Let's solve the problem step by step.
1. Sum of the Fractions:
The given fractions are [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{5}{4}\)[/tex].
To add these fractions, we need a common denominator, which is the least common multiple (LCM) of 5 and 4.
The LCM of 5 and 4 is 20.
So, we convert the fractions [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{5}{4}\)[/tex] to have a common denominator of 20:
[tex]\[
\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}
\][/tex]
[tex]\[
\frac{5}{4} = \frac{5 \times 5}{4 \times 5} = \frac{25}{20}
\][/tex]
Now, we can add these fractions:
[tex]\[
\frac{8}{20} + \frac{25}{20} = \frac{8 + 25}{20} = \frac{33}{20}
\][/tex]
2. Reciprocal of the Sum:
Next, we find the reciprocal of [tex]\(\frac{33}{20}\)[/tex].
The reciprocal of [tex]\(\frac{33}{20}\)[/tex] is:
[tex]\[
\frac{20}{33}
\][/tex]
3. Numerator and Denominator:
The numerator of the reciprocal is 20 and the denominator is 33.
4. Difference Between Numerator and Denominator:
We find the difference between the numerator and the denominator:
[tex]\[
\text{Difference} = |33 - 20| = 13
\][/tex]
5. Sum of the Digits of the Difference:
Now, we calculate the sum of the digits of the difference, which is 13.
[tex]\[
\text{Digits of 13 are 1 and 3}
\][/tex]
So, the sum of these digits is:
[tex]\[
1 + 3 = 4
\][/tex]
Therefore, the answer to the problem is:
[tex]\[
\boxed{4}
\][/tex]
