A worker at a farm supply company in Franklin County is mixing up a 420-kilogram batch of its special grain blend for horses, which costs $5.80 per kilogram to make. If the mixture includes oats that cost $9.40 per kilogram and feed corn that costs $3.40 per kilogram, how many kilograms of each should the worker include in the mixture?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
kilograms of oats
kilogram

Answer 1

Let \( x \) be the number of kilograms of oats and \( y \) be the number of kilograms of feed corn in the mixture.

We know the following:

1. The total weight of the mixture is 420 kilograms: \[ x + y = 420 \]

2. The total cost of the mixture can be expressed as: \[ 9.4x + 3.4y = 5.80 \times 420 \]

Calculating \( 5.80 \times 420 \): \[ 5.80 \times 420 = 2436 \]

So, the cost equation becomes: \[ 9.4x + 3.4y = 2436 \]

Now we have a system of equations:

1. \( x + y = 420 \) (1)
2. \( 9.4x + 3.4y = 2436 \) (2)

We can solve equation (1) for \( y \): \[ y = 420 - x \]

Substituting for \( y \) in equation (2): \[ 9.4x + 3.4(420 - x) = 2436 \] Expanding this: \[ 9.4x + 1428 - 3.4x = 2436 \] Combining like terms: \[ (9.4 - 3.4)x + 1428 = 2436 \] \[ 6.0x + 1428 = 2436 \] Subtracting 1428 from both sides: \[ 6.0x = 2436 - 1428 \] \[ 6.0x = 1008 \] Dividing by 6.0: \[ x = \frac{1008}{6.0} = 168 \]

Now we can find \( y \): \[ y = 420 - x = 420 - 168 = 252 \]

Thus, the worker should include:

• Oats: \( \boxed{168} \) kilograms
• Feed corn: \( \boxed{252} \) kilograms