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Chapter 1: Problem 514

Simplify. $$ \frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}} $$

Short Answer

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Step by step solution

01

- Identify Individual Fractions

Identify the fractions in the numerator: \(\frac{1}{6} \text{ and } \frac{3}{10}\). Identify the fraction in the denominator: \(\frac{14}{30}\).
02

- Find Common Denominator for Numerator

To add the fractions in the numerator, find a common denominator for \(\frac{1}{6}\) and \(\frac{3}{10}\). The least common denominator (LCD) of 6 and 10 is 30.
03

- Convert Fractions

Convert both fractions in the numerator to have the common denominator 30. \( \frac{1}{6} = \frac{5}{30} \) and \( \frac{3}{10} = \frac{9}{30} \).
04

- Add Fractions in the Numerator

Add the converted fractions: \(\frac{5}{30} + \frac{9}{30} = \frac{14}{30}\). The new expression now is: \(\frac{\frac{14}{30}}{\frac{14}{30}}\).
05

- Simplify the Fraction

Since both the numerator and denominator are the same: \(\frac{14}{30} \div \frac{14}{30} = 1\).

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Key Concepts

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

Common Denominator
When adding fractions, the first step is to make sure they have the same denominator. This is called finding a common denominator. If the denominators are different, you need to convert the fractions so that their denominators are the same. For example, if you have \(\frac{1}{6}\) and \(\frac{3}{10}\), you need to find the least common denominator (LCD) of the two denominators, which in this case is 30. To convert the fractions, make the denominators the same by finding equivalent fractions:
  • \( \frac{1}{6} \rightarrow \frac{5}{30} \)
  • \( \frac{3}{10} \rightarrow \frac{9}{30} \)
Addition of Fractions
Once the fractions have the same denominator, you can add them easily. To add fractions, simply add the numerators while keeping the common denominator the same. Using our earlier example: \(\frac{5}{30} + \frac{9}{30} = \frac{14}{30}\). You don鈥檛 change the denominator when adding fractions, only the numerators. In math terms, it looks like this: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\). This is an important step in any fraction-related problem.
Simplifying Fractions
After adding, you may need to simplify the result. Simplifying fractions means reducing them to their smallest form. You do this by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. Sometimes, the problem itself might simplify naturally. For instance, in our example, we ended up with \(\frac{\frac{14}{30}}{\frac{14}{30}}\). When a fraction's numerator and denominator are the same, it simplifies to 1. So, \(\frac{14}{30} \div \frac{14}{30} = 1\). Always ensure fractions are in their simplest form for final answers.

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