Final answer:
Option A and Option D represent proportional functions.
Step-by-step explanation:
A proportional function is a function in which the ratio of the output to the input remains constant. To determine which of the given options represents a proportional function, we need to check if the ratio of the dependent variable (total amount) to the independent variable (time or quantity) stays the same.
Option A:
The total amount of money in the account depends on the number of days. If $20 is spent per day for 5 days, the total amount of money spent is $20 x 5 = $100. Since the ratio of the total amount to the number of days ($100/5) is constant at $20, Option A represents a proportional function.
Option B:
The total amount of money in the piggy bank depends on the number of weeks. If $125 is added every week for 6 weeks, the total amount of money saved is $125 x 6 = $750. However, the ratio of the total amount to the number of weeks ($750/6) is not constant. Therefore, Option B does not represent a proportional function.
Option C:
The total earnings per day depend on the hourly wages and the additional amount for expenses. The ratio of total earnings to the number of days will not be constant because the additional amount for expenses varies. Therefore, Option C does not represent a proportional function.
Option D:
The total cost for installing a fence depends on the length of the fence. Since the cost is a flat fee of $100 plus $12.75 per foot of fencing installed, the ratio of the total cost to the length of the fence will be constant. Therefore, Option D represents a proportional function.
Based on the analysis above, the options that represent proportional functions are Option A and Option D.