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Chapter 7: Problem 59

Perform each indicated operation. \(\frac{1}{5}+\frac{4}{5}\)

Short Answer

Expert verified
The sum is 1.

Step by step solution

01

Identify the Fractions

The fractions involved in this operation are \( \frac{1}{5} \) and \( \frac{4}{5} \). Both have a common denominator of 5.
02

Add the Numerators

Since the fractions have a common denominator, you can add the numerators directly. Add \(1\) and \(4\) to get \(5\).
03

Retain the Denominator

The denominator remains the same during the addition of fractions. Therefore, the denominator is \(5\).
04

Write the Fraction

The result of the addition is \( \frac{5}{5} \).
05

Simplify the Fraction

Simplify \( \frac{5}{5} \) by dividing the numerator and the denominator by their greatest common divisor, which is 5. \( \frac{5}{5} = 1.\)

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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, a common denominator is essential. It is the number that is divisible by each of the denominators of the fractions you are working with. In the problem, both fractions, \( \frac{1}{5} \) and \( \frac{4}{5} \), already share a common denominator of 5. This makes the addition process straightforward.
  • A common denominator allows you to focus only on the numerators.
  • Without it, you would first need to find one to proceed with addition.
In cases where fractions have different denominators, you must find the least common multiple of the denominators to create equivalent fractions with a common denominator.
Numerator
The numerator tells you how many parts of the whole you have. In fraction addition, after establishing a common denominator, you simply add the numerators. For our addition, the numerators were 1 and 4.
  • Add the numerators directly: \( 1 + 4 = 5 \).
  • The sum of the numerators becomes the new numerator of your resulting fraction.
This new numerator over the common denominator forms a new fraction: \( \frac{5}{5} \). In this way, learning to handle the numerators with confidence is a key skill in fraction operations.
Simplifying Fractions
Once you have added the numerators over a common denominator, the next step is to simplify the fraction. Simplifying means to reduce the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  • For \( \frac{5}{5} \), both the numerator and denominator are 5, which is their GCD.
  • Divide both by 5: \( \frac{5 \div 5}{5 \div 5} = 1 \).
The resulting simplification reduces \( \frac{5}{5} \) to 1. Simplifying fractions helps in recognizing equivalent values and ensures the fraction is expressed in its most basic and understandable form.

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Most popular questions from this chapter

Does \(\frac{x+7}{x}\) simplify to \(7 ?\) Why or why not?

For which of the following equations can we immediately use cross products to solve for \(x ?\) A. \(\frac{2-x}{5}=\frac{1+x}{3}\) B. \(\frac{2}{5}-x=\frac{1+x}{3}\)

Simplify each complex fraction. See Examples 1 and 2. $$ \frac{\frac{10}{3 x}}{\frac{5}{6 x}} $$

The total revenue \(R(x)\) from the sale of a popular music compact disc is approximately given by the function $$ R(x)=\frac{150 x^{2}}{x^{2}+3} $$ where \(x\) is the number of years since the CD has been released and revenue \(R\) is in millions of dollars. a. Find the total revenue generated by the end of the first year. b. Find the total revenue generated by the end of the second year. c. Find the total revenue generated in the second year only.

Simplify. $$ \frac{x}{1-\frac{1}{1+\frac{1}{x}}} $$
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