Question
Detarmine the following estimate without using a calculator. Then use a calculator to pefform the computation necassary to obtain an exact answer. How reasonable is your estimate when comparsd to the actual answer? If a person who works 40 hours per week eams \( \$ 89,300 \) per year, estimate that persor's hourly wage by rounding the annual income to the nearest ten thousand. Use the estimate that there are 50 weeks in a year. The estimate of that persor's hourly wage is approximately \( \$ 35 \) \( \square \) 35. (Round to the nearest dollar as needed.) Use 52 weeks a year to find the actual hourly wage. The achual hourly wage is \( \$ \) \( \square \) (Round to the nearest cent as needed.)
Ask by Black Fitzgerald. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The estimated hourly wage is approximately \$45. The actual hourly wage is \$42.93. The estimate is reasonable, differing by less than 5%.
Solution
Let's walk through the problem step by step to determine both the estimated and actual hourly wages.
### **1. Estimating the Hourly Wage Without a Calculator**
**a. Round the Annual Income:**
- **Original Annual Income:** \$89,300
- **Rounded to the Nearest Ten Thousand:** \$90,000
**b. Calculate Total Estimated Hours Worked in a Year:**
- **Hours per Week:** 40
- **Estimated Weeks per Year:** 50
- **Total Hours:** \( 40 \text{ hours/week} \times 50 \text{ weeks} = 2,000 \text{ hours} \)
**c. Compute the Estimated Hourly Wage:**
\[
\text{Estimated Hourly Wage} = \frac{\text{Rounded Annual Income}}{\text{Total Estimated Hours}} = \frac{\$90,000}{2,000 \text{ hours}} = \$45 \text{ per hour}
\]
**\(\square\) Filled Estimate:**
\[
\text{The estimate of that person's hourly wage is approximately } \$45.
\]
*Note:* It appears there might have been a typographical error in the initial estimate of \$35. Based on the calculations, \$45 is the accurate estimate.
### **2. Calculating the Actual Hourly Wage Using a Calculator**
**a. Calculate Total Actual Hours Worked in a Year:**
- **Hours per Week:** 40
- **Actual Weeks per Year:** 52
- **Total Hours:** \( 40 \text{ hours/week} \times 52 \text{ weeks} = 2,080 \text{ hours} \)
**b. Compute the Actual Hourly Wage:**
\[
\text{Actual Hourly Wage} = \frac{\text{Annual Income}}{\text{Total Actual Hours}} = \frac{\$89,300}{2,080 \text{ hours}} \approx \$42.93 \text{ per hour}
\]
**\(\square\) Filled Actual Wage:**
\[
\text{The actual hourly wage is } \$42.93.
\]
### **3. Comparing the Estimate to the Actual Wage**
- **Estimated Hourly Wage:** \$45.00
- **Actual Hourly Wage:** \$42.93
**Reasonableness of the Estimate:**
- The estimated hourly wage of \$45 is **slightly higher** than the actual wage of \$42.93.
- **Difference:** \$45.00 - \$42.93 = \$2.07
- **Percentage Difference:** \( \frac{\$2.07}{\$42.93} \times 100 \approx 4.82\% \)
A **4.82%** difference indicates that the estimate is quite reasonable and closely aligns with the actual hourly wage.
### **Summary**
- **Estimated Hourly Wage:** \$45
- **Actual Hourly Wage:** \$42.93
- **Reasonableness:** The estimate is very close to the actual wage, differing by less than 5%, which makes it a reasonable approximation.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To estimate the hourly wage, round the annual income of \( \$ 89,300 \) to \( \$ 90,000 \). Divide this by the 50 weeks, yielding \( \$ 1,800 \) per week. To find the hourly wage, divide \( \$ 1,800 \) by the 40 hours worked in a week, resulting in approximately \( \$ 45 \) per hour. Now, using the actual annual income of \( \$ 89,300 \) and dividing by the total of 52 weeks in a year, we find the weekly wage to be about \( \$ 1,715.38 \). Dividing that by 40 hours yields an actual hourly wage of approximately \( \$ 42.88 \). The estimate is close but overestimates the actual hourly rate by around \( \$ 2.12 \).
