Solving the Fraction Expression: $\frac{5}{4} + \frac{1}{\left(1 + \frac{5}{7}\right)}-\frac{3}{5}$

This problem requires us to simplify a mathematical expression involving fractions. We need to perform addition and subtraction of fractions, including a compound fraction (a fraction within another fraction).

Step-by-Step Solution

1. Simplify the compound fraction part: First, focus on the term $\frac{1}{\left(1 + \frac{5}{7}\right)}$.
   • Calculate the sum inside the parenthesis: $1 + \frac{5}{7}$. To add these, find a common denominator, which is 7. Convert 1 to $\frac{7}{7}$.
   • So, $1 + \frac{5}{7} = \frac{7}{7} + \frac{5}{7} = \frac{7+5}{7} = \frac{12}{7}$.
   • Now substitute this back into the term: $\frac{1}{\left(\frac{12}{7}\right)}$.
   • Dividing 1 by a fraction is the same as taking the reciprocal of that fraction. Therefore, $\frac{1}{\frac{12}{7}} = \frac{7}{12}$.
2. Substitute the simplified term back into the original expression: The original expression $\frac{5}{4} + \frac{1}{\left(1 + \frac{5}{7}\right)}-\frac{3}{5}$ becomes $\frac{5}{4} + \frac{7}{12} - \frac{3}{5}$.
3. Find a common denominator for the fractions: We need to add and subtract $\frac{5}{4}$, $\frac{7}{12}$, and $\frac{3}{5}$. The denominators are 4, 12, and 5. The least common multiple (LCM) of 4, 12, and 5 is 60.
4. Convert fractions to the common denominator (60):
   • $\frac{5}{4} = \frac{5 \times 15}{4 \times 15} = \frac{75}{60}$
   • $\frac{7}{12} = \frac{7 \times 5}{12 \times 5} = \frac{35}{60}$
   • $\frac{3}{5} = \frac{3 \times 12}{5 \times 12} = \frac{36}{60}$
5. Perform the addition and subtraction: Now the expression is $\frac{75}{60} + \frac{35}{60} - \frac{36}{60}$.
   • Combine the numerators over the common denominator: $\frac{75 + 35 - 36}{60}$.
   • Calculate the numerator: $75 + 35 = 110$, and $110 - 36 = 74$.
   • The result is $\frac{74}{60}$.
6. Simplify the final fraction: The fraction $\frac{74}{60}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
   • $\frac{74 \div 2}{60 \div 2} = \frac{37}{30}$.

Final Answer Explanation

After performing the necessary fraction arithmetic step-by-step, simplifying the compound fraction first, finding a common denominator, and then combining the terms, the value of the expression $\frac{5}{4} + \frac{1}{\left(1 + \frac{5}{7}\right)}-\frac{3}{5}$ is $\frac{37}{30}$.