SOLUTION: I have been stuck on this word problem for forever! Please help me. Army requirements call for 55 coffee grounds per quart of coffee. The cooks have one barrel of coffee with 67.5

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Question 504591: I have been stuck on this word problem for forever! Please help me.
Army requirements call for 55 coffee grounds per quart of coffee. The cooks have one barrel of coffee with 67.5 grounds per quart in it and another barrel with 45 grounds per quart in it. How many quarts of each should they mix together to make up 1 quarts?
the answer is: eight quarts of the 67.5 grounds/quart and ten quarts of the 45 grounds/quart.
I just can't figure out how they got these answers. I've gotten a set of answers of my own that will work, but I would like to know how to get these answers.

thanks
Found 2 solutions by Theo, jorel1380:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
army requirements ae 55 coffee grounds per quart of coffee.
cooks have a barrel of coffee with 67.5 ground per quart in it.
they have another barrel with 45 grounds per quart in it.
how many quarts of each should they mix to make up 1 quart with 55 grounds per quart in it.
answer is 8 quarts of 67.5 grounds per quart and 10 quarts of 45 grounds per court.
how did they get it?
we're talking quarts as the unit of measure.
you will need x quarts of 67.5 grounds per quart mixed with y quarts of 45 grounds per quart.
you need an amount of quarts to shoot for.
i gather the amount is 18 quarts because that's what the answer shows.
you only showed 1 quart with 55 ground per quart.
that may be a typo.
i'm assuming that it is because without that figure of 18 quarts i would have difficulty solving your problem to generate the answer that was given.
anyway.
i'm assuming you need 18 quarts of 55 grounds per quart.
we let x equal the amount of quarts with 67.5 grounds in it.
we let y equal the amount of quarts wwith 45 grounds per quart.
total quarts must equal 18 so we get:
x + y = 18 (equation 1)
total grounds must equals 18 quarts times 55 grounds per quart for a total of 55 * 18 = 990 grounds so we get:
67.5x + 45y = 990 (equation 2)
the 2 equations you need to solve simultaneously are:
x + y = 18 (equation 1)
67.5x + 45y = 990 (equation 2)
if we multiply both sides of equation 1 by 67.5, then we will be able to eliminate x and solve for y as follows:
x + 6 = 18 multiplied by 67.5 on both sides equals:
67.5x + 67.5y = 1215 (equation 3)
we subtract equation 2 from equation 3 to get:
67.5x 67.5y = 1215 minus:
67.5x + 45y = 990 becomes:
67.5y - 45y = 1215 - 990 which becomes:
22.5y = 225 which becomes:
y = 10
since x + y = 18 and y = 10 then x = 8
we have:
x = 8 and y = 10
you can confirm the answer is good by substituting in the original equations.
your answer is:
x = 8
y = 10
this means that:
8 quarts of the 67.5 ground per quart plus 10 quarts of the 45 grounds per quart will equal 18 quarts of 55 grounds per quart.
i confirmed the answer is correct on this end.
it's good.

Answer by jorel1380(3719) (Show Source):
You can put this solution on YOUR website!
67.5m+45n=55(m+n)
67.5m+45n=55m+55n
12.5m=10n
1.25m=1n
m=.8n
For every part of 45 ground coffee, you need to use .8 parts of 67.5 ground coffee to get 55 ground coffee:
n+.8n=1
1.8n=1
n=1/1.8=.5556
.5556(45)+.4444(67.5)=25.002+29.997=54.9999.