Selling Hamburgers and Sodas at the Carnival: A System of Equations

Scenario 1: Current Prices and Sales Goals

Let:

• x = number of hamburgers sold
• y = number of sodas sold

We have two equations based on the given information:

1. Price equation: 1.75x + 0.75y = 117.50 (total revenue)
2. Quantity equation: x + y = 120 (total items sold)

Solving this system of equations:

• From equation 2, y = 120 – x.
• Substitute this into equation 1: 1.75x + 0.75(120 – x) = 117.50.
• Simplifying, we get 1x = 45.
• Therefore, x = 45 hamburgers and y = 75 sodas.

Scenario 2: Adjusting Prices for Equal Consumption

To encourage equal consumption, you want the per-unit price of a hamburger to be closer to the per-unit price of a soda. This could be achieved by:

• Increasing the soda price: This would make sodas seem relatively more expensive, potentially lowering soda sales and encouraging customers to buy more hamburgers. A moderate increase, like raising the soda price to $1.00, would maintain a comparable price ratio (2.75:1) and likely shift consumption closer to 60 hamburgers and 60 sodas.
• Decreasing the hamburger price: This would make hamburgers seem more affordable, potentially increasing hamburger sales and balancing out soda sales. A slight decrease, like lowering the hamburger price to $1.50, would maintain a closer price ratio (2:1) and potentially result in around 60 hamburgers and 60 sodas sold.

New Amount of Sales:

The exact number of hamburgers and sodas sold with adjusted prices would depend on customer preferences and price sensitivity. However, both options outlined above should encourage closer parity in the number of hamburgers and sodas sold compared to the original price point scenario.

Additional Considerations:

• You can use a breakeven analysis to determine the minimum sales volume needed to cover your costs with the adjusted prices.
• Offering combo deals with both a hamburger and a soda could incentivize balanced consumption at a slightly lower price point per item.
• Experimenting with different price combinations and observing customer behavior can help you find the optimal pricing strategy for your carnival.

Remember, balancing price and demand is a complex process. Adjusting prices based on your expected sales and desired consumption balance can help you maximize your revenue and satisfy your customers at the carnival.