Math / Algebra

QuestionQuestion 12 You are offered two different sales jobs. The first company offers a straight commission of $5 \%$ of the sales. The second company offers a salary of $\$ 440$ per week plus $3 \%$ of the sales. How much would you have to sell in a week in order for the straight commission offer to be at least as good?

Studdy Solution

STEP 1

What is this asking? How much do I need to sell to make the commission-only job pay at least as much as the salary plus commission job? Watch out! Don't mix up the commission percentages or forget to include the base salary in your calculations!

STEP 2

1. Define the functions
2. Set up the inequality
3. Solve for the sales amount

STEP 3

Let's define $E_1$ as the earnings from the first job (straight commission) and $E_2$ as the earnings from the second job (salary plus commission).
We'll use $x$ to represent the amount of sales in $\$$ made in a week.

STEP 4

The first company offers a straight commission of $5\%$ of sales.
So, the earnings from the first job can be represented as: $E_1 = 0.05 \cdot x$

STEP 5

The second company offers a salary of $\$440$ per week plus $3\%$ of sales.
So, the earnings from the second job can be represented as: $E_2 = 440 + 0.03 \cdot x$

STEP 6

We want to find the sales amount $x$ where the straight commission offer ( $E_1$ ) is at least as good as the salary plus commission offer ( $E_2$ ). "At least as good" means greater than or equal to.
So, we can set up the following inequality: $E_1 \ge E_2$

STEP 7

Now, substitute the expressions we found for $E_1$ and $E_2$ : $0.05 \cdot x \ge 440 + 0.03 \cdot x$

STEP 8

To isolate the $x$ terms, let's subtract $0.03 \cdot x$ from both sides of the inequality: $0.05 \cdot x - 0.03 \cdot x \ge 440 + 0.03 \cdot x - 0.03 \cdot x$ $0.02 \cdot x \ge 440$

STEP 9

Now, to find $x$ , we'll divide both sides of the inequality by 0.02: $\frac{0.02 \cdot x}{0.02} \ge \frac{440}{0.02}$ $x \ge 22000$

STEP 10

You would need to sell at least $\$22,000$ worth of products or services in a week for the straight commission offer to be at least as good as the salary plus commission offer.

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Studdy Solution

STEP 1

What is this asking? How much do I need to sell to make the commission-only job pay at least as much as the salary plus commission job? Watch out! Don't mix up the commission percentages or forget to include the base salary in your calculations!

STEP 2

1. Define the functions
2. Set up the inequality
3. Solve for the sales amount

STEP 3

Let's define $E_1$ as the earnings from the first job (straight commission) and $E_2$ as the earnings from the second job (salary plus commission).
We'll use $x$ to represent the amount of sales in $\$$ made in a week.

STEP 4

The first company offers a straight commission of $5\%$ of sales.
So, the earnings from the first job can be represented as: $E_1 = 0.05 \cdot x$

STEP 5

The second company offers a salary of $\$440$ per week plus $3\%$ of sales.
So, the earnings from the second job can be represented as: $E_2 = 440 + 0.03 \cdot x$

STEP 6

We want to find the sales amount $x$ where the straight commission offer ( $E_1$ ) is at least as good as the salary plus commission offer ( $E_2$ ). "At least as good" means greater than or equal to.
So, we can set up the following inequality: $E_1 \ge E_2$

STEP 7

Now, substitute the expressions we found for $E_1$ and $E_2$ : $0.05 \cdot x \ge 440 + 0.03 \cdot x$

STEP 8

To isolate the $x$ terms, let's subtract $0.03 \cdot x$ from both sides of the inequality: $0.05 \cdot x - 0.03 \cdot x \ge 440 + 0.03 \cdot x - 0.03 \cdot x$ $0.02 \cdot x \ge 440$

STEP 9

Now, to find $x$ , we'll divide both sides of the inequality by 0.02: $\frac{0.02 \cdot x}{0.02} \ge \frac{440}{0.02}$ $x \ge 22000$

STEP 10

You would need to sell at least $\$22,000$ worth of products or services in a week for the straight commission offer to be at least as good as the salary plus commission offer.

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Studdy Solution

STEP 1

What is this asking? How much do I need to sell to make the commission-only job pay at least as much as the salary plus commission job? Watch out! Don't mix up the commission percentages or forget to include the base salary in your calculations!

STEP 2

1. Define the functions2. Set up the inequality3. Solve for the sales amount

STEP 3

Let's define E1E_1E1​ as the earnings from the first job (straight commission) and E2E_2E2​ as the earnings from the second job (salary plus commission).We'll use xxx to represent the amount of sales in $\$$ made in a week.

STEP 4

The first company offers a straight commission of 5%5\%5% of sales.So, the earnings from the first job can be represented as: E1=0.05⋅xE_1 = 0.05 \cdot xE1​=0.05⋅x

STEP 5

The second company offers a salary of $440\$440$440 per week plus 3%3\%3% of sales.So, the earnings from the second job can be represented as: E2=440+0.03⋅xE_2 = 440 + 0.03 \cdot xE2​=440+0.03⋅x

STEP 6

We want to find the sales amount xxx where the straight commission offer (E1E_1E1​) is *at least as good* as the salary plus commission offer (E2E_2E2​). "At least as good" means greater than or equal to.So, we can set up the following inequality: E1≥E2E_1 \ge E_2E1​≥E2​

STEP 7

Now, substitute the expressions we found for E1E_1E1​ and E2E_2E2​: 0.05⋅x≥440+0.03⋅x0.05 \cdot x \ge 440 + 0.03 \cdot x0.05⋅x≥440+0.03⋅x

STEP 8

STEP 9

Now, to find xxx, we'll divide both sides of the inequality by **0.02**: 0.02⋅x0.02≥4400.02\frac{0.02 \cdot x}{0.02} \ge \frac{440}{0.02}0.020.02⋅x​≥0.02440​ x≥22000x \ge 22000x≥22000

STEP 10

You would need to sell at least $22,000\$22,000$22,000 worth of products or services in a week for the straight commission offer to be at least as good as the salary plus commission offer.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define the functions
2. Set up the inequality
3. Solve for the sales amount

Let's define $E_1$ as the earnings from the first job (straight commission) and $E_2$ as the earnings from the second job (salary plus commission).
We'll use $x$ to represent the amount of sales in $\$$ made in a week.

The first company offers a straight commission of $5\%$ of sales.
So, the earnings from the first job can be represented as: $E_1 = 0.05 \cdot x$

The second company offers a salary of $\$440$ per week plus $3\%$ of sales.
So, the earnings from the second job can be represented as: $E_2 = 440 + 0.03 \cdot x$

We want to find the sales amount $x$ where the straight commission offer ( $E_1$ ) is at least as good as the salary plus commission offer ( $E_2$ ). "At least as good" means greater than or equal to.
So, we can set up the following inequality: $E_1 \ge E_2$

Now, substitute the expressions we found for $E_1$ and $E_2$ : $0.05 \cdot x \ge 440 + 0.03 \cdot x$

Now, to find $x$ , we'll divide both sides of the inequality by 0.02: $\frac{0.02 \cdot x}{0.02} \ge \frac{440}{0.02}$ $x \ge 22000$

You would need to sell at least $\$22,000$ worth of products or services in a week for the straight commission offer to be at least as good as the salary plus commission offer.