A store mixes Kentucky bluegrass worth $11 per pound and ryegrass worth $14 per pound. The mixture is to sell for $13 per pound. Find out how much of each should be used to make a 588 pound mixture. How many pounds of the Kentucky bluegrass should be in the mixture?
add up the cost of the parts:
If there are x lbs of bluegrass, then there are 588-x pounds of rye grass, so
11x+14(588-x) = 13*588
x = 196
note that since the expensive grass is $3/lb more than the cheap grass, and the final mixture is $1/lb more expensive, since 1 is 1/3 of 3, 1/3 of the grass is cheap, and 2/3 is expensive.
To solve this problem, we can use the concept of a weighted average. Let's assign variables to the quantities we need to find.
Let's assume that x represents the pounds of Kentucky bluegrass in the mixture and y represents the pounds of ryegrass in the mixture.
Since we know that the store wants to create a 588 pound mixture, we have the equation:
x + y = 588 (Equation 1)
We also know the cost per pound of each grass and the desired price per pound of the mixture.
Kentucky bluegrass costs $11 per pound, so the cost of x pounds of Kentucky bluegrass is 11x dollars.
Ryegrass costs $14 per pound, so the cost of y pounds of ryegrass is 14y dollars.
The mixture is to be sold for $13 per pound, so the total cost of the mixture is 13 * 588 dollars.
Since cost equals price in this case, we have the equation:
11x + 14y = 13 * 588 (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) that we can solve to find the values of x and y.
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution.
From Equation 1, we can express y in terms of x:
y = 588 - x
Substitute this expression for y in Equation 2:
11x + 14(588 - x) = 13 * 588
Simplify the equation:
11x + 8232 - 14x = 7644
Combine like terms:
-3x = -588
Divide both sides by -3 to solve for x:
x = 196
Therefore, there should be 196 pounds of Kentucky bluegrass in the mixture.
Let's assume the store uses x pounds of Kentucky bluegrass and y pounds of ryegrass in the mixture.
According to the problem, the total weight of the mixture is 588 pounds. So we have the equation:
x + y = 588 (Equation 1)
We also know the cost per pound for each type of grass and the desired cost per pound for the mixture:
Kentucky bluegrass cost = $11 per pound
Ryegrass cost = $14 per pound
Mixture cost = $13 per pound
To find the cost of the mixture per pound, we can use a weighted average:
(x * 11 + y * 14) / (x + y) = 13 (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve to find the values of x and y.
We can start by rearranging Equation 1:
x = 588 - y (Equation 3)
Substituting Equation 3 into Equation 2:
((588 - y) * 11 + y * 14) / (588 - y + y) = 13
(6468 - 11y + 14y) / 588 = 13
(6468 + 3y) / 588 = 13
Multiplying both sides by 588 to get rid of the fraction:
6468 + 3y = 13 * 588
6468 + 3y = 7644
Subtracting 6468 from both sides:
3y = 7644 - 6468
3y = 1176
Dividing both sides by 3:
y = 392
So, the store should use 392 pounds of ryegrass in the mixture.
To find the amount of Kentucky bluegrass, we can substitute the value of y into Equation 3:
x = 588 - 392
x = 196
Therefore, the store should use 196 pounds of Kentucky bluegrass in the mixture.