Units Expressed Correctly

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Explain this!

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If 4 workers are paid $1.50 per hour, and they work work 3 1/2 hours shoveling 10 1/2 tons of gravel, how much is that per ton?

Here is my solution: 3 1/2 hours * 4 workers * [($1.50/hour)/worker]/10 1/2 tons

This will equal $2.00/ton or $2.00 per ton.

I'm not sure how to express $1.50 per hour per worker. Is it as I have it in my solution or is it $1.50/(hour/worker) or something else?
 
What you wrote, ($1.50/hour)/worker, is appropriate; but $1.50/(hour/worker) is not.

An alternative way to say it is $1.50/(worker*hour); this used to be called "per man-hour".

This is because [MATH]\frac{\frac{$1.50}{\text{hour}}}{\text{worker}} = \frac{$1.50}{\text{hour}}\times\frac{1}{\text{worker}} =\frac{$1.50}{\text{hour}\times\text{worker}}[/MATH], but [MATH]\frac{$1.50}{\frac{\text{hour}}{\text{worker}}} = $1.50\times\frac{\text{worker}}{\text{hour}} =\frac{$1.50\times\text{worker}}{\text{hour}}[/MATH]
 
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Explain this! said:
If 4 workers are paid $1.50 per hour, and they work work 3 1/2 hours shoveling 10 1/2 tons of gravel, how much is that per ton?

Here is my solution: 3 1/2 hours * 4 workers * [($1.50/hour)/worker]/10 1/2 tons

This will equal $2.00/ton or $2.00 per ton.

I'm not sure how to express $1.50 per hour per worker. Is it as I have it in my solution or is it $1.50/(hour/worker) or something else?
Click to expand...
Are the 4 workers each paid $1.50/hr or do the 4 workers receive a total of $1.50/hr?
 
Dr.Peterson said:
What you wrote, ($1.50/hour)/worker, is appropriate; but $1.50/(hour/worker) is not.

An alternative way to say it is $1.50/(worker*hour); this used to be called "per man-hour".

This is because [MATH]\frac{\frac{$1.50}{\text{hour}}}{\text{worker}} = \frac{$1.50}{\text{hour}}\times\frac{1}{\text{worker}} =\frac{$1.50}{\text{hour}\times\text{worker}}[/MATH], but [MATH]\frac{$1.50}{\frac{\text{hour}}{\text{worker}}} = $1.50\times\frac{\text{worker}}{\text{hour}} =\frac{$1.50\times\text{worker}}{\text{hour}}[/MATH]
Click to expand...

Thank you for expressing it this way. It is easier to see how the units cancel.
 
Explain this! said:
If 4 workers are paid $1.50 per hour, and they work work 3 1/2 hours shoveling 10 1/2 tons of gravel, how much is that per ton?

Here is my solution: 3 1/2 hours * 4 workers * [($1.50/hour)/worker]/10 1/2 tons

This will equal $2.00/ton or $2.00 per ton.

I'm not sure how to express $1.50 per hour per worker. Is it as I have it in my solution or is it $1.50/(hour/worker) or something else?
Click to expand...
I do not exactly agree with you answer of $2/ton.
If the 4 workers work for 3 1/2 hrs and shovel 10 1/2 tons then at that rate in 1 hour they shovel 3 tons. Now your answer of $2 is the total that all 4 workers get. Each individual worker gets $0.50/ton. Did you realize that?
 
Explain this! said:
If 4 workers are paid $1.50 per hour, and they work work 3 1/2 hours shoveling 10 1/2 tons of gravel, how much is that per ton?
Click to expand...
As Jomo indicated, the question could be worded better. I didn't comment on that before, but better wording (for what you are clearly taking it to mean) would be

If 4 workers are each paid $1.50 per hour, and they each work 3 1/2 hours shoveling 10 1/2 tons of gravel, how much is the total labor cost per ton?​
 
Jomo said:
I do not exactly agree with you answer of $2/ton.
If the 4 workers work for 3 1/2 hrs and shovel 10 1/2 tons then at that rate in 1 hour they shovel 3 tons. Now your answer of $2 is the total that all 4 workers get. Each individual worker gets $0.50/ton. Did you realize that?
Click to expand...
Thank you for your explanation!
 
Just to add to the misery, do the four workers dig 10 and 1/2 tons each or total?
 
Yes, I realized too late that I neglected to add "total" there. I think it's reasonable to read the original with the presumably correct meaning, but that's as much from experience with this sort of problem as from anything in the grammar.
 
Four workers work for 3.5 hours so each is paid 1.5*3.5= $5.25. That is a total of 4(5.25)= $21.00. Finally it is \(\displaystyle \frac{21}{10.5}\) dollars per pound.
 
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HallsofIvy said:
Four workers work for 3.5 hours so each is paid 1.5*3.5= $5.25. That is a total of 4(5.25)= $21.00. Finally it is \(\displaystyle \frac{21}{10.5}\) dollars per pound.
Click to expand...

You people have totally confused me by now. The answer is $2.00/per ton. Word the question so that the answer will be $2.00 per ton.
 
"Word the question"? This was your question and your wording was

Explain this! said:
If 4 workers are paid $1.50 per hour, and they work work 3 1/2 hours shoveling 10 1/2 tons of gravel, how much is that per ton?

Here is my solution: 3 1/2 hours * 4 workers * [($1.50/hour)/worker]/10 1/2 tons

This will equal $2.00/ton or $2.00 per ton.

I'm not sure how to express $1.50 per hour per worker. Is it as I have it in my solution or is it $1.50/(hour/worker) or something else?
Click to expand...
3 1/2 hours * 4 workers = 14 "worker hours" * $1.50/hour= $21. That divided by 10 1/2 tons (that's your calculation) is NOT $2.00 per ton.
 
HallsofIvy said:
3 1/2 hours * 4 workers = 14 "worker hours" * $1.50/hour= $21. That divided by 10 1/2 tons (that's your calculation) is NOT $2.00 per ton.
Click to expand...

Huh? But $21/10.5 tons IS $2.00 per ton.

(You had accidentally said per pound previously.)
 
Could please explain what you meant in post number 2 Dr.peterson? I cannot use this formula to obtain the answer.

1595057486783.png

To me the problem is as simple as separating the problem into time and cost per worker

4 (workers) * 3.5 (each working this many hours) * 1.5 (pay per hour) = $21

followed by dividing by 10.5 to answer the posters question.
 
MalikN said:
Could please explain what you meant in post number 2 Dr.peterson? I cannot use this formula to obtain the answer.

View attachment 20510

To me the problem is as simple as separating the problem into time and cost per worker

4 (workers) * 3.5 (each working this many hours) * 1.5 (pay per hour) = $21

followed by dividing by 10.5 to answer the posters question.
Click to expand...
That is not a formula at all. It is a unit: dollars per hour per worker, or dollars per worker-hour.

And, yes, there are a variety of ways to think through a problem like this; the question was about one particular method, using units to guide the work ("dimensional analysis"), so we were discussing that method. And in fact what you say is exactly what the OP did.
 
Thank you for clarifying, i misunderstood your post.

To tell you the truth i don't include units like mph in my expressions and calculations to avoid confusion but it is something i will think about; Dimensional analysis.
 
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