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Chapter 6: Problem 14

Add or subtract as indicated. $$\frac{12 t}{5}+\frac{s}{5}$$

Short Answer

Expert verified
\frac{12t + s}{5}

Step by step solution

01

Identify the common denominator

Notice that both fractions, \(\frac{12t}{5}\) and \(\frac{s}{5}\) have the same denominator, which is 5.
02

Combine the numerators

Since the denominators are the same, just add the numerators together: \(\frac{12t + s}{5}\).

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Key Concepts

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

Fractions with Common Denominators
When working with fractions, having common denominators makes the process of adding or subtracting much simpler. In this exercise, both fractions \(\frac{12t}{5}\) and \( \frac{s}{5} \) have the denominator of 5. This common denominator allows us to combine the fractions easily.

Common denominators mean that the parts of the whole are of the same size. For example, if both fractions have a denominator of 5, they are both split into 5 equal parts. This consistency is key to straightforward addition or subtraction of fractions.
Combining Numerators
Once you have established that two fractions have the same denominator, the next step is to combine their numerators. For the given exercise, the numerators are 12t and s.
  • Step 1: Confirm that the denominators are the same. Here, they are both 5.
  • Step 2: Add the numerators directly, while keeping the denominator the same.
So, \(\frac{12t}{5}\) and \( \frac{s}{5}\) become \( \frac{12t + s}{5} \).
When the denominators are the same, adding the numerators is as simple as regular addition. Just combine the numbers or expressions in the numerators over the common denominator.
Algebraic Expressions
Algebraic expressions can include variables and constants. In this case, fractions have variables like 't' and 's' in the numerators.
  • Step 1: Recognize and keep the variables as they are within the combined numerator.
  • Step 2: Simplify if possible, but here \( 12t + s \) can't be simplified further.
For example, the resulting expression \( \frac{12t + s}{5} \) is the combination of the two algebraic expressions from each fraction. Remember, combining algebraic expressions in fractions works just the same as combining numerical values, as long as the denominators match. This means students should pay careful attention to variables and treat them like terms in arithmetic operations.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{3 x+7}{3} $$

Multiply or divide as indicated. $$ \frac{p^{2}-25}{4 p} \cdot \frac{2}{5-p} $$

Multiply or divide as indicated. $$ \left(\frac{6 k^{2}-13 k-5}{k^{2}+7 k} \div \frac{2 k-5}{k^{3}+6 k^{2}-7 k}\right) \cdot \frac{k^{2}-5 k+6}{3 k^{2}-8 k-3} $$

Write each rational expression in lowest terms. $$ \frac{(x+4)(x-3)}{(x+5)(x+4)} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{5}{6} \div \frac{14}{15} $$
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