How do I approach this Algebra problem?

Benjamin7399

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Frank & Frank industries developed colors for knitting wools for special purpose applications. The color baths are mixed using two different ingredients - the color and an agent that makes the color more UV stable. Other ingredients go into the color baths that are not important to the analysis. The mix of the two ingredients depends on the product type. You must calculate the amount of blue color bath you can prepare for Product P and Product Q. You have 390 liters of blue color and 543 liters of UV agent. The following table indicates how much of the blue color (in liters) and how much of the UV agent (in liters) you need to make a liter of color bath for Product P and Product Q.

Product PProduct Q
Blue Color (in liters)0.140.407
UV agent (in liters)0.4750.444


Suppose you have to prepare the color baths for Product P and Product Q at the same time. How many liters of color bath for Product P and Product Q can you prepare using the available inventory of the color blue and the UV agent? Round your result to two decimal digit.

Calculate Amount of color bath for Product P:
Calculate Amount of color bath for Product Q:

I've been stuck with this problem for a week now. I'm not sure what the correct answer is. I approached this like a linear equation

0.14x+0.407y=390
0.475x+0.444y=543

Amount of color bath for Product P: 0.14x+0.475x
Amount of color bath for Product Q: 0.407y+0.444y

Am I missing something? Is this the correct way to approach this? If not please teach me.

Thanks in advance.


 

Dr.Peterson

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Looks good to me.

Now solve for x and y.

Also, please tell us why you feel stuck. What makes you unsure of your work so far? Why haven't you completed it?
 

Subhotosh Khan

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Frank & Frank industries developed colors for knitting wools for special purpose applications. The color baths are mixed using two different ingredients - the color and an agent that makes the color more UV stable. Other ingredients go into the color baths that are not important to the analysis. The mix of the two ingredients depends on the product type. You must calculate the amount of blue color bath you can prepare for Product P and Product Q. You have 390 liters of blue color and 543 liters of UV agent. The following table indicates how much of the blue color (in liters) and how much of the UV agent (in liters) you need to make a liter of color bath for Product P and Product Q.


Product PProduct Q
Blue Color (in liters)0.140.407
UV agent (in liters)0.4750.444


Suppose you have to prepare the color baths for Product P and Product Q at the same time. How many liters of color bath for Product P and Product Q can you prepare using the available inventory of the color blue and the UV agent? Round your result to two decimal digit.

Calculate Amount of color bath for Product P:
Calculate Amount of color bath for Product Q:

I've been stuck with this problem for a week now. I'm not sure what the correct answer is. I approached this like a linear equation

0.14x+0.407y=390
0.475x+0.444y=543

Amount of color bath for Product P: 0.14x+0.475x
Amount of color bath for Product Q: 0.407y+0.444y

Am I missing something? Is this the correct way to approach this? If not please teach me.

Thanks in advance.


0.14x + 0.407y = 390......................................................(1)
0.475x + 0.444y = 543......................................................(2)

You have two linear equations (1 & 2) for two unknowns (x & y). Solve for 'x' & 'y' using your favorite method.
 

Benjamin7399

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Looks good to me.

Now solve for x and y.

Also, please tell us why you feel stuck. What makes you unsure of your work so far? Why haven't you completed it?
Hello! Thank you for your response. I solved the equations and obtained x = 364.74 and y = 832.77. When I substitute the values and find the color baths for A and B, the system says my answer is wrong. Which is why I'm not sure if my method is correct. Also, I feel I'm missing a key information from the question. What do you think?
 

Benjamin7399

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0.14x + 0.407y = 390......................................................(1)
0.475x + 0.444y = 543......................................................(2)

You have two linear equations (1 & 2) for two unknowns (x & y). Solve for 'x' & 'y' using your favorite method.
Thank you kind Sir. When I input my x and y values in the system, it says my answers for P and Q are incorrect. Is there any other way to approach this?
 

Dr.Peterson

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Hello! Thank you for your response. I solved the equations and obtained x = 364.74 and y = 832.77. When I substitute the values and find the color baths for A and B, the system says my answer is wrong. Which is why I'm not sure if my method is correct. Also, I feel I'm missing a key information from the question. What do you think?
Is it possible that "the system" (whatever that is) is wrong? That happens.

But please state the numbers you entered, so we can check all the way to the answer.

The question is not whether there is another way to approach it, but whether you and/or the system are wrong.
 

JoeyW

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You could forego a formulaic approach altogether and just add 0.14 to 0.407, then divide 390 by this and round down. Multiply by 0.14 (or 0.407) and subtract the product from 390 to get the desired quantity for P and Q.
 

Otis

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… I solved the equations and obtained x = 364.74 and y = 832.77
Those results are correct.

When I substitute the values and find the color baths for A and B, the system says my answer is wrong …
You meant to write P and Q, not A and B, yes?

