SOLUTION: 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 wage

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Question 1209934: 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 5 solutions by CPhill, angelatoyogon, ikleyn, josgarithmetic, 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 angelatoyogon(1)   (Show Source): You can put this solution on YOUR website!
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
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 the daily wages for a technician and an apprentice
~~~~~~~~~~~~~~~~~~~~~~

Dear Angela,

as I see, you are first time at this forum.

You are welcome, we are glad to every newcomer !

You came with the problem, which was posted to the forum earlier (two or three months ago),
and for which the solution was just developed and placed at this forum by the other tutor
exactly at the same spot, from which you submitted your post.

So, I will assume that you just have read that solution. It is correct, so I do not see
any need to re-tell or to re-write it again.

At these circumstances, it is not precisely clear to me the meaning of your request.
For us, it would be much easier to communicate, if you formulate the goal of your visit
with the problem, which was just solved before.

I may to suspect that the problem's formulation is not fully clear to you.
To be honest, as written in the post, it is not clear to me, too.

In my view, the problem formulation in the post is not as clear as it should be in Math problems,
and requires to be explained.

Had I formulate this problem, I would make it differently. I would write 

    A firm hires 7 technicians and 3 apprentices for total daily wages of 820. 
    Later, one of the 3 apprentices was appointed to a position of technician, 
    and, therefore, the total daily wages became 880. 
    What are the daily wages for a technician and an apprentice ?

Now everything is clear, everything is transparent, the meaning is clear and additional clarifications are not needed.

It is how Math problem should be formulated, when it goes outside.

I grew and got my mathematical education in other country, where a clear formulation
of school Math problems was mandatory, and an unclear formulation had no chance to see a world.

Here in US it is not so. At every corner, you can find formulations of Math problems,
that do not stand up to any criticism.

Also, in the Internet era, thousands of people that have no adequate Math education,
think that they can create and disseminate their own math problems in the Internet.

But it is the reality in which we live.

In any case, I am glad you visited this website/forum, and hope that you will find here
many useful conversations/communications/information/solutions/ideas/advises for your Math education.



Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
Apprentice day pay rate, x
Technician day pay rate, y
Initially 3 apprentices and 7 technicians

                   PAY-RATE           TIME(day count)      WAGE 

APPRENTICES          x                     1                  3x

TECHNICIANS          y                     1                  7y

Total                                                       820


one less apprentice and one more technicin

                   PAY-RATE           TIME(day count)      WAGE (total)

APPRENTICES          x                    1                  2x  

TECHNICIANS          y                    1                   8y

Total                                                       880

------Solve this system.
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Here is a quick and easy informal mental solution to the problem.

The combined wages of 7 technicians and 3 apprentices is 820.
When one apprentice becomes a technician, the combined wages of 8 technicians and 2 apprentices is 880.
So when an apprentice becomes a technician, the combined wages go up by 60.

Have that happen twice more, ending up with 10 technicians and 0 apprentices. The combined wages is now 880 + 60 + 60 = 1000.

So the wage for each technician is 1000/10 = 100.

When there were 7 technicians and 3 apprentices, the total wages was 820. The wages for the 7 technicians was 7*100 = 700, so the wages for the 3 apprentices was 820-700 = 120; so the wage for each apprentice is 120/3 = 40.

Alternatively, once we find that the wages for each technician is 100, we can use the fact that the total wages increase by 60 each time an apprentice becomes a technician to determine that the wage for each apprentice is 100-60 = 40.
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