SOLUTION: Solve the problem using three variables. A company sells nuts in bulk quantities. When bought in bulk, peanuts sell for $1.50 per pound, almonds $2.25 per pound, and cashews fo

Question 134775: Solve the problem using three variables.
A company sells nuts in bulk quantities. When bought in bulk, peanuts sell for $1.50 per pound, almonds $2.25 per pound, and cashews for $3.75 per pound. Suppose a specialty shop wants a mixture of 270 pounds that will cost $2.89 per pound. Find the number of pounds of each type of nut if the sum of the number of pounds of almonds and cashews is twice the number of pounds of peanuts. Round your answer to the nearest pound.
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Solve the problem using three variables.
A company sells nuts in bulk quantities. When bought in bulk, peanuts sell for $1.50 per pound, almonds $2.25 per pound, and cashews for $3.75 per pound. Suppose a specialty shop wants a mixture of 270 pounds that will cost $2.89 per pound. Find the number of pounds of each type of nut if the sum of the number of pounds of almonds and cashews is twice the number of pounds of peanuts. Round your answer to the nearest pound.
:
Let a = lbs of almonds
Let c = lbs of cashews
Let p = lbs of peanuts
:
2.25a + 3.75c + 1.50p = 2.89(270)
and
a + c + p = 270
:
It says,"the sum of the number of pounds of almonds and cashews is twice the number of pounds of peanuts."
2p =(a+c)
p = .5(a+c)
;
Substitute .5(a+c) for p in the two equations
2.25a + 3.75c + 1.50(.5(a+c)) = 2.89(270)
2.25a + 3.75c + .75a + .75c = 780.3
3a + 4.5c = 780.3
:
a + b + .5(a+b) = 270
a + b + .5a + .5b = 270
1.5a + 1.5b = 270
:
Multiply the above by 2 and subtract from 1st equation
3a + 4.5c = 780.3
3a + 3.0c = 540
----------------subtracting eliminates a
0a + 1.5c = 240.3
c = 240.3/1.5
c = 160.2 ~ 160 lb of cashews
:
use equation 3a + 3c = 540 to find a
3a + 3(160.2) = 540
3a + 480.6 = 540
3a = 540 - 480.6
3a = 59.4
a = 59.4/3
a = 19.8 ~ 20 lb of almonds
:
Find p
20 + 160 + p = 270
p = 270 - 180
p = 90 lb of peanuts
:
Check solutions in 1st original equation:
2.25(20) + 3.75(160) + 1.5(90) = 2.89(270)
45 + 600 + 135 = 780
:
Confirms our solution of: 20 lb of almonds, 160 lb of cashews, 90 lb of peanuts
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