$3$ . Salary Increases A man gets a job with a salary of $\$ 30,000$ a year. He is promised a $\$ 2300$ raise each subsequent year. Find his total earnings for a $10$ -year period.

View step-by-step help

Home

Math Problems

Grade 6

Multiply and divide decimals: word problems

Full solution

3

. Salary Increases A man gets a job with a salary of

\$ 30,000

a year. He is promised a

\$ 2300

raise each subsequent year. Find his total earnings for a

10

-year period.

Calculate Yearly Salary: First, let's calculate the salary for each year without adding them up yet. The first year, he earns $\$30,000$ .
Yearly Salary Calculation: In the second year, he gets a $\$$ $2300$ raise, so he will earn $\$30,000 + \$2300 = \$32,300$ .
Arithmetic Series Formula: For the third year, another $\$2300$ raise, so that's $\$32,300 + \$2300 = \$34,600$ .
Calculate Last Term: We can see a pattern here; each year, the salary increases by \$$2300$. So instead of calculating each year separately, we can use the formula for the sum of an arithmetic series.
Calculate Sum of Series: The sum $S$ of the first $n$ terms of an arithmetic series is given by $S = \frac{n}{2} \times (\text{first term} + \text{last term})$. Here, $n=10$, the first term is $\$30,000$, and the last term will be the first term plus $9$ times the raise ($\$2,300$).
Calculate Sum of Series: The sum $S$ of the first $n$ terms of an arithmetic series is given by $S = \frac{n}{2} \times (\text{first term} + \text{last term})$. Here, $n=10$, the first term is $\$30,000$, and the last term will be the first term plus $9$ times the raise ($\$2,300$).So the last term is $\$30,000 + 9 \times \$2,300 = \$30,000 + \$20,700 = \$50,700$.
Calculate Sum of Series: The sum $S$ of the first $n$ terms of an arithmetic series is given by $S = \frac{n}{2} \times (\text{first term} + \text{last term})$. Here, $n=10$, the first term is $\$30,000$, and the last term will be the first term plus $9$ times the raise ($\$2,300$).So the last term is $\$30,000 + 9 \times \$2,300 = \$30,000 + \$20,700 = \$50,700$.Now we plug these values into the sum formula: $S = \frac{10}{2} \times (\$30,000 + \$50,700) = 5 \times \$80,700 = \$403,500$.