I need to know how to calculate these questions.

Question

Sondra is offered a choice as to how her wage will be calculated.

Option A: A base payment of $250 per week plus 8% commission on sales.

Option B: A base payment of $100 per week plus 12% commission on sales.

(a) Calculate Sondra's wage using Option A for sales of $5000 in a week.

(b) Calculate Sondra's wage using Option B for sales of $3500 in a week.

(c) Determine the sales amount which gives the same total payment for a week and write a brief comparison of the options.

Answer 1

Heya Tiana.

So the idea here is to try to create equations to represent the different options.

In both options, there are two parts, base payment and commission on sales.

What does base payment mean? It means no matter what happens, that base payment will be given. Even if 0 sales are made, Sondra will still get the base payment. So for example, in Option A we can represent this as:

base payment + other payments

$250 + other payments

No matter what happens in the other payment, that $250 is guaranteed.

Now, how does commission on sales work? Basically, whatever the amount of money Sondra gets in sales, she gets a percentage of that amount.

If Sondra sold $50 and she got 100% commission on sales, she would get $50.

If Sondra sold $50 and she got 50% commission on sales, she would get $25.

So how do we calculate a percentage of an amount? We multiple the percentage by the amount.

For the above examples we would calculate:

100% × $50 = $50

50% × $50 = $25

Ok so now we know how to represent the base payment and the commission as formulas. So let's put everything together:

Option A: A base payment of $250 per week plus 8% commission on sales:

Base payment + other payments

Base payment + commission on sales

$250 + commission on sales

$250 + 8% × (sales)

Option B: A base payment of $100 per week plus 12% commission on sales:

$100 + 12% × (sales)

For simplicity's sake, we are now going to rewrite these equations in math terms.

To do this, we will change three things:

1. Remove the $ so everything is in numbers only.
2. Rewrite the percentage as a decimal. This is done by putting the number over one hundred (and in fact the word "percent" comes from "per cent" which is Latin for per hundred). So 8% is represented as 8/100 or .08. And 12% = 12/100 = .12. Another way to look at this is to move the decimal to the left two spaces. So 8% is 8. → .8 → .08.
3. Sales we are going to represent as a variable, x. This means x stands for some dollar amount of sales that may change from case to case.

Option A:

$250 + 8% × (sales)

250 + (.08) × x

250 + .08x

Option B:

$100 + 12% × (sales)

100 + (.12) × x

100 + .12x

Great, now that we have represented the options as equations, the rest of the question is smooth sailing.

Part (a) says to calculate Sondra's wage using Option A for $5000 in sales.

So we can input 5000 in the sales variable, which we denoted as x.

Option A:

250 + .08x

250 + .08 × (5000)

This calculation is trivial and can be done with a calculator.

Part (b) is similar but for Option B and $3500 in sales.

The work is identical but swap for Option B and input 3500. I will leave that part for you.

Part (c) asks for the sales amount that gives the same payment for the week. So once again, we want to try to write this as an equation.

We want a sales amount (which we already determined that we are representing as x), in which the payment is the same for both options. Well in order for the two options to be the same, they must be equal. That is to say, we are looking for x where Option A = Option B. Or as an equation:

Option A = Option B

250 + .08x = 100 + .12x

So we want to find at what value of x the above equation is true. In order to find that value of x, we need to isolate it on one side of the equation. Basically if we can get x by itself on one side of the equation, then the equation will also be saying that x equals whatever is on the other side (the equation will be of the form x = some number).

250 + .08x = 100 + .12x | we subtract .08x from each side so it's gone from one side

- .08x - .08x

250 + 0x = 100 + .04x

or

250 = 100 + .04x | we get the 100 away x by subtracting 100 from each side

-100 = -100 + .04x

150 = 0 + .04x

or

150 = .04x | we want to get x alone so we divide each side by .04

/.04 = /.04

3750 = x

Wonderful! we have x isolated and thus have a value for x. So we know at x = 3750, the two options are equal. Or written as a statement, at $3750 in sales, Option A and Option B give the same total payment.

The last part asks for a brief comparison of the options. Essentially, given the two options are different, yet we know they are equal at this one value, $3750 in sales, that means there must be sales amounts in which Option A is better (gives more money) and sales amounts in which Option B is better.

To find out which is which, you can calculate the payment for a sales amount under $3750 for both options and see which is better for values less than $3750.

Then calculate the payment for a sales amount over $3750 for both options to see which is better for values greater than $3750 (and perhaps you may have already done part of these calculations).

After you have done these, you would be able to make a statement that looks something like this:

"Under $3750 in sales, Option _ gives more money than Option _. Over $3750 in sales, Option _ gives more money than Option _. At $3750, the two options give the same amount of money."

Hope that was helpful. Let me know if you have any other questions.