To determine how long it will take Terrence to fill 4 aquariums, given that he can fill [tex]\(1 \frac{1}{3}\)[/tex] aquariums in [tex]\(2 \frac{1}{2}\)[/tex] hours, we can follow these steps:
1. Convert the mixed numbers to improper fractions:
- [tex]\(1 \frac{1}{3} = \frac{4}{3}\)[/tex] aquariums
- [tex]\(2 \frac{1}{2} = \frac{5}{2}\)[/tex] hours
2. Calculate the rate at which Terrence fills the aquariums:
[tex]\[
\text{Rate} = \frac{\text{Aquariums filled}}{\text{Time taken}} = \frac{\frac{4}{3} \text{ aquariums}}{\frac{5}{2} \text{ hours}}
\][/tex]
3. To simplify the fraction for the rate, we can multiply by the reciprocal of the denominator:
[tex]\[
\text{Rate} = \frac{4}{3} \div \frac{5}{2} = \frac{4}{3} \times \frac{2}{5} = \frac{8}{15} \text{ aquariums per hour}
\][/tex]
4. Determine how long it will take him to fill 4 aquariums:
[tex]\[
\text{Time to fill 4 aquariums} = \frac{4 \text{ aquariums}}{\text{Rate}} = \frac{4 \text{ aquariums}}{\frac{8}{15} \text{ aquariums per hour}} = 4 \times \frac{15}{8} = \frac{60}{8} = 7.5 \text{ hours}
\][/tex]
Therefore, it will take Terrence [tex]\(7 \frac{1}{2}\)[/tex] hours to fill 4 aquariums. So, the correct option is:
[tex]\[ 7 \frac{1}{2} \text{ hours} \][/tex]
