Answer
Jennifer can perform 10 Routine Physicals in the morning
Solution
- The question tells us that Jennifer works 3 1/2 hours each morning and we are asked to find how many Routine physicals she can perform in the morning if 1 Routine physical takes 1/3 hours.
- In order to find the number of Routine physicals that can be performed in 3 1/2 hours, we simply need to assume that Jennifer can perform n number of Routine physicals. That is:
![\begin{gathered} \text{ Total number of Routine physicals:} \\ (1)/(3)+(1)/(3)+(1)/(3)+\cdots=n*((1)/(3)) \\ \text{ } \\ \text{ If we take this as the total number of Routine Physicals possible within }3(1)/(2)\text{ hours, and r is the number } \\ \text{ of minutes left before }3(1)/(2)\text{ hours are completed, we have:} \\ \\ (1)/(3)+(1)/(3)+(1)/(3)+\cdots+r=3(1)/(2)=((3*2)+1)/(2)=(7)/(2) \\ \\ \therefore n*((1)/(3))+r=(7)/(2) \end{gathered}]()
Thus, to find the number of Routine Physicals, n, we simply make n the subject of the formula:
![\begin{gathered} n=3((7)/(2)-r) \\ \\ \text{This implies that the number of Routine Physicals is divisible by 3. This means that n is a multiple of 3} \\ \text{Thus, we can test values with multiples of 3.} \\ \\ \text{when n = 3:} \\ 3*((1)/(3))=1\text{hour} \\ \\ \text{when n= 6}\colon \\ 6*((1)/(3))=2\text{hours} \\ \\ \text{when n = 9:} \\ 9*((1)/(3))=3\text{hours} \\ \\ \text{when n= 12:} \\ 12*((1)/(3))=4hours \\ \\ \text{This implies that the number of Routine Physicals cannot be greater than or equal to 12. It also } \\ \text{implies that the number of Routine Physicals cannot be less than 9 since 9 physicals would take} \\ 3\text{ hours.} \\ \\ This\text{ means that the number of possible hours is between 10 or 11.} \\ \\ when\text{ n = 10:} \\ 10*((1)/(3))=3+(1)/(3)\text{ hours}<3(1)/(2)\text{ hours} \\ \\ \text{when n = 11:} \\ 11*((1)/(3))=3+(2)/(3)\text{ hours}>3(1)/(2)\text{ hours} \\ \\ \text{Thus, we can conclude that Jennifer can only do 10 Routine Physicals in the morning} \end{gathered}]()
Final answer
Jennifer can perform 10 Routine Physicals in the morning