Problem 38 - Prealgebra and Introductory Algebra

Question

Simplify. $$1+\frac{1}{1+\frac{1}{2}}$$

Step-by-Step Solution

Answer

\(\frac{5}{3}\)

Step 1: Simplify the Innermost Fraction

This exercise should be solved by starting from the innermost brackets, looking at the fraction \(\frac{1}{2}\). This fraction is already in its simplest form.

Step 2: Simplify the Next Layer

Proceed to simplify the whole expression enclosed within the outer brackets; \(1+\frac{1}{2}\). Add 1 to \(\frac{1}{2}\), which is \(1.5\) or \(\frac{3}{2}\). The expression becomes \(\frac{1}{\frac{3}{2}}\).

Step 3: Invert the Fraction

To simplify \(\frac{1}{\frac{3}{2}}\), one can use the rule of division by a fraction which is equivalent to multiplication by its reciprocal. So it becomes \(1*\frac{2}{3} = \frac{2}{3}\).

Step 4: Add the Final Part

Finally, add the remaining 1 to \(\frac{2}{3}\). This should be done after converting 1 into a fraction with the same denominator as \(\frac{2}{3}\), giving \(\frac{3}{3}\). Adding \(\frac{3}{3}\) to \(\frac{2}{3}\) results in \(\frac{5}{3}\).

Key Concepts

Adding Fractions

When it comes to adding fractions, it's essential to understand that fractions must have the same denominator to be added directly. This allows you to add the numerators across and keep the denominator the same. For example, adding \( \frac{1}{2} \) and \( \frac{1}{3} \) involves finding a common denominator, usually the least common multiple of the two denominators, which is 6 in this case. So, \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \). Thus, \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).

• Ensure common denominators for straightforward addition.
• Add numerators and keep denominators unchanged.

Using this method, you can confidently add any two fractions step-by-step.

Reciprocal

The concept of a reciprocal is pivotal when dealing with division in fraction arithmetic. The reciprocal of any fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). This is useful because dividing by a fraction is the same as multiplying by its reciprocal.

• A reciprocal flips the parts of a fraction: \( \frac{a}{b} \to \frac{b}{a} \).
• Converts division into an easier multiplication.

Understanding and applying reciprocals can simplify complex expressions and are particularly handy when fractions are nested within other operations.

Mixed Numbers

Mixed numbers combine whole numbers with fractions, like \( 1 \frac{1}{2} \). While they might seem complicated, they can be easily converted into improper fractions for simpler arithmetic operations.

• Multiply the whole number by the fraction's denominator.
• Add this to the fraction's numerator: \( 1 \frac{1}{2} = \frac{3}{2} \).

This simplifies addition, subtraction, multiplication, or division with other fractions, as everything is handled uniformly as improper fractions during calculations.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division extend naturally from integers to fractions. Each operation adheres to its set of rules when fractions are involved, often requiring skills like finding common denominators or reciprocals.

• Addition/Subtraction often demand common denominators.
• Multiplication involves multiplying across numerators and denominators.
• Division simplifies to multiplying by the reciprocal.

Mastering these operations involves understanding both common procedures and tricks like reciprocal usage, ensuring accuracy and efficiency in solving fraction problems.