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To determine Ms. Bordeaux's gross earnings for the week, we need to calculate her daily wages based on two possible payment methods:

1. **Piecework Rate**: Ms. Bordeaux is paid $1.70 per problem tested.
2. **Hourly Rate**: Ms. Bordeaux is paid $12.70 per hour for an 8-hour day, which totals $ 12.70 \times 8 = 101.60 $ dollars per day.

For each day, we'll calculate earnings using both methods and select the higher amount.

Let's go through each day:

### Day-by-Day Calculation

1. **Monday**
   - Problems tested: 70
   - Piecework Earnings: $ 70 \times 1.70 = 119.00 $
   - Since $119.00 is greater than $101.60, Monday’s wage = $119.00.

2. **Tuesday**
   - Problems tested: 59
   - Piecework Earnings: $ 59 \times 1.70 = 100.30 $
   - Since $101.60 is greater than $100.30, Tuesday’s wage = $101.60.

3. **Wednesday**
   - Problems tested: 67
   - Piecework Earnings: $ 67 \times 1.70 = 113.90 $
   - Since $113.90 is greater than $101.60, Wednesday’s wage = $113.90.

4. **Thursday**
   - Problems tested: 68
   - Piecework Earnings: $ 68 \times 1.70 = 115.60 $
   - Since $115.60 is greater than $101.60, Thursday’s wage = $115.60.

5. **Friday**
   - Problems tested: 68
   - Piecework Earnings: $ 68 \times 1.70 = 115.60 $
   - Since $115.60 is greater than $101.60, Friday’s wage = $115.60.

### Total Gross Earnings for the Week

Now, we sum up the daily wages:
\[
119.00 + 101.60 + 113.90 + 115.60 + 115.60 = 565.70
\]

### Answer
Ms. Bordeaux's gross earnings for the week are **$565.70**.

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Would you like further details on any part of this calculation, or have additional questions?

### Related Questions
1. What would her earnings be if the hourly rate was $15 per hour instead?
2. How many problems would she need to test daily to always exceed the hourly rate?
3. If the piecework rate increased to $2 per problem, how would her earnings change?
4. How would her total weekly earnings change if she worked 10 hours per day instead of 8?
5. What if she only worked 4 days a week instead of 5? How would this impact her earnings?

#### Tip:
Always compare different pay calculation methods if given a choice in hourly vs. piecework rates to ensure maximum earnings.

Learn how to calculate gross earnings for an employee paid by both piecework and hourly rates. This tutorial covers calculating daily wages based on both methods and choosing the higher amount each day. Perfect for Grade 10-12 students studying wage calculations.

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