Problem 39 - Algebra A Combined Function

Question

Iceberg lettuce is grown and shipped to stores for about 40 cents a head, and consumers purchase it for about 70 cents a head. Find the percent increase.

Step-by-Step Solution

Answer

The percent increase is 75%.

Step 1: Identify the Cost Price and Selling Price

First, note the cost price of the iceberg lettuce, which is 40 cents per head, and the selling price, which is 70 cents per head. These will be used to calculate the percent increase.

Step 2: Calculate the Increase in Price

Subtract the cost price from the selling price to find the increase in price: \[ 70 \text{ cents} - 40 \text{ cents} = 30 \text{ cents} \] The increase in price is 30 cents per head.

Step 3: Use the Percent Increase Formula

To find the percent increase, use the formula:\[ \text{Percent Increase} = \left( \frac{\text{Increase in Price}}{\text{Cost Price}} \right) \times 100 \]Substitute the values:\[ \left( \frac{30}{40} \right) \times 100 \]

Step 4: Perform the Calculation

Calculate the percent increase by first dividing 30 by 40:\[ 30 \div 40 = 0.75 \]Then multiply by 100 to convert to a percentage:\[ 0.75 \times 100 = 75\% \]

Step 5: Conclusion

The percent increase from the cost price to the selling price of each iceberg lettuce head is 75%. This is the final answer.

Key Concepts

Cost Price

When dealing with the concept of cost price, think of it as the original amount spent to produce or purchase an item before any mark-up. It acts as the foundation for determining profitability. In the context of our iceberg lettuce example, the cost price is the 40 cents paid to grow and ship each head of lettuce.

It's crucial to understand the cost price when performing any calculations regarding profit margin or percent increase. Without knowing the initial cost, it's nearly impossible to gauge the success or failure of a business venture.

• Cost price is the base value from which selling strategies are developed.
• It's the amount one needs to recover to avoid a loss.
• Being aware of the cost price aids in setting a competitive selling price.

In our example, the cost price is pivotal in figuring out how much more consumers are paying compared to the production cost.

Selling Price

The selling price is the amount that consumers pay to purchase an item. It generally includes both the cost price and an additional amount, or margin, to ensure that the seller makes a profit. In our iceberg lettuce case, the selling price is 70 cents per head.

A well-calculated selling price is vital for balancing competitiveness with profitability. It directly influences consumer behavior and market positioning.

• Selling price must cover the cost price to generate profit.
• It should be attractive enough to entice buyers.
• Setting the right selling price can help achieve market dominance.

In the example given, understanding the selling price helps us figure out the amount of revenue generated per item sold, as well as the room for profit over the cost price.

Percent Increase Formula

The percent increase formula is a method used to express the change between two values as a percentage. It is extremely useful in financial assessments to determine growth or inflation. The formula itself is as follows: \[ \text{Percent Increase} = \left( \frac{\text{Increase in Price}}{\text{Cost Price}} \right) \times 100 \] This allows for straightforward calculation of the percentage by which a quantity has increased, opening a window into understanding pricing efficiency.

In our iceberg lettuce scenario, any percent increase indicates how much extra consumers are paying compared to the production cost.

• It quantifies the growth in selling price over cost price.
• The formula simplifies understanding of price dynamics.
• Essential for pricing strategy and financial forecasting.

By knowing that the percent increase is 75%, stakeholders can evaluate whether their pricing strategy is optimal for covering expenses and achieving targeted profits.