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The baynard high school boosters are selling hotdogs and pretzels at the concession stand. They have 300 hot dogs and pretzels in stock and want to earn an income of 800. They plan to sell the hot dogs for 4 each and the pretzels for 2 each. How many hot dogs and how many pretzels do the boosters need to sell to reach their goal of 800?

1 answer

To solve the problem, let's define the variables:

  • Let \( x \) = the number of hot dogs sold.
  • Let \( y \) = the number of pretzels sold.

We have two equations based on the information given.

  1. The total number of hot dogs and pretzels sold must be 300: \[ x + y = 300 \]

  2. The total income from selling the hot dogs and pretzels must be $800: \[ 4x + 2y = 800 \]

Now, we can solve these two equations.

From the first equation, we can express \( y \) in terms of \( x \): \[ y = 300 - x \]

Now, substitute this expression for \( y \) into the second equation: \[ 4x + 2(300 - x) = 800 \]

Distributing the 2: \[ 4x + 600 - 2x = 800 \]

Combining like terms: \[ 2x + 600 = 800 \]

Subtracting 600 from both sides: \[ 2x = 200 \]

Dividing both sides by 2: \[ x = 100 \]

Now we can find \( y \): \[ y = 300 - x = 300 - 100 = 200 \]

Therefore, the boosters need to sell:

  • 100 hot dogs
  • 200 pretzels

These quantities satisfy both conditions: they add up to 300 items, and they yield an income of \( 4(100) + 2(200) = 400 + 400 = 800 \).