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A farm supply company in Franklin County has developed its own blend of grains for horses. The blend includes oats, which cost per kilogram, and feed corn, which costs per kilogram. Overall, the mixture costs per kilogram. If a worker at the company is mixing up a batch and starts with kilograms of oats, how many kilograms of corn should he add to make the blend? Write your answer as a whole number or as a decimal rounded to the nearest tenth. kilograms
Full solution
Q. A farm supply company in Franklin County has developed its own blend of grains for horses. The blend includes oats, which cost per kilogram, and feed corn, which costs per kilogram. Overall, the mixture costs per kilogram. If a worker at the company is mixing up a batch and starts with kilograms of oats, how many kilograms of corn should he add to make the blend? Write your answer as a whole number or as a decimal rounded to the nearest tenth. kilograms
- Denote Corn Weight: Let's denote the number of kilograms of corn to be added as . The cost of the oats per kilogram is \\(9\).\(40\), and the cost of the corn per kilogram is \$\(3\).\(40\). The target cost of the mixture per kilogram is \$\(5\).\(80\).
- Calculate Oats Cost: The total cost of the oats already in the mixture is \(130 \, \text{kilograms} \times \$9.40 \, \text{per kilogram}, which equals \$1,222.\(\newline\)Calculation: \)\(130\) \, \text{kg} \times \$\(9\).\(40\)/\text{kg} = \$\(1\),\(222\)
- Calculate Corn Cost: The total cost of the \(x\) kilograms of corn that will be added is \(x\) times \(\$3.40\) per kilogram.\(\newline\)Expression: \(x \times \$3.40/\text{kg}\)
- Calculate Total Weight: The total weight of the mixture after adding the corn will be \(130\) kilograms plus \(x\) kilograms.\(\newline\)Expression: \(130\,\text{kg} + x\,\text{kg}\)
- Calculate Total Cost: The total cost of the mixture after adding the corn will be the sum of the cost of the oats and the cost of the corn. This total cost should equal the total weight of the mixture times the target cost per kilogram (.).Equation: .) = ( + x) * $\(5\).\(80\) becomes \(1222 + (x \times 3.40) = (130 + x) \times 5.80\)
- Distribute Target Cost: Now we solve the equation for \(x\). First, distribute \(\$5.80\) on the right side of the equation.\(\$1,222 + (x \times \$3.40) = \$754 + \$5.80x\)
- Isolate Terms with x: Subtract \(\$1,222\) from both sides to isolate the terms with x on one side.\(\newline\)\(x \times \$3.40 - \$1,222 = \$754 + \$5.80x - \$1,222\)\(\newline\)\(x \times \$3.40 - 5.80x = \$754 - \$1,222\)
- Combine Like Terms: Combine like terms and solve for \(x\).
\(3.40x - 5.80x = -468\)
\(-2.40x = -468\)
\(x = \frac{-468}{-2.40}\)
\(x = 195\)
