Problem 8 - Algebra for College Students

Question

If the price of a pound of coffee is \(\$ 3.20\) and the projected rate of inflation is \(5 \%\) per year, how much per pound should we expect coffee to cost in 5 years? Express your answer to the nearest cent. $4.08

Step-by-Step Solution

Answer

The expected cost is \( \$4.08 \) per pound.

Step 1: Understand the problem

We need to determine how much the price of coffee will be in 5 years, given a current price of \( \$3.20 \) per pound and an annual inflation rate of \( 5\% \).

Step 2: Determine the formula to use

To calculate the future price of an item considering inflation, we use the formula: \( \text{future price} = \text{current price} \times (1 + \text{inflation rate})^n \), where \( n \) is the number of years into the future.

Step 3: Insert known values into the formula

Here, the current price is \( 3.20 \), the inflation rate is \( 5\% \), which is equivalent to \( 0.05 \) as a decimal, and \( n = 5 \), so we get: \[ 3.20 \times (1 + 0.05)^5 \]

Step 4: Calculate the future price

Evaluate the expression by calculating \( 1 + 0.05 = 1.05 \) and raising it to the power of 5: \( 1.05^5 \approx 1.2762815625 \). Then, multiply by \( 3.20 \):\[ 3.20 \times 1.2762815625 \approx 4.0841 \]

Step 5: Round the result

Round the result to the nearest cent, which gives us \( \$4.08 \).

Key Concepts

Algebraic Formulas

Algebraic formulas are expressions composed of variables and constants, using operations like addition, subtraction, multiplication, and division. They are used to describe relationships between quantities and to calculate unknown values.

In the exercise, we used an algebraic formula to determine the future price of coffee considering inflation. The formula applied is:

• \( \text{future price} = \text{current price} \times (1 + \text{inflation rate})^n \)

This formula incorporates both a known value (initial coffee price) and variables (inflation rate and number of years). This is a powerful aspect of algebraic formulas, as they allow us to predict unknown quantities under specific conditions.

By inserting given values into our formula, we can solve for the future cost, reflecting how algebraic manipulation is key to solving real-world problems like inflation adjustments.

Percentage Increase

Percentage increase measures how much a quantity grows compared to its original size. It's commonly used to express growth, often in prices or populations. The inflation rate is a typical example of a percentage increase, signifying how prices are expected to rise over time.

In our scenario, inflation is projected to increase the coffee price by 5% annually. Understanding what this means is crucial:

• A 5% increase means for every dollar, an extra 5 cents is added each year.
• This rate is compounded over multiple years, adding to the previous year’s price.

To express this mathematically, we converted the percentage into a decimal (5% becomes 0.05), and incorporated it into our formula. Hence, the future price is calculated as a series of compounded percentage increases, showing how original prices are impacted incrementally each year.

Future Value Calculations

Future value calculations help determine the value of an investment or item at a future date, accounting for interest rates or inflation. This is particularly valuable in financial planning, allowing individuals to anticipate the cost of items over time.

The calculation used in the exercise predicts the future price of coffee after 5 years of inflation:

• Current price: $3.20
• Inflation rate: 5%, or 0.05 in decimal form
• Time period: 5 years

Using the formula \( \text{future price} = \text{current price} \times (1 + \text{inflation rate})^n \), we calculated the price as \( 3.20 \times 1.05^5 \), demonstrating how repeated compounding affects future value. The result—rounded to the nearest cent—provides a practical sense of how consistent inflation impacts costs, making future value calculations a crucial component in economic forecasting.