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Chapter 4: Problem 412

In the following exercises, simplify. $$\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}$$

Short Answer

Expert verified
The simplified result is 1.

Step by step solution

01

Add the fractions in the numerator

To add the fractions \(\frac{5}{8} + \frac{1}{6}\), find a common denominator. The least common multiple (LCM) of 8 and 6 is 24.
02

Convert fractions to a common denominator

Convert \(\frac{5}{8}\) and \(\frac{1}{6}\) to fractions with denominator 24: \(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\), \(\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}\)
03

Add the numerators

Now add the numerators: \(\frac{15}{24} + \frac{4}{24} = \frac{15 + 4}{24} = \frac{19}{24}\)
04

Divide the resulting fraction by the denominator

The original expression is \(\frac{\frac{19}{24}}{\frac{19}{24}}\). Dividing one fraction by another is equivalent to multiplying by the reciprocal. Thus, \(\frac{19}{24} \times \frac{24}{19}\)
05

Simplify the division

The \(19\)s and the \(24\)s cancel out, leaving \(1\). Therefore, \(\frac{19}{24} \times \frac{24}{19} = 1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adding Fractions
When adding fractions, it's crucial to have a common denominator. This means the bottom numbers (the denominators) must be the same.
For example, in the given exercise, we need to add \(\frac{5}{8}\) and \(\frac{1}{6}\). To do this, we find a common denominator.
The least common multiple (LCM) of 8 and 6 is 24. Once the denominators are the same, you can add the numerators (the top numbers) directly.
Remember: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\). So, always find the common denominator first!
Common Denominators
A common denominator is essential when adding or subtracting fractions. This step ensures that the fractions are comparable.
To find a common denominator, identify the least common multiple (LCM) of the original denominators.
In the example, we used 24 because it is the smallest number that both 8 and 6 can divide into without leaving a remainder.
Converting each fraction's denominator to this common number allows for straightforward addition or subtraction.
After finding the LCM (24 in our case), convert the numerators accordingly: \(\frac{5}{8} = \frac{15}{24}\) and \(\frac{1}{6} = \frac{4}{24}\). This makes addition simpler: \(\frac{15+4}{24} = \frac{19}{24}\).
Reciprocal of a Fraction
The reciprocal of a fraction is what you multiply it by to get 1.
If you have \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
In fraction division, instead of dividing by a fraction, you multiply by its reciprocal.
In our exercise, we ended with the fraction \(\frac{19}{24}\) divided by \(\frac{19}{24}\).
Rather than performing complex division, we multiply by the reciprocal: \(\frac{19}{24} \times \frac{24}{19}\).
This method simplifies the operation and brings clarity to finding the correct answer.
Simplifying Fractions
Simplifying a fraction means making it as simple as possible.
It involves dividing the numerator and the denominator by their greatest common divisor (GCD).
For example, consider \(\frac{15}{24}\).
Both numbers share a common factor of 3. Dividing by this GCD, we get: \(\frac{15 \/ \div 3}{24 \/ \div 3} = \frac{5}{8}\).
A simplified fraction is easier to understand and work with.
In our final step, \(\frac{19}{24} \times \frac{24}{19} = 1\), we see how both numerator and denominator cancel out perfectly, leaving us with the simplest result: 1.

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Most popular questions from this chapter

In the following exercises, perform the indicated operations and simplify. $$\frac{2}{5}+\left(-\frac{5}{9}\right)$$

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