Math  /  Algebra

QuestionAfter graduating from college, Carlos receives two different job offers. Both pay a starting salary of $63000\$ 63000 per year, but one job promises a $3150\$ 3150 raise per year, while the other guarantees a 4%4 \% raise each year.
Complete the tables below to determine what his salary will be after tt years. Round your answers to the nearest dollar. \begin{tabular}{|c|c|c|c|c|c|} \hline tt years & 1 & 5 & 10 & 15 & 20 \\ \hline Salary with \3150 & 66150 v & 78750 > & 94500 V & 110250 V & 126000 \cdot \\ \hline raise per year & 0^{8} & 00^{\circ}$ & 0 & 0 & 0 \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? Carlos has two job offers with the same starting salary, but different raise structures.
We need to figure out how much he'd make at each job over time. Watch out! Don't mix up adding and multiplying!
One job adds a fixed amount, the other multiplies by a percentage.

STEP 2

1. Calculate Salary with Fixed Raise
2. Calculate Salary with Percentage Raise

STEP 3

We already have the completed table for the job with a $3150\$3150 raise each year, so we're good to go here!
It's always good to double-check though.
Let's take year 5 as an example.
Starting at $63000\$63000, after 5 years, Carlos would have received 5$3150=$157505 \cdot \$3150 = \$15750 in raises.
Adding that to his initial salary: $63000+$15750=$78750\$63000 + \$15750 = \$78750.
Perfect, it matches the table!

STEP 4

To calculate the salary after tt years with a 4%4\% annual raise, we can use a handy formula.
Think of it this way: each year, his salary is multiplied by 1+0.04=1.041 + 0.04 = 1.04.
So, after tt years, the salary will be $63000(1.04)t\$63000 \cdot (1.04)^t.
Why 1+0.041 + 0.04?
Well, the *1* keeps the original amount, and the *0.04* adds the 4%4\% increase!

STEP 5

After 1 year (t=1t=1): $63000(1.04)1=$630001.04=$65520\$63000 \cdot (1.04)^1 = \$63000 \cdot 1.04 = \$65520.

STEP 6

After 5 years (t=5t=5): $63000(1.04)5=$630001.2166529024$76643\$63000 \cdot (1.04)^5 = \$63000 \cdot 1.2166529024 \approx \$76643.

STEP 7

After 10 years (t=10t=10): $63000(1.04)10=$630001.4802442849$93256\$63000 \cdot (1.04)^{10} = \$63000 \cdot 1.4802442849 \approx \$93256.

STEP 8

After 15 years (t=15t=15): $63000(1.04)15=$630001.8009435056$113460\$63000 \cdot (1.04)^{15} = \$63000 \cdot 1.8009435056 \approx \$113460.

STEP 9

After 20 years (t=20t=20): $63000(1.04)20=$630002.191123143$137936\$63000 \cdot (1.04)^{20} = \$63000 \cdot 2.191123143 \approx \$137936.

STEP 10

Here's the completed table for the 4%4\% raise job:
tt years | 1 | 5 | 10 | 15 | 20 ------- | -------- | -------- | -------- | -------- | -------- Salary with 4%4\% raise per year | $65520\$65520 | $76643\$76643 | $93256\$93256 | $113460\$113460 | $137936\$137936

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