Two design consultants, P and Q, started working from 8 AM for a client. The client budgeted a total of USD 3000 for the consultants. P stopped working when the hour hand moved by 210 degrees on the clock. Q stopped working when the hour hand moved by 240 degrees. P took two tea breaks of 15 minutes each during her shift, but took no lunch break. Q took only one lunch break for 20 minutes, but no tea breaks. The market rate for consultants is USD 200 per hour and breaks are not paid. After paying the consultants, the client shall have USD_remaining in the budget.
166.67
Consultant Budget Calculation: Determining Remaining Client Funds
This problem involves calculating the paid work hours for two design consultants, P and Q, considering their specific work duration based on clock hand movement and accounting for their unpaid breaks. Finally, we determine the client's remaining budget after paying the consultants at a given hourly rate.
Understanding Clock Hand Movement and Time
The hour hand of a clock completes a full circle (360 degrees) in 12 hours. Therefore, the speed of the hour hand is:
- Speed of hour hand = $\frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees per hour}$.
This information is crucial for accurately determining the total time each consultant worked.
Consultant P's Work Duration and Payment
Consultant P started working at 8 AM. The problem states P stopped working when the hour hand moved by 210 degrees.
- Total time P worked: To find the total hours P worked from 8 AM, we divide the degrees moved by the hour hand's speed:
- Time worked by P = $\frac{210 \text{ degrees}}{30 \text{ degrees/hour}} = 7 \text{ hours}$.
- So, P worked from 8 AM for 7 hours, meaning P stopped at 3 PM.
- P's breaks: P took two tea breaks of 15 minutes each.
- Total break time for P = $2 \times 15 \text{ minutes} = 30 \text{ minutes}$.
- Converting break time to hours: $30 \text{ minutes} = \frac{30}{60} \text{ hours} = 0.5 \text{ hours}$.
- P's paid working time: Breaks are not paid. So, we subtract the break time from the total time P worked.
- P's paid working time = $7 \text{ hours} - 0.5 \text{ hours} = 6.5 \text{ hours}$.
- P's earnings: The market rate for consultants is USD 200 per hour.
- P's earnings = $6.5 \text{ hours} \times \text{USD } 200/\text{hour} = \text{USD } 1300$.
Consultant Q's Work Duration and Payment
Consultant Q also started working at 8 AM. Q stopped working when the hour hand moved by 240 degrees.
- Total time Q worked: Similar to P, we calculate Q's total work hours:
- Time worked by Q = $\frac{240 \text{ degrees}}{30 \text{ degrees/hour}} = 8 \text{ hours}$.
- So, Q worked from 8 AM for 8 hours, meaning Q stopped at 4 PM.
- Q's breaks: Q took only one lunch break for 20 minutes.
- Total break time for Q = $20 \text{ minutes}$.
- Converting break time to hours: $20 \text{ minutes} = \frac{20}{60} \text{ hours} = \frac{1}{3} \text{ hours}$.
- Q's paid working time: We subtract the break time from Q's total work duration.
- Q's paid working time = $8 \text{ hours} - \frac{1}{3} \text{ hours} = \frac{24}{3} \text{ hours} - \frac{1}{3} \text{ hours} = \frac{23}{3} \text{ hours}$.
- Q's earnings: Using the same market rate of USD 200 per hour.
- Q's earnings = $\frac{23}{3} \text{ hours} \times \text{USD } 200/\text{hour} = \frac{4600}{3} \text{ USD}$.
- Converting to decimal (approximately): $\frac{4600}{3} \approx \text{USD } 1533.33$.
Total Payment to Consultants and Remaining Budget
The client budgeted a total of USD 3000 for the consultants. Now, we calculate the total amount paid to both P and Q.
- Total payment to consultants:
- Total payment = P's earnings + Q's earnings
- Total payment = $\text{USD } 1300 + \text{USD } \frac{4600}{3}$
- To add these, we find a common denominator:
- Total payment = $\frac{1300 \times 3}{3} + \frac{4600}{3} = \frac{3900}{3} + \frac{4600}{3} = \frac{3900 + 4600}{3} = \frac{8500}{3} \text{ USD}$.
- Converting to decimal (approximately): $\frac{8500}{3} \approx \text{USD } 2833.33$.
- Remaining budget: We subtract the total payment from the client's initial budget.
- Remaining budget = Client's total budget - Total payment to consultants
- Remaining budget = $\text{USD } 3000 - \text{USD } \frac{8500}{3}$
- Remaining budget = $\frac{3000 \times 3}{3} - \frac{8500}{3} = \frac{9000}{3} - \frac{8500}{3} = \frac{9000 - 8500}{3} = \frac{500}{3} \text{ USD}$.
- Converting to decimal (rounded to two decimal places): $\frac{500}{3} \approx \text{USD } 166.67$.
Summary of Consultant Payments
| Consultant | Total Work Hours | Break Hours | Paid Work Hours | Earnings (USD) |
|---|---|---|---|---|
| P | 7 | 0.5 | 6.5 | 1300.00 |
| Q | 8 | $\frac{1}{3}$ | $\frac{23}{3}$ | $\frac{4600}{3} \approx 1533.33$ |
| Total Payment | $\frac{8500}{3} \approx 2833.33$ | |||
Final Budget Remaining
The client started with a budget of USD 3000. After paying both consultants, the total expenditure is approximately USD 2833.33.
- Client's Remaining Budget = $\text{USD } 3000 - \text{USD } 2833.33 = \text{USD } 166.67$.
Therefore, after paying the consultants, the client shall have USD 166.67 remaining in the budget.
Important Questions from Numerical Computation
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