\frac{4}{5}+\left(\frac{1}{1+\frac{5}{8}}\right)-\frac{3}{5}

The correct option is 53/65.

Let's solve the expression step-by-step to see why this option is correct. The given expression is:

$$\frac{4}{5}+\left(\frac{1}{1+\frac{5}{8}}\right)-\frac{3}{5}$$

Step 1: Simplify the denominator in the middle term.
We first calculate the value of the term in the denominator: 1 + 5/8.
To add these, find a common denominator, which is 8:

$$1+\frac{5}{8}=\frac{8}{8}+\frac{5}{8}=\frac{8+5}{8}=\frac{13}{8}$$

Step 2: Simplify the middle fraction.
Now we substitute this back into the middle term:

$$\frac{1}{1+\frac{5}{8}}=\frac{1}{\left(\frac{13}{8}\right)}=\frac{8}{13}$$

Step 3: Substitute this value back into the original expression.
Now, the expression becomes:

$$\frac{4}{5}+\frac{8}{13}-\frac{3}{5}$$

Step 4: Group and simplify the terms with the same denominator.
We can group the first and the last terms because they share the common denominator of 5:

$$(\frac{4}{5}-\frac{3}{5})+\frac{8}{13}=\frac{1}{5}+\frac{8}{13}$$

Step 5: Perform the final addition.
To add 1/5 and 8/13, find the least common multiple (LCM) of the denominators 5 and 13. Since both are prime numbers, their LCM is 5 * 13 = 65.
Convert both fractions to have this common denominator:

$$\frac{1}{5}=\frac{1\times 13}{5\times 13}=\frac{13}{65}$$

And for the second fraction:

$$\frac{8}{13}=\frac{8\times 5}{13\times 5}=\frac{40}{65}$$

Now, add the two converted fractions together:

$$\frac{13}{65}+\frac{40}{65}=\frac{13+40}{65}=\frac{53}{65}$$

Thus, the final simplified value is indeed 53/65.