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Magazine Chapter 8: Problem 63
(Section 5.6) Find the value \(\frac{6-\frac{1}{4}}{6+\frac{1}{4}}\).
Short Answer
Expert verified
The value is \(\frac{23}{25}\).
Step by step solution
01
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:\[6 - \frac{1}{4} = \frac{24}{4} - \frac{1}{4} = \frac{24-1}{4} = \frac{23}{4}.\]
02
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:\[6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{24+1}{4} = \frac{25}{4}.\]
03
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:\[\frac{23}{4} \times \frac{4}{25} = \frac{23 \times 4}{25 \times 4} = \frac{23}{25}.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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: \(\frac{24 - 1}{4} = \frac{23}{4}\)
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: \(\frac{24 + 1}{4} = \frac{25}{4}\)
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: \(\frac{23 \times 4}{25 \times 4} = \frac{23}{25}\)
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
