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Chapter 8: Problem 74

You are investigating two employment opportunities. Company A offers \(\$ 30,000\) the first year. During the next four years, the salary is guaranteed to increase by \(6 \%\) per year. Company B offers \(\$ 32,000\) the first year, with guaranteed annual increases of \(3 \%\) per year after that. Which company offers the better total salary for a five-year contract? By how much? Round to the nearest dollar.

Short Answer

Expert verified
Company B offers the better total salary for a five-year contract. The difference is \$782.

Step by step solution

01

Calculate the Total Salary for Company A

The initial salary is $30,000 and there is a $6\%$ annual increase. We can calculate the salary per year as follows:Year 1: $30,000Year 2: $30,000 * 1.06 = \$31,800Year 3: \$31,800 * 1.06 = \$33,708Year 4: \$33,708 * 1.06 = \$35,730Year 5: \$35,730 * 1.06 = \$37,874The total salary for the 5 years is therefore \$30,000 + \$31,800 + \$33,708 + \$35,730 + \$37,874 = \$169,112.
02

Calculate the Total Salary for Company B

The initial salary is $32,000 and there is a $3\%$ annual increase. We can calculate the salary per year as follows:Year 1: $32,000Year 2: $32,000 * 1.03 = \$32,960Year 3: \$32,960 * 1.03 = \$33,949Year 4: \$33,949 * 1.03 = \$34,968Year 5: \$34,968 * 1.03 = \$36,017The total salary for the 5 years is therefore \$32,000 + \$32,960 + \$33,949 + \$34,968 + \$36,017 = \$169,894.
03

Compare the Total Salaries

We can see that the total 5 years salary from Company B is \$169,894, which is higher than the total 5 years salary from Company A which is \$169,112.
04

Find Out the Salary Difference

The difference between the total salaries is \$169,894 - \$169,112 = \$782.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Salary Increase Calculation
Understanding how to calculate salary increases is essential when comparing job offers. With an initial salary and a fixed percentage increase year over year, you can determine future earnings. Suppose you start with a salary of \(30,000 and expect a 6% annual raise. You calculate the next year's salary by multiplying the current salary by 1.06 (since 6% as a decimal is 0.06, and you add this to the original 100%, which is 1.00). This calculation is repeated for each subsequent year, always taking the most recent salary to apply the increase.

For example:
  • Year 1: \)30,000
  • Year 2: \(30,000 脳 1.06 = \)31,800
  • Year 3: \(31,800 脳 1.06 = \)33,708
Repeat this process until you get the salary for each year, and then sum them up to find the total compensation over the period you're considering.
Percent Increase
The concept of percent increase is pivotal in various real-world applications, like salary negotiations. It represents the relative growth compared to the original amount. To express a salary increase as a percent, you calculate the difference between the new and old salary and then divide by the original salary. Multiply by 100 to get the percentage.

For example, if your salary rises from \(30,000 to \)31,800, the increase in dollar terms is \(1,800. The percent increase is calculated as follows:
  • Percent Increase = ((\)31,800 - \(30,000) / \)30,000) 脳 100 = 6%
Accurately calculating this increase helps to compare different job offers where the salary growth rates might differ even if the starting salaries are close.
Total Salary Comparison
When comparing job offers, it's not enough to simply look at the starting salary; the overall earnings over a given period are a more accurate measure. After calculating the yearly salary including increases for each year of the contract, you sum all the annual salaries to get a total. This total salary reflects the actual earnings from each job over the contract period.

In the provided example, the total salary for a five-year period includes each year's compensation:
  • Company A: Year 1 + Year 2 + Year 3 + Year 4 + Year 5
  • Company B: Year 1 + Year 2 + Year 3 + Year 4 + Year 5
In the end, you would compare the totals from both companies to determine which offer is financially better. Remember to consider factors such as benefits, work-life balance, and career growth opportunities in addition to the salary comparison.

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Most popular questions from this chapter

Enough curiosities involving the Fibonacci sequence exist to warrant a flourishing Fibonacci Association, which publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the Internet or the research department of your library, and find one property that interests you. After doing this research, get together with your group to share these intriguing properties.

Use the formula for \(_{n} C_{r}\) to solve An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

Solve by the method of your choice. Using 15 flavors of ice cream, how many cones with three different flavors can you create if it is important to you which flavor goes on the top, middle, and bottom?

Follow the outline below and use mathematical induction to prove the Binomial Theorem: $$\begin{aligned}(a+b)^{n} &-\left(\begin{array}{c}n \\\0\end{array}\right) a^{n}+\left(\begin{array}{c}n \\\1\end{array}\right) a^{n-1} b+\left(\begin{array}{c}n \\\2\end{array}\right) a^{n-2} b^{2} \\\&+\cdots+\left(\begin{array}{c}n \\\n-1\end{array}\right) a b^{n-1}+\left(\begin{array}{c}n \\\n\end{array}\right) b^{n}\end{aligned}$$ a. Verify the formula for \(n-1\) b. Replace \(n\) with \(k\) and write the statement that is assumed true. Replace \(n\) with \(k+1\) and write the statement that must be proved. c. Multiply both sides of the statement assumed to be true by \(a+b .\) Add exponents on the left. On the right, distribute \(a\) and \(b,\) respectively. d. Collect like terms on the right. At this point, you should have $$\begin{array}{l}(a+b)^{k+1}-\left(\begin{array}{c}k \\\0\end{array}\right)a^{k+1}+\left[\left(\begin{array}{c}k \\\0\end{array}\right)+\left(\begin{array}{c}k \\\1\end{array}\right)\right] a^{k} b \\\\+\left[\left(\begin{array}{c}k \\\1\end{array}\right)+\left(\begin{array}{c}k \\\2\end{array}\right)\right] a^{k-1} b^{2}+\left[\left(\begin{array}{c}k \\\2\end{array}\right)+\left(\begin{array}{c}k \\\3\end{array}\right)\right] a^{k-2} b^{3} \\\\+\cdots+\left[\left(\begin{array}{c}k \\\k-1\end{array}\right)+\left(\begin{array}{c}k \\\k\end{array}\right)\right] a b^{k}+\left(\begin{array}{c}k \\\k\end{array}\right) b^{k+1}\end{array}$$ e. Use the result of Exercise 84 to add the binomial sums in brackets. For example, because \(\left(\begin{array}{l}n \\\ r\end{array}\right)+\left(\begin{array}{c}n \\ r+1\end{array}\right)\) $$\begin{aligned}&-\left(\begin{array}{l}n+1 \\\r+1\end{array}\right), \text { then }\left(\begin{array}{l}k \\\0\end{array}\right)+\left(\begin{array}{l}k \\\1\end{array}\right)-\left(\begin{array}{c}k+1 \\\1\end{array}\right) \text { and }\\\&\left(\begin{array}{l}k \\\1\end{array}\right)+\left(\begin{array}{l}k \\\2\end{array}\right)-\left(\begin{array}{c}k+1 \\\2\end{array}\right)\end{aligned}$$ f. Because \(\left(\begin{array}{l}k \\\ 0\end{array}\right)-\left(\begin{array}{c}k+1 \\ 0\end{array}\right)(\text { why? })\) and \(\left(\begin{array}{l}k \\\ k\end{array}\right)-\left(\begin{array}{l}k+1 \\ k+1\end{array}\right)\) (why?), substitute these results and the results from part (e) into the equation in part (d). This should give the statement that we were required to prove in the second step of the mathematical induction process.

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