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Chapter 9: Problem 61

In Exercises \(1-62,\) perform the indicated operations and simplify your answer as completely as possible. $$\frac{\frac{1}{t+1}-\frac{1}{t+2}}{1-\frac{1}{t+2}}$$

Short Answer

Expert verified
\( \frac{1}{t+1} \)

Step by step solution

01

- Simplify the numerator

The numerator is \(\frac{1}{t+1}-\frac{1}{t+2}\). To subtract these fractions, find a common denominator. The common denominator is \((t+1)(t+2)\). Rewrite each fraction with the common denominator: \(\frac{(t+2)-(t+1)}{(t+1)(t+2)} = \frac{t+2-t-1}{(t+1)(t+2)} = \frac{1}{(t+1)(t+2)}\).
02

- Simplify the denominator

The denominator is \(1-\frac{1}{t+2}\). Rewrite \(1\) as \(\frac{t+2}{t+2}\): \(1-\frac{1}{t+2} = \frac{t+2}{t+2} - \frac{1}{t+2} = \frac{t+2-1}{t+2} = \frac{t+1}{t+2}\).
03

- Divide the simplified numerator by the simplified denominator

Now, divide the simplified numerator \(\frac{1}{(t+1)(t+2)}\) by the simplified denominator \(\frac{t+1}{t+2}\). This is done by multiplying by the reciprocal of the denominator: \(\frac{1}{(t+1)(t+2)} \times \frac{t+2}{t+1} = \frac{1}{(t+1)(t+2)} \times \frac{t+2}{t+1} = \frac{1}{(t+1)} \times 1 = \frac{1}{t+1}\).

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Key Concepts

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

Fraction Subtraction
When working with fraction subtraction, the key is to have a common base or denominator. This simplifies the subtraction process. Consider the fractions: \( \frac{1}{t+1} \) and \( \frac{1}{t+2} \). To subtract these fractions, you need a common denominator. The easiest way is to multiply the denominators together, giving \( (t+1)(t+2) \). Rewrite both fractions using this common denominator:

\( \frac{1}{t+1} - \frac{1}{t+2} = \frac{(t+2)-(t+1)}{(t+1)(t+2)} = \frac{1}{(t+1)(t+2)} \).

Subtracting the fractions becomes straightforward because both now have a shared base.
Common Denominator
A common denominator is fundamental when adding or subtracting fractions. In the example above, the denominators were \( t+1 \) and \( t+2 \). The least common denominator (LCD) is their product: \( (t+1)(t+2) \). Rewrite each fraction with the common denominator:

  • \( \frac{1}{t+1} \) becomes \( \frac{t+2}{(t+1)(t+2)} \)
  • \( \frac{1}{t+2} \) becomes \( \frac{t+1}{(t+1)(t+2)} \)
Using the common denominator helps consolidate the fractions into a single fraction, making it easier to perform operations.
Simplifying Expressions
After finding the common denominator and rewriting the fractions, simplify the expressions to a more manageable form.

In our example, the numerator became \( \frac{1}{(t+1)(t+2)} \) and the denominator simplified to \( \frac{t+1}{t+2} \). Now, the operation needed is division, which turns into multiplying by the reciprocal of the denominator.

Multiplication by the reciprocal simplifies the final result:

\( \frac{\frac{1}{(t+1)(t+2)}}{\frac{t+1}{t+2}} = \frac{1}{(t+1)(t+2)} \times \frac{t+2}{t+1} \). This further simplifies to \( \frac{1}{t+1} \).
Reciprocal
The reciprocal of a number is obtained by flipping its numerator and denominator. For fraction operations, especially division, using reciprocals transforms the problem into multiplication. Here鈥檚 how it works with our example:

The simplified denominator was \( \frac{t+1}{t+2} \). The reciprocal of this is \( \frac{t+2}{t+1} \). So we must multiply:

\( \frac{1}{(t+1)(t+2)} \times \frac{t+2}{t+1} \). The terms \( t+2 \) and \( t+1 \) cancel out what they can:

\( \frac{1}{(t+1)(t+2)} \times \frac{t+2}{t+1} = \frac{1}{t+1} \).

Understanding reciprocals is crucial for performing division with fractions, making operations far simpler.

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