A food store makes a 11–pound mixture of peanuts, almonds, and raisins. The cost of peanuts is $1.50 per pound, almonds cost $3.00 per pound, and raisins cost $1.50 per pound. The mixture calls for twice as many peanuts as almonds. The total cost of the mixture is $21.00. How much of each ingredient did the store use?

Question

3 lbs peanuts, 6 lbs almonds, 2 lbs raisins
8 lbs peanuts, 1 lbs almonds, 2 lbs raisins
6 lbs peanuts, 3 lbs almonds, 2 lbs raisins
8 lbs peanuts, 2 lbs almonds, 1 lbs raisins

Answer 1

Let the amount of peanuts be P, the amount of almonds A, and the amount of raisins R.

From the given information, we have the following equations:
P + A + R = 11 (equation 1)
P = 2A (equation 2)
1.5P + 3A + 1.5R = 21 (equation 3)

Substituting equation 2 into equation 1:
2A + A + R = 11
3A + R = 11 (equation 4)

Multiplying equation 4 by 1.5 to match the coefficients in equation 3:
4.5A + 1.5R = 16.5 (equation 5)

Subtracting equation 5 from equation 3:
1.5P + 3A + 1.5R - (4.5A + 1.5R) = 21 - 16.5
1.5P - 1.5A = 4.5
P - A = 3 (equation 6)

Adding equation 6 to equation 5:
4A = 19.5
A = 4.875 (approximately 4.9 lbs)

Substituting A back into equation 6:
P - 4.9 = 3
P = 7.9 (approximately 7.9 lbs)

Substituting A = 4.9 and P = 7.9 into equation 4:
3(4.9) + R = 11
R = 11 - 14.7
R = -3.7

Since R cannot be negative, this solution is not valid.

Therefore, the correct answer is: 8 lbs peanuts, 1 lb almonds, and 2 lbs raisins.