Let's solve the problem step by step, starting with converting the mixed numbers to improper fractions:
1. Convert [tex]\( 8 \frac{2}{5} \)[/tex] to an improper fraction:
[tex]\[
8 \frac{2}{5} = 8 + \frac{2}{5} = \frac{8 \times 5 + 2}{5} = \frac{40 + 2}{5} = \frac{42}{5}
\][/tex]
2. Convert [tex]\( 3 \frac{7}{10} \)[/tex] to an improper fraction:
[tex]\[
3 \frac{7}{10} = 3 + \frac{7}{10} = \frac{3 \times 10 + 7}{10} = \frac{30 + 7}{10} = \frac{37}{10}
\][/tex]
3. Find a common denominator:
The denominators are 5 and 10. The least common multiple (LCM) of 5 and 10 is 10. So, we'll convert both fractions to have this common denominator.
4. Convert [tex]\( \frac{42}{5} \)[/tex] to have the common denominator 10:
[tex]\[
\frac{42}{5} = \frac{42 \times 2}{5 \times 2} = \frac{84}{10}
\][/tex]
Now we have the fractions in a common denominator:
[tex]\[
\frac{84}{10} \quad \text{and} \quad \frac{37}{10}
\][/tex]
5. Perform the subtraction:
[tex]\[
\frac{84}{10} - \frac{37}{10} = \frac{84 - 37}{10} = \frac{47}{10}
\][/tex]
6. Simplify the result, if possible:
[tex]\[
\frac{47}{10}
\][/tex]
Since 47 and 10 have no common factors other than 1, this fraction cannot be simplified further. So, we convert it into a mixed number:
[tex]\[
\frac{47}{10} = 4 \frac{7}{10}
\][/tex]
The whole number part of the answer is:
[tex]\[
\boxed{4}
\][/tex]
