The monthly performance bonus of a salesman depends on

[M7MBA]

The monthly performance bonus of a salesman depends on the number of units of a product sold by him in a particular month.
Till he sells 10 units of the product, the amount of his monthly performance bonus is given by the formula ax^2+ax, where x is the number of units sold, x ≤ 10 and a is a positive number. If he sells more than 10 units, he gets an added bonus equivalent to ab^2+50b where b is the number of additional units sold. If the salesman sells 28 units in the month of June, how much will he receive as his monthly performance bonus?

(1) Had the salesman sold 5 units he would have received a bonus of $300
(2) Had the salesman sold 15 units he would have received an added bonus of $500

The OA is the option D.

Experts, can you help me here? I don't know how to show that each statement is sufficient.

[ErikaPrepScholar]

Statement 1
Since the salesman sold 5 units in this scenario, his performance bonus will be given by ax^2+ax. His bonus is $300, so ax^2+ax = 300 when x = 5. By plugging in 5 for x, we should be able to solve for a.

If we know a, we can plug it into our equation ax^2+ax and set x equal to 10 to solve for the salesman's bonus for the first 10 units. Then, we can plug a into the equation for selling more than 10 units (ab^2+50b).

Then, since 28 units is 18 additional units past 10 units, we'll set b equal to 18 and solve for the salesman's additional bonus.

Finally, we can add the bonus for the first 10 units to the bonus for the additional 18 units to get the salesman's total bonus. We don't actually need to do any of these calculations (and we shouldn't because it's a huge waste of time) - we only need to know that we can. Sufficient.

Statement 2
Since the salesman sold 15 units in this scenario, he would have sold 5 additional units, which means b = 5, and his additional performance bonus will be given by ab^2+50b. His additional bonus is $500, so ab^2+50b = 500 when b = 5. By plugging in 5 for b, we should be able to solve for a.

Once we have a, we can do the exact same process we did with Statement 1 to solve for the salesman's total bonus. Again, we don't actually need to do the calculations - knowing we could if we wanted to makes this statement sufficient.