SOLUTION: Two mechanics worked on a car. The first mechanic charged 105 per hour, and the second mechanic charged 85 per hour. The mechanics worked for a combined total of 20 hours, and toge

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Two mechanics worked on a car. The first mechanic charged 105 per hour, and the second mechanic charged 85 per hour. The mechanics worked for a combined total of 20 hours, and toge      Log On


   

Question 1204237: Two mechanics worked on a car. The first mechanic charged 105 per hour, and the second mechanic charged 85 per hour. The mechanics worked for a combined total of 20 hours, and together they charged a total of 1800. How long did each mechanic work?

Found 3 solutions by ikleyn, MathLover1, greenestamps:
Answer by ikleyn(52886) About Me  (Show Source):
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.
Two mechanics worked on a car. The first mechanic charged 105 per hour,
and the second mechanic charged 85 per hour. The mechanics worked for a combined total of 20 hours,
and together they charged a total of 1800. How long did each mechanic work?
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Let x = hours of the first mechanics; y = hours of the second mechanics.


Equations

       x +   y =   20      hours     (1)    combined time

    105x + 85y = 1800    dollars     (2)    combined dollars


To solve, express y = 20-x  from equation (1) and substitute it into equation (2).
You will get then


    105x + 85(20-x) = 1800

    105x + 1700 - 85y = 1800

    105x - 85y = 1800 - 1700

        20x    =     100

          x    =     100/20 = 5.


ANSWER.  First mechanics worked 5 hours; second mechanics worked 20-5 = 15 hours.


CHECK.   105*5 + 85*15 = 1800  dollars total combined.   ! correct !

Solved.



Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

the first mechanic charged 105 per hour, cost of his work is 105x where x+is number of ours
and the second mechanic charged+85 per hour, cost of his work is 85y where y is number of ours
if the mechanics worked for a combined total of+20 hours, we have
+x%2By=20....solve for x
x=20-y.....eq.1

and, if together they charged a total of 1800, we have
105x%2B85y=1800......eq.2, substitute+x from eq.1
105%2820-y%29%2B85y=1800....solve for+y
2100+-+105+y%2B85y=1800
2100-20y=1800
2100-1800=20y
300=20y
y=300%2F20
y=15
go to eq.1
x=20-y
x=20-15
x=5
the mechanic who charged 105 per hour, worked 5 hours
the mechanic who charged 85 per hour, worked 15 hours

Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


The two responses you have received show basically the same solution, using two equations with two unknowns solved using substitution.

The other standard algebraic way of solving a system of two equations in two unknowns is elimination.

Let x be the number of hours worked at $85 per hour and y be the number of hours worked at $105 per hour. Then

the total time was 20 hours: x%2By=20
the total charge was $1800: 85x%2B105y=1800

Simplify the second equation by dividing by the greatest common factor, 5.

17x%2B21y=360

Multiply the first equation by 17:

17x%2B17y=340

Eliminate x by subtracting one equation from the other.

4y=20
y=5

Use y=5 in the first equation x%2By=20 to find x=15.

ANSWER: The mechanic who charges $85 per hour worked 15 hours on the car; the mechanic who charged $105 per hour worked 5 hours.

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Here is a quick and easy informal solution that uses exactly the same calculations as the formal elimination method above.

If all 20 hours were by the mechanic who charged $85 per hour, the total charge would be $1700. The actual charge was $100 more than that. Since the second mechanic charged $20 more per hour than the first, the number of hours he worked was $100/$20 = 5; so the first mechanic worked 15 hours.

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And here is another, very different, informal way of solving any 2-part "mixture" problem like this.

The average charge per hour was $1800/20 = $90. Use a number line if it helps to observe/calculate that $90 is one-fourth of the way from $85 to $105. That means the mechanic who charged $105 per hour worked one-fourth of the total of 20 hours -- i.e., 5 hours; meaning the other mechanic worked 15 hours.