If June $1^{\text{st}}$ is a Saturday, what day of the week will July $5^{\text{th}}$ fall on in the same year?

Let's count the total number of days between June $1^{\text{st}}$ and July $5^{\text{th}}$:
* Total days in June = $30$ days.
* Remaining days in June after June $1^{\text{st}}=30-1=29$ days.
* Days in July up to July $5^{\text{th}}=5$ days.

$$\text{Total days}=29+5=34\text{ days}$$

Now, find the number of odd days by dividing by $7$:

$$34÷7=4\text{ weeks with a remainder of }6\text{ days}$$

Number of odd days = $6$.
Add $6$ days to the initial day (Saturday):

$$\text{Saturday}+6\text{ days}=\text{Friday}$$

Therefore, July $5^{\text{th}}$ will be a Friday.