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Sagot :
To calculate Vincent's benefit rate, we need to determine the total contributions from the employer and then express these contributions as a percentage of Vincent's salary. Here’s the detailed, step-by-step solution:
1. Employer Contribution to Health Insurance:
- The total cost of the health insurance is [tex]\( \$10,000 \)[/tex].
- The employer subsidizes [tex]\( 70\% \)[/tex] of this cost.
- So, the employer's contribution is:
[tex]\[ 10,000 \text{ dollars} \times 0.70 = 7,000 \text{ dollars} \][/tex]
2. Employer Contribution to Medicare and Social Security (FICA):
- The FICA rate is [tex]\( 7.65\% \)[/tex] of Vincent’s annual salary.
- Vincent's annual salary is [tex]\( \$65,000 \)[/tex].
- So, the employer's FICA contribution is:
[tex]\[ 65,000 \text{ dollars} \times 0.0765 = 4,972.5 \text{ dollars} \][/tex]
3. Employer Contribution for Paid Vacation (PTO):
- Vincent receives 10 days of paid vacation.
- There are 260 working days in a year.
- The daily salary is:
[tex]\[ \frac{65,000 \text{ dollars}}{260 \text{ days}} = 250 \text{ dollars/day} \][/tex]
- The total value of paid vacation is:
[tex]\[ 250 \text{ dollars/day} \times 10 \text{ days} = 2,500 \text{ dollars} \][/tex]
4. Employer Retirement Matching Contribution:
- The retirement matching rate is [tex]\( 8\% \)[/tex].
- So, the employer's contribution is:
[tex]\[ 65,000 \text{ dollars} \times 0.08 = 5,200 \text{ dollars} \][/tex]
5. Total Employer Contributions:
- Adding all the employer contributions, we get:
[tex]\[ 7,000 \text{ dollars} \text{ (Health Insurance)} + 4,972.5 \text{ dollars} \text{ (FICA)} + 2,500 \text{ dollars} \text{ (PTO)} + 5,200 \text{ dollars} \text{ (Retirement Matching)} = 19,672.5 \text{ dollars} \][/tex]
6. Benefit Rate:
- The benefit rate is calculated as the total employer contributions divided by Vincent's salary, expressed as a percentage:
[tex]\[ \left( \frac{19,672.5 \text{ dollars}}{65,000 \text{ dollars}} \right) \times 100 \approx 30.265\% \][/tex]
7. Rounding:
- Rounding [tex]\( 30.265\% \)[/tex] to the nearest percentage point, we get:
[tex]\[ 30\% \][/tex]
Thus, Vincent's benefit rate is [tex]\( \boxed{30\%} \)[/tex].
1. Employer Contribution to Health Insurance:
- The total cost of the health insurance is [tex]\( \$10,000 \)[/tex].
- The employer subsidizes [tex]\( 70\% \)[/tex] of this cost.
- So, the employer's contribution is:
[tex]\[ 10,000 \text{ dollars} \times 0.70 = 7,000 \text{ dollars} \][/tex]
2. Employer Contribution to Medicare and Social Security (FICA):
- The FICA rate is [tex]\( 7.65\% \)[/tex] of Vincent’s annual salary.
- Vincent's annual salary is [tex]\( \$65,000 \)[/tex].
- So, the employer's FICA contribution is:
[tex]\[ 65,000 \text{ dollars} \times 0.0765 = 4,972.5 \text{ dollars} \][/tex]
3. Employer Contribution for Paid Vacation (PTO):
- Vincent receives 10 days of paid vacation.
- There are 260 working days in a year.
- The daily salary is:
[tex]\[ \frac{65,000 \text{ dollars}}{260 \text{ days}} = 250 \text{ dollars/day} \][/tex]
- The total value of paid vacation is:
[tex]\[ 250 \text{ dollars/day} \times 10 \text{ days} = 2,500 \text{ dollars} \][/tex]
4. Employer Retirement Matching Contribution:
- The retirement matching rate is [tex]\( 8\% \)[/tex].
- So, the employer's contribution is:
[tex]\[ 65,000 \text{ dollars} \times 0.08 = 5,200 \text{ dollars} \][/tex]
5. Total Employer Contributions:
- Adding all the employer contributions, we get:
[tex]\[ 7,000 \text{ dollars} \text{ (Health Insurance)} + 4,972.5 \text{ dollars} \text{ (FICA)} + 2,500 \text{ dollars} \text{ (PTO)} + 5,200 \text{ dollars} \text{ (Retirement Matching)} = 19,672.5 \text{ dollars} \][/tex]
6. Benefit Rate:
- The benefit rate is calculated as the total employer contributions divided by Vincent's salary, expressed as a percentage:
[tex]\[ \left( \frac{19,672.5 \text{ dollars}}{65,000 \text{ dollars}} \right) \times 100 \approx 30.265\% \][/tex]
7. Rounding:
- Rounding [tex]\( 30.265\% \)[/tex] to the nearest percentage point, we get:
[tex]\[ 30\% \][/tex]
Thus, Vincent's benefit rate is [tex]\( \boxed{30\%} \)[/tex].
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