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Deb Cook is given the choice of two​ positions, one paying ​$3 comma 100 per month and the other paying ​$2 comma 300 per month plus a​ 5% commission on all sales made during the month. What amount must she sell in a month for the second position to be more​ profitable?

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Part 1
The second position is more profitable if Deb sells more than ​$
  
enter your response here per month.

1 answer

To find the amount Deb must sell for the second position to be more profitable than the first position, we can set up an inequality.

Let \( x \) be the amount of sales Deb makes in a month.

  1. The salary for the first position is: \[ 3100 \text{ dollars/month} \]

  2. The salary for the second position is: \[ 2300 + 0.05x \text{ dollars/month} \]

To find the break-even point where the two positions are equal in pay: \[ 2300 + 0.05x = 3100 \]

Now, solve for \( x \):

  1. Subtract \( 2300 \) from both sides: \[ 0.05x = 3100 - 2300 \] \[ 0.05x = 800 \]

  2. Divide both sides by \( 0.05 \): \[ x = \frac{800}{0.05} \] \[ x = 16000 \]

Thus, Deb must sell more than \( 16000 \) dollars in a month for the second position to be more profitable.

So, the answer is: \[ \text{Deb must sell more than } $16000 \text{ per month.} \]