convert whole number 6 into fractions that share the same denominator as the fractional part with \(\frac{1}{4}\). This was done by expressing 6 as \(\frac{24}{4}\), so that addition and subtraction of fractions becomes direct and error-free. When converting fractions:

• Determine a common denominator that facilitates easier arithmetic operations (in this problem, the denominator was 4)
• Convert whole numbers or other fractions by multiplying their figures to have numbers aligned with the desired denominator

Converting to a common denominator throughout ensures coherence, leading to more simplified and uniform expressions in calculations.