Company A pays $\mathrm{\$}43,000$ yearly with a guaranteed raise of $\mathrm{\$}1,000$ per year. Company B pays $\mathrm{\$}44,500$ yearly with a guaranteed raise of $\mathrm{\$}800$ per year. Which company will pay more for the first 10 years of employment, and how much more?

Which company pays more for the first 10 years of employment? Select the correct answer below and if necessary, fill in the answer box to complete your choice.
Company B pays $\mathrm{\$}$ more.
Company A pays $ more.
Both companies pay the same amount, $
There is not enough information.

Step 1 ：We are given that Company A pays $43,000yearlywithaguaranteedraiseof$1,000 per year, and Company B pays $44,500yearlywithaguaranteedraiseof$800 per year. We are asked to find out which company will pay more for the first 10 years of employment, and how much more.

Step 2 ：We can calculate the total salary for each company over 10 years using the formula for the sum of an arithmetic series. The formula is $S=\frac{n}{2}\cdot (a_1+a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.

Step 3 ：For Company A, $n=10$, $a_1=\mathrm{\$}43,000$, and the common difference $d=\mathrm{\$}1,000$. So, the last term $a_n=a_1+(n-1)\cdot d=\mathrm{\$}43,000+9\cdot \mathrm{\$}1,000=\mathrm{\$}52,000$.

Step 4 ：Substituting these values into the formula, we get the total salary for Company A over 10 years: $S_A=\frac{10}{2}\cdot (\mathrm{\$}43,000+\mathrm{\$}52,000)=\mathrm{\$}475,000$.

Step 5 ：For Company B, $n=10$, $a_1=\mathrm{\$}44,500$, and the common difference $d=\mathrm{\$}800$. So, the last term $a_n=a_1+(n-1)\cdot d=\mathrm{\$}44,500+9\cdot \mathrm{\$}800=\mathrm{\$}51,700$.

Step 6 ：Substituting these values into the formula, we get the total salary for Company B over 10 years: $S_B=\frac{10}{2}\cdot (\mathrm{\$}44,500+\mathrm{\$}51,700)=\mathrm{\$}481,000$.

Step 7 ：Comparing the totals, we find that Company B pays more over the first 10 years of employment. The difference between the total amounts paid by the two companies is $481,000 - $475,000 = $6,000.

Step 8 ：Final Answer: Company B pays $\boxed{{\displaystyle \mathrm{\$}6,000}}$ more.