Inverse Proportion

E

Explain this!

Junior Member
Joined
Feb 7, 2019
Messages
170
If 5 men can perform a job in 10 hours, 10 men can perform the same job working at the same rate in how many hours?

Solution:

5/10 = 10/? One of the ratios needs to be inverted to determine the answer.

5/10 = ?/10

10 X ? = 5 X 10 divide both side by 10. This equals 5 or 5 hours.

Which one of the ratios should be inverted 5/10 or 10/?, or does it make a difference? Is there a mathematical reason for one or the other?
 
If you invert both sides of an equation, the resulting equation is equivalent. 5/10 = ?/10 means the same thing as 10/5 = 10/? .

But it was wrong to even write 5/10 = 10/? in the first place. That just isn't true.

I personally would rather say that the amount of work done is found by multiplying the number of men times the number of hours (giving man-hours), so the equation to write is

5 men * 10 hrs = 10 men * ? hrs​

The answer is obvious: 5 hrs.
 
Isn't it obvious that if have twice the number of people working that it will take half the time? So the answer is ___?

I prefer to think of people-hours.
If 5 men can perform a job in 10 hours then it take 50 people hours to do the job.
Now if you have 10 people and want to know how many hours then solve 10people * x hours = 10x people hours = 50 people hours. Now divide both sides by 10 people hours to get x alone on the lhs. What does the rhs equal??
 
Dr.Peterson said:
If you invert both sides of an equation, the resulting equation is equivalent. 5/10 = ?/10 means the same thing as 10/5 = 10/? .

But it was wrong to even write 5/10 = 10/? in the first place. That just isn't true.

I personally would rather say that the amount of work done is found by multiplying the number of men times the number of hours (giving man-hours), so the equation to write is

5 men * 10 hrs = 10 men * ? hrs​

The answer is obvious: 5 hrs.
Click to expand...

If you invert both sides of an equation, the resulting equation is equivalent. 5/10 = ?/10 means the same thing as 10/5 = 10/? .

You inverted both of the fractions. Yes, that would make the both equal, but in my question only one gets inverted. I am asking is there a mathematical reason to invert one or the other--not both! I indicated the proportion as 5/10 = 10/?, but the answer cannot be determined until one of the fractions in inverted. I inverted 10/? to ?/10 to determine the answer.
 
Explain this! said:
If you invert both sides of an equation, the resulting equation is equivalent. 5/10 = ?/10 means the same thing as 10/5 = 10/? .

You inverted both of the fractions. Yes, that would make the both equal, but in my question only one gets inverted. I am asking is there a mathematical reason to invert one or the other--not both! I indicated the proportion as 5/10 = 10/?, but the answer cannot be determined until one of the fractions in inverted. I inverted 10/? to ?/10 to determine the answer.
Click to expand...
It is mathematically WRONG to invert only one-side of an equation.
 
Explain this! said:
If you invert both sides of an equation, the resulting equation is equivalent. 5/10 = ?/10 means the same thing as 10/5 = 10/? .

You inverted both of the fractions. Yes, that would make the both equal, but in my question only one gets inverted. I am asking is there a mathematical reason to invert one or the other--not both! I indicated the proportion as 5/10 = 10/?, but the answer cannot be determined until one of the fractions in inverted. I inverted 10/? to ?/10 to determine the answer.
Click to expand...
My point was that if you start with your WRONG equation 5/10 = 10/?, then the equation you get by inverting only the LHS, 10/5 = 10/?, and the equation you get by inverting only the RHS, 5/10 = ?/10, are EQUIVALENT, so it makes no difference. That is the question you asked, isn't it?

But, again, you shouldn't have started with a false equation.

Explain this! said:
No it is not! This is one method that is seen in arithmetic books to determine the answer!
Click to expand...
Maybe some books teach that way, but at best, they are showing how to correct a wrong equation. Can you quote one?
 
Explain this! said:
No it is not! This is one method that is seen in arithmetic books to determine the answer!
Click to expand...
Books !! - plural!

Please cite the name and publisher of such book/s - along with the page number where this one-sided operation has been recommended!
 
Explain this! said:
If 5 men can perform a job in 10 hours, 10 men can perform the same job working at the same rate in how many hours?

Solution:

5/10 = 10/? One of the ratios needs to be inverted to determine the answer.

5/10 = ?/10

10 X ? = 5 X 10 divide both side by 10. This equals 5 or 5 hours.

Which one of the ratios should be inverted 5/10 or 10/?, or does it make a difference? Is there a mathematical reason for one or the other?
Click to expand...
5/10 = 10/?
You should see that 5/10 is 1/2. When is a fraction equal to 1/2? When the denominator is twice the numerator. When does 10/? = 1/2? When ?=20.

Even if inverting one of the fractions is correct, you should know ?=20 without inverting.

Is converting correct? Let's see. I know that 10/3 is more than 1. Clearly 10/3 = 10/3. But does 10/3 = 3/10? Since 3/10 is less than 1 the answer is NO!
 
