Problem 56 - 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

Company B will pay more over a span of 10 years. The difference will be \$10,000.

Step 1: Find Total Salary for Company A

For company A, the first term (a) of the arithmetic sequence is $23,000 and the common difference (d) is $1200. The total salary (S) over 10 years can be calculated using the formula for the sum of an arithmetic series \(S = n/2 * (2a + (n-1)d)\), where n is the number of terms, which is 10 in this case. Substitute the values noting that the salary is compounded annually, we get \(S_A = 10/2 * (2*23000 + (10-1)*1200)\).

Step 2: Calculate Total Salary for Company A

After performing the calculation, company A's total salary over 10 years is \(S_A = 5 * (46000 + 9*1200) = \$265,000\).

Step 3: Find Total Salary for Company B

For company B, the first term \(a\) is $26,000 and the common difference \(d\) is $800. The total salary over the 10 years can be computed using the same formula \(S = n/2 * (2a + (n-1)d)\), replacing variable values: \(S_B = 10/2 * (2*26000 + (10-1)*800)\).

Step 4: Calculate Total Salary for Company B

After doing the math, company B's total salary over 10 years ends up being \(S_B = 5 * (52000 + 9*800) = \$275,000\).

Step 5: Compare Salaries and Calculate the Difference

Taking the difference between the two total salaries will show which company is paying more and by how much. So, \(Difference = S_B - S_A = 275000 - 265000 = \$10,000\).