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Chapter 7: Problem 60

Perform each indicated operation. $$ \frac{3}{15}+\frac{6}{15} $$

Short Answer

Expert verified
\( \frac{3}{15} + \frac{6}{15} = \frac{3}{5} \) after simplification.

Step by step solution

01

Identify Common Denominator

The fractions given are \( \frac{3}{15} \) and \( \frac{6}{15} \). Both have the same denominator, which is 15.
02

Add the Numerators

Since the denominators are the same, you can add the numerators directly: \( 3 + 6 = 9 \). This gives you \( \frac{9}{15} \).
03

Simplify the Fraction

To simplify \( \frac{9}{15} \), find the greatest common divisor of 9 and 15, which is 3. Divide both the numerator and the denominator by 3 to get \( \frac{3}{5} \).

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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, a common denominator is essential for combining or comparing them. The denominator is the bottom number of a fraction and represents how many equal parts the whole is divided into. For two fractions to be added or subtracted, their denominators must be the same. This concept is called the 'common denominator'.

In our example, both fractions \( \frac{3}{15} \) and \( \frac{6}{15} \) already have the same denominator, which is 15. Therefore, these fractions are already prepared for direct addition since their denominators match, allowing you to focus directly on the numerators.
Numerator
The numerator is the top number in a fraction, and it tells you how many parts of the whole or group you have. In order to add fractions with a common denominator, simply add their numerators while keeping the denominator the same.

In the expression given, the numerators are 3 and 6. They essentially indicate how many parts are being considered from each fraction. When adding \( \frac{3}{15} \) and \( \frac{6}{15} \), you only deal with the numerators: \( 3 + 6 = 9 \). The result is \( \frac{9}{15} \). By maintaining the same denominator, the total parts considered from your fractions remain consistent over the same sized sections.
Simplifying Fractions
Simplifying fractions means making the fraction as simple as possible, often by reducing the numbers to their smallest whole numbers while maintaining the same value. The process involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our example, we need to simplify \( \frac{9}{15} \). The GCD of both 9 and 15 is 3, as it is the largest number that divides evenly into both. Therefore, divide both the numerator and the denominator by 3:
  • Numerator: \( 9 \div 3 = 3 \)
  • Denominator: \( 15 \div 3 = 5 \)
The simplified fraction is \( \frac{3}{5} \). By simplifying fractions, you often make them easier to understand and use, particularly when further computations involving fractions are required.

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Most popular questions from this chapter

Simplify. See Sections 5.1 and 5.5. $$ \frac{144 x^{5} y^{5}}{-16 x^{2} y} $$

Perform each indicated operation. See Section 1.3. $$ \frac{7}{8} \div \frac{1}{2} $$

Simplify. See Sections 5.1 and 5.5. $$ \frac{3 x^{3} y^{2}}{12 x} $$

In the study of calculus, the difference quotient \(\frac{f(a+h)-f(a)}{h}\) is often found and simplified. Find and simplify this quotient for each function \(f(x)\) by following steps a through \(d\). a. Find \((a+h)\) b. Find \(f(a)\) c. Use steps a and b to find \(\frac{f(a+h)-f(a)}{h}\) d. Simplify the result of step \(\mathbf{c}\). $$ f(x)=\frac{1}{x} $$

Decide whether each rational expression equals \(1,-1,\) or neither. a. \(\frac{x+5}{5+x}\) b. \(\frac{x-5}{5-x}\) c. \(\frac{x+5}{x-5}\) d. \(\frac{-x-5}{x+5}\) e. \(\frac{x-5}{-x+5}\) f. \(\frac{-5+x}{x-5}\)
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