System of Equations Word Problem - My answer isn't coming out the same as the textbooks

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r1ch.r1cky

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A merchant has two kinds of paint. If 9 gal of the inexpensive paint is mixed with 7 gal of the expensive paint, the mixture will be worth $19.70 per gallon. If 3 gal of the inexpensive paint is mixed with 5 gal of the expensive paint, the mixture will be worth $19.825 per gallon. What is the price per gallon of each type of paint?
 

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If you include units in your calculations it will be easier to see errors/problems.
E.g. what exactly are x and y? Prices? Amounts? What are 9x and 7y? If it's gal * price, the result of the sum would be the total cost of the first mix, not the price.
 
I rewrote it with units but doesn't look right to me still. Since we are trying to find the price per gallon of each paint type, I figured x and y would be the prices. I don't know if I'm translating the question right. I know I'm missing something but don't know what...
 

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r1ch.r1cky said:
I rewrote it with units but doesn't look right to me still. Since we are trying to find the price per gallon of each paint type, I figured x and y would be the prices. I don't know if I'm translating the question right. I know I'm missing something but don't know what...
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The point of using units is to check the validity of your expression/equation.
If x and y are prices:
9gal*x$/gal + 7gal*y$/gal
What are the units of the result? $
What do you have? $/gal
As I wrote earlier, the left hand side of your equation is the total cost. You can't make it equal to the price. How can you change the left hand side to make it the price instead of total cost?
 
r1ch.r1cky said:
x $/gal + y $/gal = $19.7/gal
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Ok, units match. But, I'm afraid, it still does not work. On the left you have the sum of prices, which is the cost of 1 gallon of paint 1 and 1 gallon of paint 2. What do you have on the right? The price of the first mix (cost per 1 gallon). Completely different thing.
I would go back to the original equation and make its left hand side the price of the first mix instead of cost. If you have the cost of a certain amount of mix, how do you calculate its price?
 
9gal * x $/gal + 7gal * y $/gal (gallon units cancel?)

9 gallon * x $/gallon + 7 gallon * y $/gallon ...................................... [ = (9 + 7) gallon * 19.7 $/gallon]
 
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r1ch.r1cky said:
A merchant has two kinds of paint. If 9 gal of the inexpensive paint is mixed with 7 gal of the expensive paint, the mixture will be worth $19.70 per gallon. If 3 gal of the inexpensive paint is mixed with 5 gal of the expensive paint, the mixture will be worth $19.825 per gallon. What is the price per gallon of each type of paint?
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Suppose you purchased two types of paint. You purchased 7 gallons of paint 1 at $5/gal and 11 gallons of paint 2 at $7/gal. How would you find the average price per gallon? You certainly will not be able to do your problem until you see how to do this one. Seriously, do this problem, show us your work, let us confirm you did it correctly or we will tell you where you made any mistakes and then you can attempt your problem.
 
A couple of comments: In your first post you have
x: expensive paint
y: inexpensive paint
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If I were being really hard-nosed I would say that is non-sense! x and y are numbers not kinds of paint. Since I am in a better mood (I just had a good night's sleep) I will say "okay, you mean that x is the cost of one gallon of expensive paint and y is the cost of one gallon of inexpensive paint."

But then you have a real error: you write
9x+ 7y= 19.7
NO! on the left is the cost of 9 gallons of the expensive paint and 7 gallons of the inexpensive paint but on the right is the cost of one gallon of the mixture! Since buying 9 gallons of expensive paint and 7 gallons of inexpensive paint is the same as buying 9+ 7= 16 gallons of the mixture your equation should be 9x+ 7y= 19.7(16) and similarly for the second equation.
 
r1ch.r1cky said:
A merchant has two kinds of paint. If 9 gal of the inexpensive paint is mixed with 7 gal of the expensive paint, the mixture will be worth $19.70 per gallon. If 3 gal of the inexpensive paint is mixed with 5 gal of the expensive paint, the mixture will be worth $19.825 per gallon. What is the price per gallon of each type of paint?
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This was posted on Tuesday - and today is Saturday. I think we can provide full solution now. If I were to do this problem, I would define:

I = price/gallon of inexpensive paint

E = price/gallon of expensive paint

Then we have:

"If 9 gal of the inexpensive paint is mixed with 7 gal of the expensive paint, the mixture will be worth $19.70 per gallon" \(\displaystyle \to \ \ \)

7 * E + 9 * I = (7+9) * 19.70 = 315.2 .................................................................................(1)

"If 3 gal of the inexpensive paint is mixed with 5 gal of the expensive paint, the mixture will be worth $19.825 per gallon" \(\displaystyle \to \ \ \)

5 * E + 3 * I = (5+3) * 19.825 = 158.6.................................................................................(2)

35 * E + 45 * I = 1576 ...........................Multiply eqn. 1 by 5 ..............................................(3)

35 * E + 21 * I = 1110.2..........................Multiply eqn. 2 by 7................................................(4)

subtract (4) from (3)

24*I = 465.8 \(\displaystyle \to \ \ \) I = 465.8/24 = 19.40833

Inserting this value of I into eqn. (1)

7 * E + 9 * 465.8/24 = 315.2 .. \(\displaystyle \to \ \ \)E = (315.2 - 9*465.8/24)/7 = 20.075

So (correcting to cents)

E = $ 20.08 ...........and......... I = $ 19.41

check

5 * E + 3 * I = 5 * 20.08 + 3 * 19.41 = 158.63 .............................................. checks with (2)

Since I used (1) to calculate 'E' - I checked the answer first against (2).
 
Again we do not have the original words of the problem.

But based on the student's wording, confusion is almost inevitable. We are told about the "worth" of the paint. Is that fair market value or something else? We are told about the "price" of the paint. I am not sure whether we are talking about cost of purchase or sales price. If what is meant is sales price, there is absolutely no way to tell how the sales price of the mixture should relate to the sales price of the separate ingredients unless that relationship is specified.
 
HallsofIvy said:
All that is true but I think overly picky for a secondary school math problem.
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Inexact language is always bad, but it is especially bad when addressing students who may already be struggling. Of course, I do not know that the problem was badly worded. Students are perfectly capable of misreading even a perfectly worded problem. But we cannot know whether or not the problem itself was the source of the student's confusion if we do not get to see it.

Moreover, any motivation that may be hoped for from giving a "practical" business problem is eliminated if it is not clear whether it involves sales price or cost.
 
JeffM said:
Inexact language is always bad, but it is especially bad when addressing students who may already be struggling.
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I agree with you here. It is one thing for a college math professor to say that 7*8=46 but is much more severe when an elementary school teacher, who is teaching the time tables to their students, to say this.
 
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