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Chapter 11: Problem 82

Perform the indicated operation and, if possible, simplify. $$ \frac{t}{t-1}-\frac{t}{1-t} $$

Short Answer

Expert verified
\( \frac{2t}{t-1} \)

Step by step solution

01

Observe the Denominators

The given expression is \( \frac{t}{t-1} - \frac{t}{1-t} \). Notice that \( t-1 \) and \( 1-t \) are very similar but with opposite signs.
02

Simplify the Second Fraction

Rewrite \( 1-t \) as \( -(t-1) \). This means \( \frac{t}{1-t} = \frac{t}{-(t-1)} = -\frac{t}{t-1} \). This step helps in getting common denominators.
03

Combine the Fractions

Rewrite the expression with a common denominator. Doing this gives \( \frac{t}{t-1} - (-\frac{t}{t-1}) = \frac{t}{t-1} + \frac{t}{t-1} \).
04

Add the Fractions

Since the denominators are now the same, combine the numerators: \( \frac{t + t}{t-1} = \frac{2t}{t-1} \).
05

Simplify the Expression

The simplified form of the expression \( \frac{2t}{t-1} \). Check if any further simplification is possible, but in this case, \( \frac{2t}{t-1} \) is already simplified.

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Key Concepts

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

common denominators
To solve the problem of combining fractions, the first important concept is understanding common denominators. In the example given, we have two fractions: \( \frac{t}{t-1} \) and \( \frac{t}{1-t} \). Denominators are the bottom parts of fractions, and they need to be the same for easy combination. However, notice that \( t-1 \) and \( 1-t \) are, in fact, the same value but with opposite signs. \( 1-t \) can be rewritten as \( -(t-1) \). This step helps us get common denominators, making the next steps easier. It's crucial to identify and adjust differences in denominators to proceed efficiently with fraction operations.
combining fractions
Once common denominators are found, the next step is combining fractions. After rewriting \( \frac{t}{1-t} = -\frac{t}{t-1} \), the problem transforms into \( \frac{t}{t-1} - (-\frac{t}{t-1}) \). Notice how the denominators are now the same. We can then combine the two fractions by adding their numerators. Essentially, you're combining the tops of the fractions while keeping the bottom the same: \( \frac{t}{t-1} - (-\frac{t}{t-1}) = \frac{t + t}{t-1} = \frac{2t}{t-1} \). Combining fractions correctly depends on having common denominators, so always ensure this step before adding or subtracting.
simplifying expressions
Lastly, let's talk about simplifying expressions. After combining the fractions in our example, we ended up with \( \frac{2t}{t-1} \). Simplification means reducing an expression to its most manageable form. We ask if further reduction or factoring is possible. In this example, \( \frac{2t}{t-1} \) is already in its simplest form since the numerator and the denominator have no common factors other than 1. We look for common factors or cancellation opportunities in both the numerator and denominator. Simplifying will make it easier to work with and understand expressions.

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