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Chapter 1: Problem 513

Simplify. $$ \frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}} $$

Short Answer

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Step by step solution

01

- Simplify the numerator

To simplify the numerator, add the fractions \(\frac{5}{8}\) and \(\frac{1}{6}\). Find a common denominator for these fractions.
02

- Find the Common Denominator

The least common denominator for 8 and 6 is 24. Convert each fraction: \(\frac{5}{8} = \frac{15}{24}\) and \(\frac{1}{6} = \frac{4}{24}\).
03

- Add the Fractions

Now, add the fractions with the common denominator: \(\frac{15}{24} + \frac{4}{24} = \frac{19}{24}\).
04

- Simplify the Expression

The expression now is \(\frac{\frac{19}{24}}{\frac{19}{24}}\). This is a fraction where the numerator and denominator are the same.
05

- Final Simplification

A fraction where the numerator and denominator are the same is equal to 1. Thus, \(\frac{19}{24} / \frac{19}{24} = 1\).

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Key Concepts

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

Common Denominator
In order to add fractions, they need to have the same denominator, known as a common denominator.

Let's look at the fractions \(\frac{5}{8}\) and \(\frac{1}{6}\). The numbers 8 and 6 are the denominators.
To find a common denominator, we look for the smallest number that both denominators divide into evenly.
The smallest common multiple of 8 and 6 is 24. Therefore, 24 is our common denominator.

To convert the fractions to have this common denominator, we multiply the numerator and denominator of each fraction by a number that will make the denominators equal to 24.
  • \(\frac{5}{8} \times \frac{3}{3} = \frac{15}{24}\)
  • \(\frac{1}{6} \times \frac{4}{4} = \frac{4}{24}\)
This is how we convert each fraction to have the denominator 24 before we can add them.
Adding Fractions
Once the fractions have the same denominator, adding them is straightforward.

You simply add the numerators together and keep the common denominator:
\(\frac{15}{24} + \frac{4}{24} = \frac{19}{24}\).
In this step:
  • Keep the common denominator (24).
  • Add the numerators (15 + 4 = 19).
  • The result is \(\frac{19}{24}\).
Adding fractions is all about having a common denominator first, then combining the numerators.
Simplifying Fractions
The last step involves simplifying the fraction if possible.

In our example, we found \(\frac{19}{24}\), and we reframe it as \(\frac{\frac{19}{24}}{\frac{19}{24}}\).
Any fraction where the numerator and the denominator are the same simplifies to 1.
This is due to the property \(\frac{a}{a} = 1\), where both the numerator and denominator are identical.
  • The simplified form of \(\frac{\frac{19}{24}}{\frac{19}{24}}\) is 1.
The fraction \(\frac{19}{24}\) perfectly divides by itself to give us 1. Thus, the final answer is 1.

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