Math question…help please?
A coffee manufacturer uses Colombian and Brazilian coffee beans, to produce two blends, robust and mild.A pound of the robust blend requires 12 oz of Colombian beans and 4 oz of Brazilian beans. A pound of the mild blend requires 6 oz of the Colombian beans and 10 oz of the Brazilian beans. Coffee is shipped in 99-pound burlap bags. The company has 47 bags of Colombian beans and 25 bags of Brazilian beans on hand. How many pounds of each blend should they produce in order to use all the available beans?
Using:
1 robust = (3/4) columbian + (1/4) brazian
1 mild = (3/8) columbian + (5/8) brazilian
Given:
47 * 99lbs of columbian = 4653 columbian
25 * 99lbs of brazilian = 2475 brazilian
Thus:
Total coffee = (47+25)*99 = 7128
Let X = pounds of robust to make.
Let (7128-X) = pounds of mild to make.
X * (robust) + (7128-X) * (mild) = 4653 columbian + 2475 brazilian
Let C = pounds of columbian
Let B = pounds of brazilian
Rewriting using C and B (for simplicity):
X * ((3/4)C + (1/4)B) + ((7128-X) * ((3/8)C + (5/8)B)) = 4653C + 2475B
Solve for X:
(3/4)CX + (1/4)BX + 2673C -(3/8)CX + 4455B – (5/8)BX = 4653C + 2475B
(3/4)CX + (1/4)BX -(3/8)CX -(5/8)BX + 2673C + 4455B = 4653C + 2475B
x[(3/4)C + (1/4)B -(3/8)C -(5/8)B] + 2673C + 4455B = 4653C + 2475B
x[(3/4)C + (1/4)B -(3/8)C -(5/8)B] + 2673C + 4455B = (4653-2673)C + (2475-4455)B
x[(3/8)C - (3/8)B] = (4653-2673)C + (2475-4455)B
x[(3/8)C - (3/8)B] = 1980C -1980B
x = (1980C -1980B)/((3/8)C – (3/8)B)
x = 8(1980C -1980B)/(3C – 3B)
x = 8(660C -660B)/(C – B)
x = 5280(C -B)/(C – B)
x = 5280
Solve for mild:
mild = 7128-5280 = 1848
To verify, check the use of columbian coffee:
5280*(3/4) + 1848*(3/8) = 47*99
3960 + 693 = 4653
4653 = 4653
check.
Now check the use of brazilian coffee:
5280 * (1/4) + 1848*(5/8) = 25*99
2475 = 2475
check.
Answer:
Produce 5280 pounds of robust coffee and 1848 pounds of mild
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