Question 1209933: a firm hires 7 technicians and 3 apprentices for total daily wages of 820. if one apprentice is appointed to a technician, the total daily wages become 880. What are thr daily wages for a technician and an apprentice
Found 2 solutions by CPhill, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's use variables to represent the daily wages:
* Let 't' represent the daily wage of a technician.
* Let 'a' represent the daily wage of an apprentice.
We can set up two equations based on the given information:
**Equation 1:** 7 technicians and 3 apprentices earn 820.
* 7t + 3a = 820
**Equation 2:** If one apprentice becomes a technician, there are 8 technicians and 2 apprentices, and they earn 880.
* 8t + 2a = 880
Now we have a system of two linear equations with two variables. We can solve for 't' and 'a'.
**Solve the System of Equations:**
We can use the substitution or elimination method. Let's use the elimination method.
1. Multiply Equation 1 by 2 and Equation 2 by 3 to make the 'a' coefficients match:
* (7t + 3a = 820) * 2 => 14t + 6a = 1640
* (8t + 2a = 880) * 3 => 24t + 6a = 2640
2. Subtract the first modified equation from the second modified equation to eliminate 'a':
* (24t + 6a) - (14t + 6a) = 2640 - 1640
* 10t = 1000
3. Solve for 't':
* t = 1000 / 10
* t = 100
4. Substitute the value of 't' back into either Equation 1 or Equation 2 to solve for 'a'. Let's use Equation 1:
* 7(100) + 3a = 820
* 700 + 3a = 820
* 3a = 820 - 700
* 3a = 120
5. Solve for 'a':
* a = 120 / 3
* a = 40
**Answers:**
* The daily wage of a technician (t) is $100.
* The daily wage of an apprentice (a) is $40.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The AI solution from the other "tutor" solves the problem using a standard formal algebraic process, which is of course fine.
This is an example of a problem that can be solved by logical analysis that corresponds to a non-standard algebraic process.
When there were 7 technicians and 3 apprentices, the total wages was 820; when one of the apprentices became a technician, the total wages was 880.
So when one apprentice becomes a technician, the total wages increases by 60.
With that start, suppose now the other two apprentices became technicians. When there are 9 technicians and 1 apprentice the total wages will be 940, and when there are 10 technicians and no apprentices, the total wages will be 1000.
That of course means the wage of each technician is 1000/10 = 100.
So when the wages for 7 technicians and 3 apprentices was 820, the wages for the 7 technicians was 700, which means the wages for the 3 apprentices was 820-700 = 120; so the wages for each apprentice is 120/3 = 40.
ANSWER: 100 for each technician; 40 for each apprentice.
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