Question
Amy and Ben each get paid an hourly wage of $12 per hour. This week, Ben will make an additional $46 in overtime pay. They want to calculate the total amount of money they will make this week if Amy works
hours and Ben works
hours.
Which of the following statements are true? Select two that apply.
A.
The expression 12 (a+b)+46
shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
B.
The expression 12a +58b
shows the sum of the amount of money Amy made and the amount of money Ben made.
C.
The expression 46(a+b)+12
shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
D.
The expression (a+12)+(b+58)
shows the sum of the amount of money Amy made and the amount of money Ben made.
E
.
The expression 12a+(b+58)
shows the sum of the amount of money Amy made and the amount of money Ben made.
hours and Ben works
hours.
Which of the following statements are true? Select two that apply.
A.
The expression 12 (a+b)+46
shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
B.
The expression 12a +58b
shows the sum of the amount of money Amy made and the amount of money Ben made.
C.
The expression 46(a+b)+12
shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
D.
The expression (a+12)+(b+58)
shows the sum of the amount of money Amy made and the amount of money Ben made.
E
.
The expression 12a+(b+58)
shows the sum of the amount of money Amy made and the amount of money Ben made.
Answers
GPT-4o mini
To determine which statements are true, let's break down the situation:
- Amy earns $12 per hour and works \( a \) hours, so her earnings are \( 12a \).
- Ben also earns $12 per hour and works \( b \) hours, so his earnings from regular hours are \( 12b \).
- Additionally, Ben makes an extra $46 in overtime pay.
Now, we can express their total earnings this week as:
\[
\text{Total Earnings} = \text{Amy's Earnings} + \text{Ben's Earnings} + \text{Ben's Overtime}
\]
\[
= 12a + (12b + 46)
\]
\[
= 12a + 12b + 46
\]
Now let's examine each statement:
A. The expression \( 12(a+b) + 46 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
- This is true. \( 12(a+b) + 46 \) correctly represents their combined pay including overtime.
B. The expression \( 12a + 58b \) shows the sum of the amount of money Amy made and the amount of money Ben made.
- This is false. The expression should include Ben's regular earnings as \( 12b \) (not \( 58b \) which is incorrect).
C. The expression \( 46(a+b) + 12 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
- This is false. The expression incorrectly multiplies the overtime amount by \( a+b \), and it also adds 12 (which doesn't fit).
D. The expression \( (a+12) + (b+58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.
- This is false. The expression incorrectly adds 12 and 58 instead of the correct earnings.
E. The expression \( 12a + (b + 58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.
- This is false. The expression incorrectly uses \( (b + 58) \) instead of \( (12b + 46) \).
The true statements are:
**A**: The expression \( 12(a+b) + 46 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
(Note: There is only one true statement among the given options.)
- Amy earns $12 per hour and works \( a \) hours, so her earnings are \( 12a \).
- Ben also earns $12 per hour and works \( b \) hours, so his earnings from regular hours are \( 12b \).
- Additionally, Ben makes an extra $46 in overtime pay.
Now, we can express their total earnings this week as:
\[
\text{Total Earnings} = \text{Amy's Earnings} + \text{Ben's Earnings} + \text{Ben's Overtime}
\]
\[
= 12a + (12b + 46)
\]
\[
= 12a + 12b + 46
\]
Now let's examine each statement:
A. The expression \( 12(a+b) + 46 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
- This is true. \( 12(a+b) + 46 \) correctly represents their combined pay including overtime.
B. The expression \( 12a + 58b \) shows the sum of the amount of money Amy made and the amount of money Ben made.
- This is false. The expression should include Ben's regular earnings as \( 12b \) (not \( 58b \) which is incorrect).
C. The expression \( 46(a+b) + 12 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
- This is false. The expression incorrectly multiplies the overtime amount by \( a+b \), and it also adds 12 (which doesn't fit).
