Question

Three salesmen work for the same company, selling the same product. And, although they are all paid on a weekly basis, each salesman earns his paycheck differently. Salesman A works strictly on commission. He earns $65 per sale, with a maximum weekly commission of $1,300. Salesman B earns a weekly base salary of $300, plus a commission of $40 per sale. There are no limits on the amount of commission he can earn. Salesman C does not earn any commission. His weekly salary is $900. The weekly paycheck amount for each salesman, p, is a function of the number of sales, s, they had in that week. Is the function representing the weekly paycheck amount of Salesman A a proportional relationship? Use complete sentences to explain your reasoning. Is the function representing the weekly paycheck amount of Salesman B a proportional relationship? Use complete sentences to explain your reasoning. Is the function representing the weekly paycheck amount of Salesman C a proportional relationship? Use complete sentences to explain your reasoning.

Answer 1

Q1)
is the function representing the weekly paycheck of salesman A a proportional relationship? 
p - function
s - number of sales
Salesman A earns £65 per sale. Therefore if he earns s number of sales, the total amount he earns is 65*s = 65s
Lets assume for s number of sales he receives his maximum weekly commission, therefore the equation for A's weekly payment is; 
p = 65s +1300 
to determine if p and s are proportional to each other, it should take the form of equation p=ks
where k is a constant. If we graph p=ks, then k is the slope that is a constant and the line should pass through the origin, which means that y intercept is 0. 
however in this equation there is an intercept of +1300, therefore p and s are not proportional values.

Q2)
For salesman B he has a weekly base salary of £300 and commission of £40 per sale. Then the salary is fixed however for s number of sales, the commission he will receive is 40*s = 40s
the function for B's payment is as follows;
p = 40s + 300
This equation if we take it as a graph equation with p and s as y and x coordinates respectively, the equation we get is of one with an intercept of +300
 therefore if we graph it, the line will not pass through the origin which means that the p and s values are not proportional to each other. 

Q3)
Salesman C receives a weekly salary of £900. This is a fixed amount and does not vary depending on the sales. Therefore this is anyway not proportional as p function is not affected by s value. 
therefore p and s are not related, hence they aren't proportional. 
p = 900