The manager of a farmer's market has 500lb of grain that costs $1.20 per pound. How many pounds of meal costing $.80 per pound should be mixed with the 500lb of grain to produce a mixture that costs $1.05 per pound?
let the amount to be added be x lb
1.2(500) + .8x = 1.05(500+x)
600 + .8x = 525 + 1.05x
-025x = -75
x = -75/-.25 = 300
300 lb of the cheaper grain should be added
Why did the farmer go to the math class? Because he needed help with his crops!
Now, let's solve this math problem together.
Let's say the amount of meal costing $0.80 per pound that needs to be mixed with the 500lb of grain is x pounds.
We want the resulting mixture to have a cost of $1.05 per pound. So the total cost of the grain and meal combined should be the product of the weight and cost per pound.
For the grain: 500lb * $1.20 = $600
For the meal: x pounds * $0.80 = $0.80x
The total cost: $600 + $0.80x
Since we want the resulting mixture to have a cost of $1.05 per pound, we can set up the equation:
($600 + $0.80x) / (500lb + x) = $1.05
Now, let me calculate the solution for you.
To solve this problem, we can set up an equation based on the concept of a weighted average:
Let x be the number of pounds of meal costing $0.80 per pound.
The total cost of the meal and grain mixture will be the sum of the costs of the individual ingredients:
Total Cost = (Cost of Grain) + (Cost of Meal)
Total Cost = (500 lb) * ($1.20/lb) + (x lb) * ($0.80/lb)
The total weight of the meal and grain mixture will be the sum of the weights of the individual ingredients:
Total Weight = (Weight of Grain) + (Weight of Meal)
Total Weight = 500 lb + x lb
We also know that the average cost per pound of the mixture should be $1.05 per pound. So we can set up the equation:
Average Cost = Total Cost / Total Weight
$1.05 = [(500 lb * $1.20/lb) + (x lb * $0.80/lb)] / (500 lb + x lb)
To solve for x, we will multiply through by (500 lb + x lb) to get rid of the denominator:
$1.05 * (500 lb + x lb) = (500 lb * $1.20/lb) + (x lb * $0.80/lb)
Now, we can solve for x.
$1.05 * 500 lb + $1.05 * x lb = $1.20 * 500 lb + $0.80 * x lb
525 lb + $1.05 * x lb = $600 lb + $0.80 * x lb
$1.05 * x lb - $0.80 * x lb = $600 lb - 525 lb
$0.25 * x lb = $75 lb
x lb = $75 lb / $0.25
x lb = 300 lb
Therefore, 300 pounds of the meal costing $0.80 per pound should be mixed with the 500 pounds of grain to produce a mixture that costs $1.05 per pound.
To solve this problem, we need to use a weighted average formula. Let's break down the steps to find the answer:
1. Assign variables: Let's use the variable "x" to represent the number of pounds of meal that needs to be mixed.
2. Set up equations: Since we want to find the number of pounds of meal, we can set up an equation based on the cost per pound. The equation would be:
Total cost of grain + Total cost of meal = Total cost of mixture
500 lb * $1.20/lb + x lb * $0.80/lb = (500 + x) lb * $1.05/lb
Note: We're using the total cost formula, which is: Total cost = Total quantity x Cost per unit
3. Simplify the equation: Multiply the pounds symbol with the respective cost per pound values:
600 + 0.8x = 525 + 1.05x
4. Rearrange the equation to isolate the unknown variable, x:
0.25x = 75
Divide both sides of the equation by 0.25:
x = 75 / 0.25
x = 300
Therefore, to produce a mixture that costs $1.05 per pound, the manager should mix 300 pounds of meal costing $0.80 per pound with the 500 pounds of grain.