It is important for Bob to have a gross salary in his fifth year of at least $100,000. He is promised salary increases of 3 percent per year in the second and third years and 5 percent thereafter. What must be his starting salary be to achieve his goal? My original answer was $87,000, but I think $86,000 might work. All of the percents are running together in my head. I think I will go get a Coke. Thanks in advance for your help.

Question

It is important for Bob to have a gross salary in his fifth year of at least $100,000. He is promised salary increases of 3 percent per year in the second and third years and 5 percent thereafter. What must be his starting salary be to achieve his goal? My original answer was $87,000, but I think $86,000 might work. All of the percents are running together in my head. I think I will go get a Coke. Thanks in advance for your help.

Answer 1

salary in 1st year --- x

salary in 3rd year = x(1.03)^2
salary in 4th year = x(1.03)^2 (1.05)
salary in 5th year = x(1.03)^2 (1.05)^2

x(1.03)^2 (1.05)^2 = 100000
x = 100000/((1.03)^2 (1.05)^2)
= $87,180.84

Answer 2

WOW! Thanks, I had trouble setting up the equation to solve. But it makes sense, like when solving an interest problem.

Answer 3

To find Bob's starting salary, we can work backwards using the given information and set up an equation.

Let's break down the problem step by step:

1. In the second and third years, Bob receives a 3% salary increase annually.
- Starting with the initial salary, after the first year, Bob's salary will be increased by 3%.
- After the second year, Bob's salary will be increased by another 3% on the new salary.

2. In the fourth and subsequent years, Bob receives a 5% salary increase annually.
- After the third year, Bob's salary will be increased by 5% each year.
- This will continue for the remaining years.

Based on these conditions:

- After the first year: Bob's salary remains the same.
- After the second year: Bob's salary is increased by 3%.
- After the third year: Bob's salary is increased by another 3% on the new salary.
- After the fourth year and beyond: Bob's salary is increased by 5% each year.

To find the starting salary that would achieve a gross salary of at least $100,000 in Bob's fifth year, let's work backwards:

Let's assume Bob's starting salary is "S."
- After the first year, Bob's salary remains the same: S.
- After the second year, his salary increases by 3%: S + 0.03S = S(1 + 0.03) = 1.03S.
- After the third year, his salary increases by another 3% on the new salary (1.03S): 1.03S + 0.03(1.03S) = 1.03S(1 + 0.03) = 1.03S(1.03).
- After the fourth year, his salary increases by 5% each year: (1.03S)(1.05).
- After the fifth year: (1.03S)(1.05)^2 (since the increase only applies for the fourth year and beyond).

So, we can set up an equation to find the starting salary:

S * (1.03) * (1.03) * (1.05)^2 >= $100,000.

Now we can solve this equation to find the minimum starting salary that would achieve a gross salary of at least $100,000 in Bob's fifth year:

S * 1.09215 >= $100,000.

S >= $100,000 / 1.09215.

S >= $91,494.68.

Therefore, Bob's starting salary must be at least $91,494.68 to achieve his goal of a gross salary in his fifth year of at least $100,000.

Based on this calculation, your original answer of $87,000 is not sufficient. However, $86,000 is even lower, so it would not meet the requirement either.

Feel free to double-check the calculations or ask any further questions if needed. Enjoy your Coke!