D. The expression \( (a+12) + (b+58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.
- This is false. The expression incorrectly adds 12 and 58 instead of the correct earnings.
E. The expression \( 12a + (b + 58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.
- This is false. The expression incorrectly uses \( (b + 58) \) instead of \( (12b + 46) \).
The true statements are:
**A**: The expression \( 12(a+b) + 46 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.
(Note: There is only one true statement among the given options.)
There needs to be two.
GPT-4o mini
I apologize for the oversight. Let's carefully reassess the options, keeping in mind that we need to find two true statements.
Given:
- Amy earns \( 12a \) for \( a \) hours worked.
- Ben earns \( 12b \) for \( b \) hours worked and receives an additional $46 in overtime.
The total earnings can be represented as:
\[
\text{Total Earnings} = 12a + 12b + 46
\]
Now, let's evaluate each statement again:
A. **The expression \( 12(a+b) + 46 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.**
- **True**: This expression expands to \( 12a + 12b + 46 \), which correctly represents the total earnings including overtime.
B. **The expression \( 12a + 58b \) shows the sum of the amount of money Amy made and the amount of money Ben made.**
- **False**: This expression incorrectly states Ben's earnings; it should be \( 12b + 46 \) instead of \( 58b \).
C. **The expression \( 46(a+b) + 12 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.**
- **False**: This expression incorrectly multiplies the overtime amount and does not correctly represent their earnings.
D. **The expression \( (a + 12) + (b + 58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.**
- **False**: The earnings are incorrect and do not represent their hourly wages.
E. **The expression \( 12a + (b + 58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.**
- **False**: This expression wrongly adjusts Ben's pay. It should be \( 12b + 46 \).
After re-evaluating, it appears I misunderstood the requirement for the correct identification. The only true statement found is:
1. **A**: The expression \( 12(a+b) + 46 \).
Upon reflection, it seems I miscalculated the second true statement being sought. If you clarify if there might be slight variations or if additional parameters could be included, I could assist in deeper detail. Otherwise, based on the current analysis, only option A holds true in this assessment.
If two true options must absolutely be selected despite the analysis indicating otherwise, I can only restate option A while noting the discrepancies in all presented alternatives.
Given:
- Amy earns \( 12a \) for \( a \) hours worked.
- Ben earns \( 12b \) for \( b \) hours worked and receives an additional $46 in overtime.
The total earnings can be represented as:
\[
\text{Total Earnings} = 12a + 12b + 46
\]
Now, let's evaluate each statement again:
A. **The expression \( 12(a+b) + 46 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.**
- **True**: This expression expands to \( 12a + 12b + 46 \), which correctly represents the total earnings including overtime.
B. **The expression \( 12a + 58b \) shows the sum of the amount of money Amy made and the amount of money Ben made.**
- **False**: This expression incorrectly states Ben's earnings; it should be \( 12b + 46 \) instead of \( 58b \).
C. **The expression \( 46(a+b) + 12 \) shows the combined amount of money Amy and Ben made for their hourly work plus the overtime amount.**
- **False**: This expression incorrectly multiplies the overtime amount and does not correctly represent their earnings.
D. **The expression \( (a + 12) + (b + 58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.**
- **False**: The earnings are incorrect and do not represent their hourly wages.
E. **The expression \( 12a + (b + 58) \) shows the sum of the amount of money Amy made and the amount of money Ben made.**
- **False**: This expression wrongly adjusts Ben's pay. It should be \( 12b + 46 \).
After re-evaluating, it appears I misunderstood the requirement for the correct identification. The only true statement found is:
1. **A**: The expression \( 12(a+b) + 46 \).
Upon reflection, it seems I miscalculated the second true statement being sought. If you clarify if there might be slight variations or if additional parameters could be included, I could assist in deeper detail. Otherwise, based on the current analysis, only option A holds true in this assessment.
If two true options must absolutely be selected despite the analysis indicating otherwise, I can only restate option A while noting the discrepancies in all presented alternatives.