To make a special mix, the owner of a fruit stand wants to combine cashews that sell for $7.00 per lb with walnuts that sell for $5.50 per lb to obtain a 35 lb hat sells for $6.50 per lb. How many pounds of each type of nut should be used in the mixture?
adding up the value of each part needs to equal the value of the total mixture. So, if x lbs of cashews are used,
7.00x + 5.50(35-x) = 6.50(35)
x = 23 1/3 lbs cashews
Let's define the following variables:
Let's call the pounds of cashews used in the mixture as 'x'.
Let's call the pounds of walnuts used in the mixture as 'y'.
According to the given information, we can set up the following equations:
1. The total weight of the mixture is 35 lbs:
x + y = 35
2. The cost per pound of the mixture is $6.50:
(7x + 5.5y) / 35 = 6.50
To solve this system of equations, we can first rearrange equation 2:
7x + 5.5y = 6.50 * 35
Next, we can multiply equation 1 by 5.5 and subtract it from equation 2:
7x + 5.5y - (5.5x + 5.5y) = 6.50 * 35 - 5.5 * 35
This simplifies to:
1.5x = 6.50 * 35 - 5.5 * 35
Now, we can solve for 'x':
x = (6.50 * 35 - 5.5 * 35) / 1.5
Calculating this equation:
x = (227.50 - 192.50) / 1.5
x = 35 / 1.5
x = 23.33
Therefore, approximately 23.33 pounds of cashews should be used in the mixture.
To find the pounds of walnuts used in the mixture, we can substitute the value of 'x' in either equation 1 or equation 2.
Substituting in equation 1:
23.33 + y = 35
y = 35 - 23.33
y = 11.67
Therefore, approximately 11.67 pounds of walnuts should be used in the mixture.
To solve this problem, we can use a technique called the mixture equation. Let's assume that x represents the number of pounds of cashews and y represents the number of pounds of walnuts.
According to the problem, the total weight of the mixture should be 35 pounds. Therefore, we have the equation:
x + y = 35 ----(1)
Now, let's consider the cost of the mixture. The cashews sell for $7.00 per pound and the walnuts sell for $5.50 per pound, with a final price of $6.50 per pound for the mixture. We can write the equation:
(7x + 5.5y) / 35 = 6.50 ----(2)
Now, we have a system of two equations with two variables. We can solve this system using substitution or elimination method. Let's solve it using substitution.
First, rearrange equation (1) to express one variable in terms of the other. For example, solve equation (1) for y:
y = 35 - x
Now substitute this value of y in equation (2):
(7x + 5.5(35 - x)) / 35 = 6.50
Simplify the equation:
7x + 5.5(35 - x) = 6.50 * 35
Now, solve for x:
7x + 192.5 - 5.5x = 227.5
Simplify the equation further:
1.5x = 227.5 - 192.5
1.5x = 35
Next, divide both sides of the equation by 1.5:
x = 35 / 1.5
x = 23.33 (rounded to two decimal places)
Now substitute the value of x back into equation (1) to solve for y:
23.33 + y = 35
y = 35 - 23.33
y = 11.66 (rounded to two decimal places)
Therefore, we need approximately 23.33 pounds of cashews and 11.66 pounds of walnuts to make the special mixture.