Problem 69 - Algebra and Trigonometry

Question

You are considering two job offers. Company A will start you at \(\$ 19,000\) a year and guarantee a raise of \(\$ 2600\) per year. Company B will start you at a higher salary, \(\$ 27,000\) a year, but will only guarantee a raise of \(\$ 1200\) per year. Find the total salary that each company will pay over a ten- year period. Which company pays the greater total amount?

Step-by-Step Solution

Answer

On calculating, company A pays a total of \$295,000 and company B pays \$327,000 over ten years. Therefore, company B pays the greater total amount over ten years.

Step 1: Calculating cumulative salary from company A

Consider an annual raise of \(2600\). Thus, for each year, the salary increments by that amount. In the first year, the salary will be \(19000\). So the total payment for 10 years will be \(19000 + (19000+2600) + (19000+2600*2) + ... + (19000+2600*9)\). Applying the formula for the sum of an arithmetic sequence: \[ a * n + (n * (n - 1) / 2) * d\] where \( a\) is the initial salary, \( n\) is the number of years, and \( d\) is the difference (annual raise). After substituting, the total pay over ten years can be calculated.

Step 2: Calculating cumulative salary from company B

Similarly, perform the same calculations for company B. Here, the starting salary is higher (\$27000), but the annual raise is lower (\$1200). Using the arithmetic sequence sum formula and the given values, discover the total pay over ten years.

Step 3: Comparing the salaries

Upon obtaining the total salaries from both companies over the 10-year period, it's time to make a comparison. The company that offers the higher total salary will be the better choice financially.