Problem 41 - Prealgebra and Introductory Algebra

Question

Simplify. $$1-\frac{1}{1-\frac{1}{y+1}}$$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(-\frac{1}{y}\).

Step 1: Identify the Given Expression

The given mathematical expression is \(1-\frac{1}{1-\frac{1}{y+1}}\)

Step 2: Simplify the Innermost Fraction

Let's start by simplifying the denominator of the larger fraction, \((1-\frac{1}{y+1})\). First, find a common denominator for 1 and \(\frac{1}{y+1}\), which is \((y+1)\). Therefore, we rewrite 1 as \(\frac{y+1}{y+1}\), and the expression now reads: \(1-\frac{1}{\frac{y+1}{y+1}-\frac{1}{y+1}}\) or \(1-\frac{1}{\frac{y}{y+1}}\).

Step 3: Reverse the Fraction in the Denominator

Reverse the fraction \(\frac{y}{y+1}\) in the denominator of the larger fraction. This is because dividing by a fraction is equivalent with multiplying by its reciprocal. Thus we are left with: \(1 - (1\cdot(y+1)/y)\) or \(1 - \frac{y+1}{y}\).

Step 4: Convert the Mixed Number to a Simple Fraction

We now have two fractions being subtracted from each other. To subtract them, they need to have the same denominator, which is \(y\). So, 1 is rewritten as \(\frac{y}{y}\) and the final expression is \(\frac{y}{y}-\frac{y+1}{y}\).

Step 5: Subtract the Fractions

Now that both terms have the same denominator, we can subtract the numerators: \(\frac{y-(y+1)}{y}\) which simplifies to \(\frac{-1}{y}\).

Key Concepts

Fraction Subtraction

When subtracting fractions, the goal is to create a situation where each fraction shares a common denominator, making the operation straightforward. For instance, consider the subtraction \(\frac{a}{b} - \frac{c}{d}\). To subtract these, we need both denominators to be the same. This often involves finding the least common multiple of the two denominators, in this case, \(b\) and \(d\). Once we've found a common denominator, we transform each fraction by multiplying the numerator and denominator by whatever value will result in that common denominator.

In the educational exercise of simplifying \(1-\frac{1}{1-\frac{1}{y+1}}\), you encounter a need to subtract fractions within fractions, which challenges one’s understanding of this principle. By ensuring each subtraction occurs with like denominators, the process is streamlined. For the inner fraction, finding a common denominator is key to simplifying the expression further.

• Transform integer parts to fractions with a common denominator.
• Ensure that each term within the subtraction is written as a fraction over this common denominator.
• Subtract the numerators across the common denominator.

Simplifying the expression step by step by honoring the subtraction rules keeps the process organized and helps avoid common errors.

Common Denominator

A common denominator refers to a shared multiple of the original denominators of two or more fractions. It is an essential part of arithmetic with fractions including addition, subtraction, comparison, and ordering.

The example \(1-\frac{1}{1-\frac{1}{y+1}}\) demonstrates the crucial role of a common denominator when working with complex fractions. The denominator \(y+1\) in the expression \(1-\frac{1}{y+1}\) is adopted for the integer 1, transforming it into \(\frac{y+1}{y+1}\) to create fractions with like denominators.

• Identify the least common multiple of the denominators to find the common denominator.
• Transform each fraction to an equivalent one with the common denominator found.
• Proceed with the necessary operation (addition, subtraction, etc) with these equivalent fractions.

Understanding that the common denominator unifies the denominators allows harmonizing the process of combining fractions, ensuring accuracy in the resulting expression.

Reciprocal of a Fraction

The reciprocal of a fraction is simply another fraction in which the numerator and denominator swap roles. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\), provided that neither \(a\) nor \(b\) is zero, as division by zero is undefined. Reciprocals are fundamental when it comes to division involving fractions.

To divide by a fraction, you multiply by its reciprocal. This is illustrated in the original problem \(1-\frac{1}{1-\frac{1}{y+1}}\), where in Step 3, the complex fraction's denominator \(\frac{y}{y+1}\) must be inverted to simply the equation.

• Understand that dividing by a fraction is the same as multiplying by its reciprocal.
• Find the reciprocal of a fraction by flipping its numerator and denominator, ensuring none are zero.
• Apply this in division problems involving fractions to transform them into multiplication problems.

Recognizing when and how to use reciprocals can significantly ease the complexity of dealing with fraction division in algebraic expressions.