Simplifying the Mathematical Expression

We need to find the value of the given mathematical expression involving fractions:

$ \frac{5}{3} + \left(\frac{1}{1+\frac{5}{6}}\right) - \frac{5}{3} $

Let's break down the calculation step-by-step:

1. Identify the terms: The expression consists of three main parts: $\frac{5}{3}$, the complex fraction part $\left(\frac{1}{1+\frac{5}{6}}\right)$, and $-\frac{5}{3}$.
2. Notice cancellations: We can observe that the first term, $\frac{5}{3}$, and the last term, $-\frac{5}{3}$, are additive inverses. This means they will cancel each other out when added together ($ \frac{5}{3} - \frac{5}{3} = 0 $).
3. Simplify the complex fraction: Now, we focus on simplifying the middle part: $ \left(\frac{1}{1+\frac{5}{6}}\right) $.
   • First, simplify the denominator: $ 1+\frac{5}{6} $.
   • To add these, find a common denominator, which is 6. So, $ 1 = \frac{6}{6} $.
   • The sum becomes: $ \frac{6}{6} + \frac{5}{6} = \frac{6+5}{6} = \frac{11}{6} $.
   • Now substitute this back into the complex fraction: $ \frac{1}{\frac{11}{6}} $.
   • To divide by a fraction, we multiply by its reciprocal: $ 1 \times \frac{6}{11} = \frac{6}{11} $.
4. Combine the results: Substitute the simplified complex fraction back into the original expression: $ \left(\frac{5}{3} - \frac{5}{3}\right) + \frac{6}{11} $
5. Final Calculation: The first part is 0, so we are left with: $ 0 + \frac{6}{11} = \frac{6}{11} $

Therefore, the value of the given expression is $ \frac{6}{11} $.

Key Concepts Used

• Fraction Arithmetic: Addition and subtraction of fractions.
• Complex Fractions: Simplifying fractions within fractions.
• Additive Inverses: Recognizing terms that cancel each other out.