Problem 63 - College Algebra

Question

Company A pays \(\$ 24,000\) yearly with raises of \(\$ 1600\) per year. Company B pays \(\$ 28,000\) yearly with raises of \(\$ 1000\) per year. Which company will pay more in year \(10 ?\) How much more?

Step-by-Step Solution

Answer

Without performing the actual calculations, it can be seen Company B offers a higher starting pay while Company A offers larger yearly raises. Depending on the specific numbers this may result in either the company with the higher starting pay or the larger raises having the higher total pay after 10 years. The actual calculations need to be performed to determine this.

Step 1: Calculate Total Pay for Company A

In an arithmetic sequence, the nth term (n being the year in this case) can be found using the formula \(a_n = a_1 + (n - 1) * d\) where \(a_1\) is the first term (initial annual pay), d is the common difference (annual raise) and n is the term number (year). For company A, \(a_1 = \$24,000\), \(d = \$1600\), and n = 10. Then, plug these values into the arithmetic sequence formula to get the 10th term.

Step 2: Calculate Total Pay for Company B

For company B, \(a_1 = \$28,000\), \(d = \$1000\), and n = 10. Repeat the same process as used for company A to find the 10th term.

Step 3: Compare the Total Pays

Next, compare the 10th terms (total pay in the 10th year) for companies A and B to determine which company pays more in the 10th year. The difference between the two pays will give the amount by which one exceeds the other.