## find normal and overtime rates of pay for carpenter

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little_dragon
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### find normal and overtime rates of pay for carpenter

A carpenter can make a cabinet in 30 hours. If he works for 21 hours at the normal rate of pay and 9 hours at the overtime rate, he will be paid $69. But if he works for 27 hours at the normal rate and 3 hours at the overtime rate, his earnings will be$63. Find his normal and overtime rates of pay.

If I times the second one by 3, I get 81 normal hours and 9 overtime hours for $189. This is 60 more normal hours and$120 more dollars, for $2 for each normal hour. Working backwards, I get 69 - 21*2 = 27, so each overtime hour is$3.

This answer works, but it seems too easy. Am I doing this wrong? Thanks!

stapel_eliz
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This answer works, but it seems too easy. Am I doing this wrong?
You're only "doing it wrong" if you were told to use some particular other method (such as a system of equations).

What you have done is logical, and it leads to the correct answer. I would think that, as long as you show your work on your hmework (as you did here), you would get full credit, regardless of the particular method you used.

Eliz.

DAiv
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### Re: find normal and overtime rates of pay for carpenter

If I times the second one by 3, I get 81 normal hours and 9 overtime hours for $189. This is 60 more normal hours and$120 more dollars, for $2 for each normal hour. Working backwards, I get 69 - 21*2 = 27, so each overtime hour is$3.

This answer works, but it seems too easy. Am I doing this wrong? Thanks!
You're actually doing quite a bit of calculation there, but the concise nature of your answer may make it appear simpler than it is. I see from a post in another thread that you're already doing algebra, so I would imagine that an algebraic solution may be expected. If so, here's one way of laying things out algebraically. I'll explain each step as I go, so that anyone not familiar with algebra can follow along. I've changed the figures in the question, but the method would be similar.
A carpenter can make a cabinet in 10 hours. If he works for 8 hours at the normal rate of pay and 2 hours at the overtime rate, he will be paid $34. But if he works for 6 hours at the normal rate and 4 hours at the overtime rate, his earnings will be$38. Find his normal and overtime rates of pay.
This example can be solved using 'simultaneous equations'. Basically, we are given enough information to form two separate equations. Neither equation is enough to give us the answer by itself, but together, they can.

To make things simpler to handle, we use letter variables as stand-ins for our unknown quantities. In this case, we don't know the number of normal hours or the number of overtime hours. We can choose any letters we like, but it's best to use something that's memorable. I'll choose $n$ for 'normal hours' and $e$ for 'extra' or 'overtime' hours. (I could have used $o$ for 'overtime hours', but it would likely be confused with a zero at some point, so I decided against it.)

$\begin{eqnarray} \text{Let }n&=&\text{rate in \ per normal hour} \\ \text{Let }e&=&\text{rate in \ per overtime hour} \end{eqnarray}$

$\begin{eqnarray} 8n + 2e &=& 34 & \qquad \small\text{Equation 1}&\hspace{20}\small\text{(8 normal hours + 2 overtime hours = \34)}\normalsize\\ 6n + 4e &=& 38 & \qquad \small\text{Equation 2}&\hspace{20}\small\text{(6 normal hours + 4 overtime hours = \38)\\ \end{eqnarray}$

I've numbered the equations, so that we can refer to them later. As things stand, we can't solve either equation, since each has two variables - so, we need to eliminate one of the variables. Looking at the equations, I see that the simplest way would be to eliminate $e$.

Now, $2e$ could become $4e$ if we double it, or alternatively, $4e$ could become $2e$ if we halve it. In algebra, we can modify an equation in numerous ways and not change its equality, just so long as whatever we do to it, we do it equally to everything on both sides of the equals sign. The $6$ and the $38$ of equation 2 are both even, so they'll halve nicely, so we'll choose to halve all of equation 2. (We chose to halve rather than double, because it's usually easier to work with smaller rather than larger numbers.)

$\begin{eqnarray} \frac{6n + 4e}2 &=& \frac{38}2 \\ 3n + 2e &=& 19 & \qquad \small\text{Equation 3} \end{eqnarray}$

So, to eliminate the $e$, we substract one equation from the other, so the $e$ term disappears. It doesn't matter which way round we do the subtraction (equation 2 - equation 3, or equation 3 - equation 2), but it's usually easier if we choose the one that results in the number attached to the remaining variable ($n$) being positive.

$\begin{eqnarray} &&&&8n &+& 2e &=& 34 & \qquad \small\text{Equation 1}\normalsize\\ -&&&&3n &+& 2e &=& 19 & \qquad \small\text{Equation 3}\normalsize\\ \hline &&&&5n && &=& 15& \qquad \small\text{Equation 4}\normalsize\\ \end{eqnarray}$

Rearranging equation 4 so that $n$ is on its own, we get:

$\begin{eqnarray} 5n &=& 15& \qquad \small\text{Equation 4}\normalsize\\ \frac{5n}5 &=& \frac{15}5\\ n &=& 3 \end{eqnarray}$

Great! We've got the value of one of the variables, $n$. Now, all we have to do is plug $n=3$ back into either of our original equations that contain both $n$ and $e$, and solve for $e$. We'll choose equation 2.

$\text Substituting n=3 into equation 2 gives:$

$\begin{eqnarray} 6n &+& 4e &=& 38\\ 6(3) &+& 4e &=& 38\\ 18 &+& 4e &=& 38\\ 18+4e &-& 18 &=& 38 &-& 18\\ && 4e &=& 20\\ && \frac{4e}4 &=& \frac{20}4\\ &&e &=& 5 \end{eqnarray}$

So, now we know $e=5$.

We should check that we have the correct values, so we'll plug $n=3$ and $e=5$ into the other original equation (equation 1).

$\text{Check: Substituting }n=3\text{ and }e=5 \text{ into equation 1 gives:$

$\begin{eqnarray} 8n &+& 2e &=& 34\\ 8(3) &+& 2(5) &?=?& 34\\ 24 &+& 10 &?=?& 34\\ && 34 &=& 34 &\smal\text{... Yes! It checks out.}\\ \end{eqnarray}$

Now, all that remains to be done is to re-read the original question, check we've calculated what was asked of us, and write up (and underline) our final answer:

$\underline{\text{The carpenter's normal rate of pay is \3 per hour,}}$
$\underline{\text{and his overtime rate of pay is \5 per hour.}}$

Your teacher / tutor / book may lay things out a little differently, but it'll probably be something along these lines.

As you can see, to write all that out is quite a bit more work than writing your explanation, but the actual method used in each case is actually quite similar. Here, I've written every little step to explain what I'm doing, but in practice, a few minor steps can be missed out if we are comfortable with manipulating equations.

For instance,

$\begin{eqnarray} 18 &+& 4e &=& 38\\ 18+4e &-& 18 &=& 38 &-& 18\\ && 4e &=& 20\\ && \frac{4e}4 &=& \frac{20}4\\ &&e &=& 5 \end{eqnarray}$

... could become...

$\begin{eqnarray} 18 &+& 4e &=& 38\\ && 4e &=& 20\\ &&e &=& 5 \end{eqnarray}$

But, as Eliz has said, as long as you show a logical progression from start to finish (and end up with the correct answer ), you should get full marks.

DAiv

little_dragon
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### Re: find normal and overtime rates of pay for carpenter

i've been doing this stuff for a few grades now, and sometimes its easier to do it the way i learned back before i started doing variables. Thank you for showing how to do it the algebra way!