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?
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).