To find the total amount of wood Harry needs for his project, we need to determine the total length of wood required for each type of piece and then sum these amounts.
1. Convert the mixed numbers to improper fractions:
- For [tex]$2 \frac{3}{8}$[/tex] feet:
[tex]\[
2 \frac{3}{8} = 2 + \frac{3}{8} = \frac{16}{8} + \frac{3}{8} = \frac{19}{8} = 2.375 \text{ feet}
\][/tex]
- For [tex]$1 \frac{1}{3}$[/tex] feet:
[tex]\[
1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} = 1.3333\overline{3} \text{ feet}
\][/tex]
2. Calculate the total amount of wood needed for each type of piece:
- Total length for three pieces of [tex]$2 \frac{3}{8}$[/tex] feet:
[tex]\[
3 \times 2.375 = 7.125 \text{ feet}
\][/tex]
- Total length for five pieces of [tex]$1 \frac{1}{3}$[/tex] feet:
[tex]\[
5 \times 1.3333\overline{3} = 6.6666\overline{6} \text{ feet}
\][/tex]
3. Sum the total lengths to get the total wood needed:
- Total wood required:
[tex]\[
7.125 + 6.6666\overline{6} = 13.7916\overline{6} \text{ feet}
\][/tex]
4. Convert the total wood needed into a mixed number:
- We can express [tex]\( 13.7916\overline{6} \)[/tex] as a fraction to better identify the mixed number:
[tex]\[
13.7916\overline{6} = 13 + 0.7916\overline{6}
= 13 + \frac{19}{24}
= 13 \frac{19}{24} \text{ feet}
\][/tex]
Therefore, the total amount of wood that Harry will need for the project is:
[tex]\[
\boxed{13 \frac{19}{24} \text{ feet}}
\][/tex]
