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Chapter 6: Problem 11

Add or subtract as indicated. $$\frac{7}{t}+\frac{2}{t}$$

Short Answer

Expert verified
\[\frac{9}{t}\]

Step by step solution

01

Identify the common denominator

Both fractions, \(\frac{7}{t}\) and \(\frac{2}{t}\), have the same denominator, \(t\).
02

Add the numerators

Since the denominators are the same, add the numerators directly: \(7 + 2\).
03

Write the sum over the common denominator

Combine the sum of the numerators over the common denominator: \(\frac{7 + 2}{t} = \frac{9}{t}\).

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Key Concepts

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

Common Denominator
Understanding the common denominator is key in operations like adding or subtracting fractions. A common denominator means both fractions share the same denominator, which is the bottom part of a fraction. This makes combining the fractions straightforward. In our example, the fractions \(\frac{7}{t}\) and \(\frac{2}{t}\) already have the same denominator: \('t'\).

Because they share this common denominator, we don't need to adjust the fractions before adding them. Whenever you work with fractions, always check if they have a common denominator first鈥攖hey make the addition or subtraction process much simpler.
Adding Fractions
Adding fractions might seem tricky, but it's straightforward with a common denominator. Once you've confirmed that two fractions have the same denominator, you can directly add their numerators. The numerator is the top part of the fraction.

For our example, both fractions share \('t'\) as the denominator. To add \(\frac{7}{t}\) and \(\frac{2}{t}\), simply add the numerators 7 and 2. This would be written mathematically as:
  • \(7 + 2 = 9\)
The next step is to write this sum over the common denominator, \(\frac{9}{t}\). This is the final result of adding the two fractions.
Numerator and Denominator
Understanding the roles of the numerator and denominator is crucial for working with fractions. The numerator (top number) represents how many parts you have, while the denominator (bottom number) represents how many parts make up a whole.

In our example, \(\frac{7}{t}\) and \(\frac{2}{t}\):
  • The numerators are 7 and 2.
  • The common denominator is 't'.
By adding the numerators (7 + 2), we get a new numerator, 9. The denominator remains the same during this process. Thus, the combined fraction is \(\frac{9}{t}\).

Remember: The denominator tells you into how many parts the whole is divided, while the numerator shows how many parts you have.

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Most popular questions from this chapter

Identify the two terms in the numerator and the two terms in the denominator of the rational expression \(\frac{x^{2}+4 x}{x+4},\) and write it in lowest terms.

A student was asked to simplify the rational expression $$ \frac{2 x+y}{4 x+2 y} $$ and wrote the incorrect answer \(\frac{x+y}{4 x+y}\) WHAT WENT WRONG? What is the correct answer?

Add or subtract as indicated. $$\frac{2 x+4}{x+3}+\frac{3}{x}-\frac{6}{x^{2}+3 x}$$

Multiply or divide as indicated. $$ \frac{a^{3}-8 b^{3}}{a^{2}-a b-6 b^{2}} \cdot \frac{a^{2}+a b-12 b^{2}}{a^{2}+2 a b-8 b^{2}} $$

Multiply or divide as indicated. $$ \frac{(6 x+5)(x-3)}{(x-1)(x-3)} \div \frac{(2 x+7)(x+9)}{(x-1)(x+9)} $$
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