Problem 53 - Algebra 2

Question

SALARY Mr. Vacarro's salary this year is \(\$ 41,000 .\) If he gets a raise of \(\$ 2500\) each year, how much will Mr. Vacarro earn in ten years?

Step-by-Step Solution

Answer

Mr. Vacarro will earn \( \$66,000 \) in ten years.

Step 1: Determine Initial Salary

Mr. Vacarro's current salary is given as \( \$41,000 \). We will start our calculation from this amount.

Step 2: Calculate Total Raise Over Ten Years

Mr. Vacarro receives a \( \\(2,500 \) raise each year. Over ten years, the total raise will be \( 2,500 \times 10 = \\)25,000 \).

Step 3: Calculate Salary in Ten Years

To find Mr. Vacarro's salary in ten years, we need to add the total raise to his current salary. This will be \( 41,000 + 25,000 = \$66,000 \).

Key Concepts

Understanding Linear Growth

Linear growth refers to the process by which something increases at a constant rate over time. In the context of Mr. Vacarro's salary, this means that his salary increases by the same amount each year. This consistent increase is what makes the growth linear.

Linear growth is easy to predict because you know exactly how much will be added over time. In Mr. Vacarro's case, his salary grows by \(\$2,500\) each year. You can visualize this as a straight line on a graph, where each point on the line is equally spaced from the next.

Understanding linear growth is crucial for anyone dealing with budgets, investments, or expenses that change consistently over time. By knowing your initial amount and the rate of increase, you can quickly calculate future values.

Simple Steps in Salary Calculation

Calculating a future salary with a known rate of increase is straightforward once you understand the concept of linear growth. For example, Mr. Vacarro's current salary is \(\\(41,000\), and he receives a yearly raise of \(\\)2,500\).

To determine his salary in 10 years:

• First, multiply the annual raise by the number of years: \(2,500 \times 10 = 25,000\).
• Next, add that total raise to his current salary: \(41,000 + 25,000 = \$66,000\).

In this way, you'll arrive at a future salary amount effortlessly.

This method works best when the increase each year remains consistent, as it does with Mr. Vacarro's raises.

Introducing the Recursion Formula

A recursion formula can be a handy tool for working with arithmetic sequences like Mr. Vacarro's salary increments. It helps you determine the value of each new term based on the previous term.

For an arithmetic sequence, the recursion formula has the form:\[ a_{n} = a_{n-1} + d \]where

• \(a_{n}\) is the term you're trying to find,
• \(a_{n-1}\) is the last known term, and
• \(d\) is the common difference (in our case, the annual raise of \(\\(2,500\)).

To find Mr. Vacarro's salary for any given year using this formula:

• Start with this year's salary \(a_{1} = \\)41,000\)
• Apply the formula repeatedly, using \(d = \\(2,500\). For instance, the salary for the second year would be: \[a_{2} = a_{1} + 2,500 = 41,000 + 2,500 = \\)43,500\]

Repeat this process until you reach the desired term. This formula makes it simple to calculate future values systematically and ensures consistent growth calculations.