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Chapter 4: Problem 413

In the following exercises, simplify. $$\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$$

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

- Simplify the numerator

First, simplify the numerator: \(\frac{1}{6} + \frac{3}{10}\). To do this, find a common denominator. The least common multiple (LCM) of 6 and 10 is 30. Hence, convert \(\frac{1}{6}\) and \(\frac{3}{10}\) to fractions with a denominator of 30.
02

- Convert fractions to common denominator

Convert \(\frac{1}{6}\) to a denominator of 30: \(\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}\). Similarly, convert \(\frac{3}{10}\) to a denominator of 30: \(\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}\).
03

- Add the fractions

Add the fractions: \(\frac{5}{30} + \frac{9}{30} = \frac{5 + 9}{30} = \frac{14}{30}\). So, the simplified form of the numerator is \(\frac{14}{30}\).
04

- Simplify the entire expression

Now, simplify the entire expression: \(\frac{\frac{14}{30}}{\frac{14}{30}}\). Divide the two fractions: \(\frac{14}{30} \div \frac{14}{30} = 1\).

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Key Concepts

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

Common Denominator
When working with fractions, especially during addition or subtraction, having a common denominator is crucial. A common denominator is a shared multiple of the denominators of two or more fractions. This allows you to combine the fractions more easily.
For example, in the given problem, we have \(\frac{1}{6} \) and \(\frac{3}{10} \). The least common multiple (LCM) of 6 and 10 is 30. Thus, 30 becomes our common denominator. We convert each fraction to an equivalent fraction with a denominator of 30:
  • \(\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \)
  • \(\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} \)
Using a common denominator, all fractions involved are expressed in a uniform way, making the next steps in our operations much simpler.
Addition of Fractions
Once fractions share a common denominator, adding them becomes straightforward. Simply add their numerators and keep the common denominator. Let's look at our example:
After converting \(\frac{1}{6} \) and \(\frac{3}{10} \) to \(\frac{5}{30} \) and \(\frac{9}{30} \) respectively, we can add them easily.
So, combining these two fractions: \(\frac{5}{30} + \frac{9}{30} = \frac{5 + 9}{30} = \frac{14}{30} \).
This process avoids confusion and errors, ensuring accuracy in fraction calculations.
Fraction Division
Dividing fractions may sound complex, but it is simplified through multiplication by the reciprocal. The reciprocal of a fraction flips its numerator and denominator.
Consider our simplified problem: \(\frac{\frac{14}{30}}{\frac{14}{30}} \).
This can be transformed into a multiplication problem by flipping the second fraction: \(\frac{14}{30} \times \frac{30}{14} \).
The numerators and denominators cancel each other out, resulting in:
  • \(\frac{14}{30} \times \frac{30}{14} = 1 \)
Thus, \(\frac{\frac{14}{30}}{\frac{14}{30}} = 1 \).
This technique highlights how division of fractions can be reduced to a multiplication problem, simplifying calculations significantly.

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Most popular questions from this chapter

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{a+b}{a-b}\) when \(a=-3, b=8\)

In the following exercises, simplify. $$ -\frac{3}{13}+\left(-\frac{4}{13}\right) $$

Convert the mixed number to an improper fraction. $$ 3 \frac{2}{7} $$

In the following exercises, perform the indicated operation. $$ 8 \frac{6}{11}-2 \frac{9}{11} $$

In the following exercises, simplify. $$ -\frac{5}{6} \div \frac{5}{12} $$
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