Problem 64 - College Algebra

Question

Company A pays \(\$ 23,000\) yearly with raises of \(\$ 1200\) per year. Company B pays \(\$ 26,000\) yearly with raises of \(\$ 800\) per year. Which company will pay more in year \(10 ?\) How much more?

Step-by-Step Solution

Answer

In the 10th year, Company A will pay $600 more than Company B.

Step 1: Understand Salary Increment

First, understand that the salaries offered by both companies increase by a fixed amount every year. Company A's salary increases by $1200 per year and Company B's salary increases by $800 per year. These are examples of arithmetic sequences.

Step 2: Find the 10th Term for Company A

We can use the formula for the nth term of an arithmetic sequence: \(a_n = a + (n-1)d\). For Company A, the first term \(a\) is $23000, and the common difference \(d\) is $1200. Substituting these values into the formula: \(a_{10} = 23000 + (10-1)*1200 = 23000 + 9*1200 = 33800\). Therefore, Company A will pay $33800 in the 10th year.

Step 3: Find the 10th Term for Company B

For Company B, the first term \(a\) is $26000, and the common difference \(d\) is $800. Substituting these values into the formula: \(a_{10} = 26000 + (10-1)*800 = 26000 + 9*800 = 33200\). Therefore, Company B will pay $33200 in the 10th year.

Step 4: Compare the 10th Term Salaries

To find out who pays more in the 10th year, simply subtract Company B's 10th year salary from Company A's 10th year salary: $33800 - $33200 = $600. Company A pays $600 more than Company B in the 10th year.