To determine the unit rate at which Martin paints in walls per gallon, we need to find out how much of the wall he can paint with one gallon of paint. Let's follow the steps in a detailed manner:
1. Identify the quantities given:
- Martin uses [tex]\(\frac{5}{8}\)[/tex] of a gallon of paint.
- This amount of paint covers [tex]\(\frac{4}{5}\)[/tex] of a wall.
2. Determine the unit rate:
The unit rate is found by dividing the fraction of the wall covered by the fraction of the gallon of paint used. Mathematically, this can be expressed as:
[tex]\[
\text{unit rate} = \frac{\text{wall covered}}{\text{paint used}} = \frac{\frac{4}{5}}{\frac{5}{8}}
\][/tex]
3. Perform the division:
Dividing by a fraction is the same as multiplying by its reciprocal. So, we need to multiply [tex]\(\frac{4}{5}\)[/tex] by the reciprocal of [tex]\(\frac{5}{8}\)[/tex], which is [tex]\(\frac{8}{5}\)[/tex]:
[tex]\[
\frac{4}{5} \div \frac{5}{8} = \frac{4}{5} \times \frac{8}{5} = \frac{4 \cdot 8}{5 \cdot 5} = \frac{32}{25}
\][/tex]
Therefore, the unit rate in simplest fraction form is:
[tex]\[
\frac{32}{25} \text{ walls per gallon}
\][/tex]
4. Convert the fraction to a mixed number (if needed):
We can also convert [tex]\(\frac{32}{25}\)[/tex] to a mixed number for comparison with the other choices given.
[tex]\[
\frac{32}{25} = 1 \frac{7}{25} \text{ walls per gallon}
\][/tex]
This is because [tex]\(32 \div 25 = 1\)[/tex] with a remainder of [tex]\(7\)[/tex], leading us to [tex]\(1 \frac{7}{25}\)[/tex].
5. Verify the multiple-choice options:
- [tex]\(1 \frac{7}{25}\)[/tex] walls per gallon is equivalent to [tex]\(\frac{32}{25}\)[/tex] walls per gallon.
- [tex]\(1 \frac{3}{32}\)[/tex] walls per gallon is a completely different number and does not match [tex]\(\frac{32}{25}\)[/tex].
- [tex]\(\frac{32}{25}\)[/tex] walls per gallon is exactly the same as the simplified fraction we calculated.
- [tex]\(\frac{1}{2}\)[/tex] of a wall per gallon is clearly a much lower rate and does not match our result.
Hence, the correct answers are:
[tex]\[
\frac{32}{25} \text{ walls per gallon}
\][/tex]
[tex]\[
1 \frac{7}{25} \text{ walls per gallon}
\][/tex]
