Two mechanics worked on a car. (rest in description)

Question

Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 15 hours. Together they charged a total of $1400. What was the rate charged per hour by each mechanic if the sum of the two rates was $115 per hour?

First mechanic:

Second mechanic:

Answer 1

Hey Conan, please consider this answer. It is lengthy but I did it to outline every step I would take to solve these problems.. When I was studying math, I would walk myself through each problem the way I outlined here. Only by practicing will you be able to QUICKLY find the answer to these problems. I hope it helps.

To understand this better carefully read through the question. You realize that they are actually giving you two formulas.

The first one has to do with the total amount of money the charged for working on the car, $1400.

Let us say that the hourly rate for the first mechanic is represented by the variable 'x' and the hourly rate for the second mechanic is represented by the variable 'y'.

The first equation then becomes: 10x + 15y = 1400 (Equation 1)

The first term, 10x, represents the total amount charged for the first mechanic while the second term represents the total amount charged for the second mechanic. Together, they charged a total of 1400 dollars.

The second equation is outlined towards the end of the question of the problem (the sum of the two rates was $115 per hour).

Thus, the second equation is: x + y = 115 (Equation 2)

You now have a system of equations which you can solve because there are two unknowns (x and y) and two equations which relate those unknown variables. There are multiple ways to solve for the answers. One way is to solve for one variable and then plug that equation into another OR you can cancel out a term by adding/subtracting the equations so you are left with one variable. I'll go over both.

METHOD 1

Equation 1: 10x + 15y = 1400

Equation 2: x + y = 115

I will choose to solve for y in Equation 2 (you can choose to work with Equation 1 if you want, but Equation 2 looks much simpler)

I get: y = 115 - x

Now, I can substitute this equation in Equation 1.

I get: 10x + 15(115 - x) = 1400

This is the same equation shown in the answer given by Mark M.

Now, working with that equation, simplify it to solve for x.

First I distribute the 15, then combine like terms, and finally solve for x.

10x + 1725 - 15x = 1400

-5x + 1725 = 1400

-5x = -325

x = 65

We have found the rate for the first mechanic, which is $65/hour.

Plug this into x for Equation 1 or Equation 2, and you can solve for y (I chose to use Equation 2 again for its simplicity)

65 + y = 115

y = 50

We have found that the rate for the second mechanic was $50/hour.

METHOD 2

Equation 1: 10x + 15y = 1400

Equation 2: x + y = 115

Again, Equation 2 is much simpler so I will work with this one first. You can choose to work with Equation 1 if you prefer.

I want to get rid of one variable. I can do this by multiplying Equation 2 by 10 (always remember, whatever you do to one side of the equation, you must do to the other)

Then, Equation 2 becomes: 10x + 10y = 1150

Subtracting this new equation from Equation 1:

10x + 15y = 1400

- (10x + 10y = 1150)

The 10x's go away, and you are left with: 5y = 250

Solving for y, you get y = 50.

This means that the rate for the second mechanic was $50/hour.

As in METHOD 1, plug the value of y into Equation 1 or Equation 2.

x + 50 = 115

x = 65

This means that the rate for the first mechanic was $65/hour.

These are two methods for which you can solve this question. I hoped this helped, Conan.

Answer 2

You have a system

10x + 15y = 1400

x + y = 115

x is rate of the first

y is the rate of the second

solve by multiplying the second equation by -10 you then have two equations to add were the x drops out

10x + 15y = 1400

-10x - 10y = -1150. Add them

5y = 250. Divide by 5

y the second rate equals $50 per hour

substitute in for y to find x rate

Answer 3

f represents the hourly charge of the first mechanic

115 - f represents the hourly charge of the second mechanic

10f + 15(115 - f) = 1400

Can you solve for f and answer?