buford, the farm hand, is preparing feed for the animals. He wants to mix the contents of a bag containing 50% corn with one containing 90% corn to obtain a 30-pound bag containing 80% corn. How many pounds of the 50% bag should he use? (write your answer as a decimal)

Answer 1

To determine how many pounds of the 50% corn bag Buford should use, let's set up the problem with algebraic equations.

Let \( x \) be the number of pounds of the 50% corn bag used.
Let \( y \) be the number of pounds of the 90% corn bag used.

We have two main conditions:
1. The total weight of the mix is 30 pounds.
2. The percentage of corn in the mix should be 80%.

First, write the first condition as an equation:
\[ x + y = 30 \]

Next, write the second condition using the percentages:
The total amount of corn from both bags should equal 80% of the total weight of the mix, which is 30 pounds. This can be expressed as:
\[ 0.50x + 0.90y = 0.80 \times 30 \]

Simplify the right side of the equation:
\[ 0.50x + 0.90y = 24 \]

Now we have a system of two equations:
1. \( x + y = 30 \)
2. \( 0.50x + 0.90y = 24 \)

We solve the first equation for \( y \):
\[ y = 30 - x \]

Substitute \( y = 30 - x \) into the second equation:
\[ 0.50x + 0.90(30 - x) = 24 \]

Distribute 0.90 through the parenthesis:
\[ 0.50x + 27 - 0.90x = 24 \]

Combine like terms:
\[ -0.40x + 27 = 24 \]

Subtract 27 from both sides:
\[ -0.40x = -3 \]

Divide by -0.40:
\[ x = \frac{-3}{-0.40} \]
\[ x = 7.5 \]

Therefore, Buford should use \( 7.5 \) pounds of the 50% corn bag.