To determine how many days it will take Emilio to run a total of [tex]\(7 \frac{1}{2}\)[/tex] miles, given that he runs [tex]\(1 \frac{1}{2}\)[/tex] miles every day, follow these steps:
1. Convert the mixed fractions to improper fractions for easier calculation:
- [tex]\(7 \frac{1}{2}\)[/tex] can be written as [tex]\(\frac{15}{2}\)[/tex] (since [tex]\(7 \times 2 + 1 = 15\)[/tex]).
- [tex]\(1 \frac{1}{2}\)[/tex] can be written as [tex]\(\frac{3}{2}\)[/tex] (since [tex]\(1 \times 2 + 1 = 3\)[/tex]).
2. Set up the equation to find the number of days (d):
We know that the total miles run (in miles) is equal to the number of days times the miles run per day.
[tex]\[
7 \frac{1}{2} \text{ miles} = d \times 1 \frac{1}{2} \text{ miles per day}
\][/tex]
3. Substitute the improper fractions into the equation:
[tex]\[
\frac{15}{2} = d \times \frac{3}{2}
\][/tex]
4. Isolate [tex]\( d \)[/tex] (the number of days):
To solve for [tex]\( d \)[/tex], divide both sides of the equation by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
d = \frac{\frac{15}{2}}{\frac{3}{2}}
\][/tex]
5. Simplify the division:
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
d = \frac{15}{2} \times \frac{2}{3} = \frac{15 \cdot 2}{2 \cdot 3} = \frac{30}{6} = 5
\][/tex]
Therefore, it will take Emilio [tex]\(\boxed{5}\)[/tex] days to run [tex]\(7 \frac{1}{2}\)[/tex] miles.