Did you enter 224.31 liters of color bath for P and 708.69 liters for Q?

224.31 + 708.69 = 390 + 543

Seems to check out.

\(\;\)
 

Otis

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You could forego a formulaic approach altogether and just add 0.14 to 0.407, then divide 390 by this and round down …
Hi Joey. By rounding down, I think you introduce round-off error. The answers need to be reported accurate to two decimal places.

… Multiply by 0.14 (or 0.407) and subtract the product from 390 to get the desired quantity for P and Q.
Are you continuing with the result you rounded down above? I'm having trouble following your steps. If your final step is to subtract a product from 390, how do you get two results (for P and Q)?

🤔
 

Jomo

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Possibly you switched your results.
If I was in a bad mood I would complain that you never defined your variables so no one can really say if you are correct or not. Seriously you always need to defined your variables. But since I am in a good mood tonight I will not complain about your not defining your variables.
 

HallsofIvy

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If I were in a bad mood I would complain that you should have said "were in a bad mood", not "was in a bad mood", because that clause is in subjunctive mode!
 

Subhotosh Khan

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Thank you kind Sir. When I input my x and y values in the system, it says my answers for P and Q are incorrect. Is there any other way to approach this?
Your calculated values of 'x' and 'y' are CORRECT. Good job.

Please show us (step-by-step) how you calculated 'P' and 'Q' and the answers you got.
 

Subhotosh Khan

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If I were in a bad mood I would complain that you should have said "were in a bad mood", not "was in a bad mood", because that clause is in subjunctive mode!
That's right Jomo! Don't change the "mood" from interrogative to imperative.

Now go back to the corner.....
 

JoeyW

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@Otis, I'm sorry for not being clearer, and thank you for pointing out the round-off error. The quantity for P is approximately \(\displaystyle 390 - 0.14 \bigg \lfloor \frac{390}{0.547} \bigg \rfloor \), while Q is \(\displaystyle 390 - 0.407 \bigg \lfloor \frac{390}{0.547} \bigg \rfloor \). It isn't necessary to round down \(\displaystyle \frac{390}{0.547} \), it's just a bad habit I've fallen into. 😁
 

Subhotosh Khan

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@Otis, I'm sorry for not being clearer, and thank you for pointing out the round-off error. The quantity for P is approximately \(\displaystyle 390 - 0.14 \bigg \lfloor \frac{390}{0.547} \bigg \rfloor \), while Q is \(\displaystyle 390 - 0.407 \bigg \lfloor \frac{390}{0.547} \bigg \rfloor \). It isn't necessary to round down \(\displaystyle \frac{390}{0.547} \), it's just a bad habit I've fallen into. 😁
You did not complete your calculations to tell us what "numbers" did you get for P and Q. In OP you said:

Amount of color bath for Product P: 0.14x+0.475x................................. = x * (0.14 + 0.475) = 364.74 * 0.615 = 224.315

Amount of color bath for Product Q: 0.407y+0.444y............................... = y * (0.407 + 0.444) = 832.77 * (0.851) = 708.68

What is it that you needed to calculate?

And going back to Jomo's complain:

what is it that you calculated when you calculated 'x'?​
what is it that you calculated when you calculated 'y'?​
 

JoeyW

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For \(\displaystyle 390 - 0.14 \bigg( \frac{390}{0.547} \bigg) \) I get 290.18L, and for \(\displaystyle 390 - 0.407 \bigg( \frac{390}{0.547} \bigg) \) I get 99.82L. Maybe there's a mistake I'm overlooking? As for the variables everyone keeps mentioning, \(\displaystyle x = \frac{290.18}{0.14} \) and \(\displaystyle y = \frac{99.82}{0.407} \).
 

Otis

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… you should have said "were in a bad mood" …
I never knew that, until I studied German in college (were/would), so I have to give Jomo a break.

😶
 

HallsofIvy

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Yeah , Americans just don't know how to speak English. Ask any Englishman!
 

Subhotosh Khan

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For \(\displaystyle 390 - 0.14 \bigg( \frac{390}{0.547} \bigg) \) I get 290.18L, and for \(\displaystyle 390 - 0.407 \bigg( \frac{390}{0.547} \bigg) \) I get 99.82L. Maybe there's a mistake I'm overlooking? As for the variables everyone keeps mentioning, \(\displaystyle x = \frac{290.18}{0.14} \) and \(\displaystyle y = \frac{99.82}{0.407} \).
I am not asking about 'everyone' - I am asking about you. When you wrote:

0.14x + 0.407y = 390......................................................(1)

0.475x + 0.444y = 543......................................................(2)

What did x & y stand for?
 

JoeyW

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They're just unit-less numbers that represent how you can partition the given amount of material into concurrent parts of size 0.14 and 0.407.
 
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