Dr.Peterson said:
My point was that if you start with your WRONG equation 5/10 = 10/?, then the equation you get by inverting only the LHS, 10/5 = 10/?, and the equation you get by inverting only the RHS, 5/10 = ?/10, are EQUIVALENT, so it makes no difference. That is the question you asked, isn't it?

But, again, you shouldn't have started with a false equation.


Maybe some books teach that way, but at best, they are showing how to correct a wrong equation. Can you quote one?
Click to expand...

I cannot locate a book now, but I found a PDF that will illustrate: See https://www.education.pa.gov/K-12/C... T-Chart - Direct and Inverse Proportions.pdf

See example 2 on the right hand side of the page under "Inverse Proportions." As far as "But, again, you shouldn't have started with a false equation." Tell that to people who write the the textbooks using this method!
 
I don't like it, but I'll be generous, and fill in the gaps in what they say there:

First, write a proportion as if it were direct (which is not what we want).​
Then, change that proportion to an inverse proportion by flipping either side (so it becomes the correct equation).​
Then solve that.​

It would be far better if they explicitly said that!

But observe how you stated it:
Explain this! said:
Solution:

5/10 = 10/? One of the ratios needs to be inverted to determine the answer.
Click to expand...

You don't invert a ratio to determine the answer, but to make the equation correct in the first place. This is the main reason everyone has objected. Once you have the correct equation, then you can solve it, and inverting one side is illegal in that process.
 
The bottom line is when you write those two horizontal line (an equal sign, = ) what is on the left hand side and what is on the right hand side must be equal. After all why put an equal sign if it is not equal!.

In your example you have on the left hand side [math]\dfrac{1 person}{4 persons} = 1/4[/math] since the persons cancel out. Now on the right hand side you have [math]\dfrac{8 hours}{x hours} = 8/x[/math] since the hours cancel out

The problem is if x=2, and it does, it is NOT true that 1/4 = 8/2.

There is nothing to debate. Although this method will get you the correct answer it is not a valid method! You could not pay me enough to teach this 'method' in my classroom as I would get physically sick.
 
Honestly, I am starting to get sick after just reading the solution to that problem. Yuck!!
 
Dr.Peterson said:
I don't like it, but I'll be generous, and fill in the gaps in what they say there:

First, write a proportion as if it were direct (which is not what we want).​
Then, change that proportion to an inverse proportion by flipping either side (so it becomes the correct equation).​
Then solve that.​

It would be far better if they explicitly said that!

But observe how you stated it:


You don't invert a ratio to determine the answer, but to make the equation correct in the first place. This is the main reason everyone has objected. Once you have the correct equation, then you can solve it, and inverting one side is illegal in that process.
Click to expand...
Jomo said:
The bottom line is when you write those two horizontal line (an equal sign, = ) what is on the left hand side and what is on the right hand side must be equal. After all why put an equal sign if it is not equal!.

In your example you have on the left hand side [math]\dfrac{1 person}{4 persons} = 1/4[/math] since the persons cancel out. Now on the right hand side you have [math]\dfrac{8 hours}{x hours} = 8/x[/math] since the hours cancel out

The problem is if x=2, and it does, it is NOT true that 1/4 = 8/2.

There is nothing to debate. Although this method will get you the correct answer it is not a valid method! You could not pay me enough to teach this 'method' in my classroom as I would get physically sick.
Click to expand...

Take some Pepto-Bismol !
 
Dr.Peterson said:
I don't like it, but I'll be generous, and fill in the gaps in what they say there:

First, write a proportion as if it were direct (which is not what we want).​
Then, change that proportion to an inverse proportion by flipping either side (so it becomes the correct equation).​
Then solve that.​

It would be far better if they explicitly said that!

But observe how you stated it:


You don't invert a ratio to determine the answer, but to make the equation correct in the first place. This is the main reason everyone has objected. Once you have the correct equation, then you can solve it, and inverting one side is illegal in that process.
Click to expand...

I think that you are being too critical here regarding my wording. I simply wanted to know which ratio should be inverted or whether or not there is a preference as to which ratio should be inverted. I was not asking whether or not the solution is favorable.
 
Explain this! said:
I think that you are being too critical here regarding my wording. I simply wanted to know which ratio should be inverted or whether or not there is a preference as to which ratio should be inverted. I was not asking whether or not the solution is favorable.
Click to expand...

I wasn't criticizing your wording, so much as explaining why people reacted against it. You appeared to be saying that you wanted to invert one side of an equation as part of solving it, which is utterly wrong; what you meant was that you wanted to do so as part of creating the right equation.

The page you referred to didn't make that distinction clear; probably you have been taught similarly. Everyone wants to make sure you know what the rules are in solving an equation; you probably do. But in mathematics, wording makes a big difference, and it's important to learn to communicate clearly about it.

I more or less understood what you were talking about, and answered your question about which should be inverted, by saying it doesn't matter, due to the properties of equations. If it had been taught more clearly, you wouldn't have needed to ask.
 
You must log in or register to reply here.
Top