The problem is to evaluate the expression $5 + \{4 - (\frac{2}{5} + \frac{1}{4})\}$.

1. Problem Description

The problem is to evaluate the expression $5 + \{4 - (\frac{2}{5} + \frac{1}{4})\}$.

2. Solution Steps

First, we evaluate the expression within the parentheses: $\frac{2}{5} + \frac{1}{4}$. To add these fractions, we need to find a common denominator. The least common multiple of 5 and 4 is

2

0. So, we rewrite the fractions with the common denominator of 20:

$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$

$\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$

Now we can add the fractions:

$\frac{8}{20} + \frac{5}{20} = \frac{8+5}{20} = \frac{13}{20}$

Now we can substitute this back into the original expression:

$5 + \{4 - (\frac{13}{20})\}$

Next, we evaluate the expression within the curly braces: $4 - \frac{13}{20}$. We can rewrite 4 as a fraction with a denominator of 20:

$4 = \frac{4 \times 20}{20} = \frac{80}{20}$

Now we subtract:

$\frac{80}{20} - \frac{13}{20} = \frac{80-13}{20} = \frac{67}{20}$

Finally, we add 5 to this result:

$5 + \frac{67}{20}$. We can rewrite 5 as a fraction with a denominator of 20:

$5 = \frac{5 \times 20}{20} = \frac{100}{20}$

Now we add:

$\frac{100}{20} + \frac{67}{20} = \frac{100+67}{20} = \frac{167}{20}$

3. Final Answer

$\frac{167}{20}$