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Understanding the percentage formula is essential in solving problems related to commissions. This formula helps us calculate an amount based on a given percentage and a base value. We express the formula as: \[ \text{Amount} = \left( \frac{\text{Percentage}}{100} \right) \times \text{Base} \] This means that when we know the base amount and the percentage, we can find out what portion the percentage represents of the base. For instance, in the given exercise:

• The base is the first \(1,000 in sales.
• The percentage is 9.5%.

Using the formula, we find the commission from the first \)1,000: \[ \text{Amount} = \left( \frac{9.5}{100} \right) \times 1,000 = \$95 \] This calculation helps in figuring out how much money is earned from a specific portion of sales.

Sales commission is a financial incentive given to sales staff, encouraging them to sell more products. It is usually a percentage of the sales they make. In the problem, it is structured in tiers:

• First tier: 9.5% commission on the first $1,000 of sales.
• Second tier: 15.5% commission on any sales beyond $1,000.

This tiered commission system rewards staff for reaching higher sales brackets. Alex, the sales staff in the problem, receives a commission for his sales according to these percentages. This system motivates employees to increase their sales volume because they earn a higher commission as their sales grow.

Algebraic problem-solving involves using equations and expressions to find unknown variables. To calculate Alex's total sales, we approached the problem step-by-step, using algebraic equations. First, we calculate the commission of the first $1,000 of sales. Then, using the total commission Alex earned, we subtract the portion from the first tier to find the commission for the subsequent sales. In solving for the rest of the sales, we use the formula: \[ \text{Base} = \frac{\text{Amount}}{\left( \frac{\text{Percentage}}{100} \right)} \] Here, the amount is Alex's remaining commission, and the percentage corresponds to the second commission tier (15.5%). This algebraic approach helps in breaking down complex problems into manageable parts to find solutions.

Financial mathematics is the field of study that applies mathematical methods to solve financial problems. Calculating commissions involves simple principles of financial mathematics, including handling percentages and calculating totals. In the exercise, we apply these principles to determine the total sales based on the given commission rates. We consider different percentages for varying sales amounts, showing how financial mathematics helps in real-world applications, such as calculating earnings based on sales. This branch of mathematics supports a variety of financial analyses, including budgeting, tax computation, and investment analysis. Mastery of financial mathematics empowers individuals to make informed financial decisions in both personal and professional contexts.