Problem 1 - Algebra for College Students

Question

Assuming that the rate of inflation is \(4 \%\) per year, the equation \(P=P_{0}(1.04)^{t}\) yields the predicted price \((P)\) of an item in \(t\) years that presently costs \(P_{0}\). Find the predicted price of each of the following items for the indicated years ahead: (a) \(\$ 0.77\) can of soup in 3 years \(\$ 0.87\) (b) \(\$ 3.43\) container of cocoa mix in 5 years \(\$ 4.17\) (c) \(\$ 1.99\) jar of coffee creamer in 4 years \(\$ 2.33\) (d) \(\$ 1.05\) can of beans and bacon in 10 years \(\$ 1.55\) (e) \(\$ 18,000\) car in 5 years (nearest dollar) \(\$ 21,900\) (f) \(\$ 120,000\) house in 8 years (nearest dollar) \(\$ 164,228\) (g) \(\$ 500 \mathrm{TV}\) set in 7 years (nearest dollar) \(\$ 658\)

Step-by-Step Solution

Answer

Prices will be \$0.87, \$4.17, \$2.33, \$1.55, \$21,900, \$164,228, \$658.

Step 1: Understanding the Formula

The equation \( P = P_0(1.04)^t \) is used to calculate the future price \( P \) of an item after \( t \) years, given its present price \( P_0 \) with a consistent annual inflation rate of \( 4\% \).

Step 2: Calculate Future Price for Can of Soup

To find the price after 3 years for a \\(0.77 can of soup: substitute \( P_0 = 0.77 \) and \( t = 3 \) into the formula. Calculate: \[ P = 0.77 \times (1.04)^3 \approx 0.77 \times 1.124864 = 0.866 \] Hence, the price is approximately \\)0.87.

Step 3: Calculate Future Price for Cocoa Mix

Substitute \( P_0 = 3.43 \) and \( t = 5 \) into the formula to calculate the price in 5 years for cocoa mix:\[ P = 3.43 \times (1.04)^5 \approx 3.43 \times 1.2166529 = 4.1749 \] Hence, the price is approximately \$4.17.

Step 4: Calculate Future Price for Coffee Creamer

Using \( P_0 = 1.99 \) and \( t = 4 \), find the price in 4 years for the jar of coffee creamer:\[ P = 1.99 \times (1.04)^4 \approx 1.99 \times 1.16985856 = 2.32808 \] Hence, the price is approximately \$2.33.

Step 5: Calculate Future Price for Can of Beans and Bacon

Substitute \( P_0 = 1.05 \) and \( t = 10 \) into the formula for can of beans and bacon:\[ P = 1.05 \times (1.04)^{10} \approx 1.05 \times 1.48024 = 1.55425 \] Hence, the price is approximately \$1.55.

Step 6: Calculate Future Price for Car

Using \( P_0 = 18000 \) and \( t = 5 \), calculate the price of the car in 5 years:\[ P = 18000 \times (1.04)^5 \approx 18000 \times 1.2166529 = 21900 \] Rounded to the nearest dollar, the price is \$21,900.

Step 7: Calculate Future Price for House

Substitute \( P_0 = 120000 \) and \( t = 8 \) into the formula to calculate the price in 8 years for the house:\[ P = 120000 \times (1.04)^8 \approx 120000 \times 1.3685696 = 164228.352 \] Rounded to the nearest dollar, the price is \$164,228.

Step 8: Calculate Future Price for TV Set

For the TV set, use \( P_0 = 500 \) and \( t = 7 \):\[ P = 500 \times (1.04)^7 \approx 500 \times 1.3168727 = 658.436 \] Rounded to the nearest dollar, the price is \$658.

Key Concepts

Inflation Rate

Inflation refers to the general increase in prices of goods and services over time, which effectively reduces the purchasing power of money. An inflation rate is typically represented as a percentage and indicates how much prices are expected to rise annually. For instance, if the inflation rate is 4%, this means that, on average, prices are expected to be 4% higher one year from now compared to the current year.
Tracking inflation is crucial because it affects everything from household budgets to the broader economy. Understanding how inflation works can help people make better financial decisions, such as saving for the future or adjusting incomes accordingly.

Future Price Calculation

The calculation of a future price of an item in light of inflation involves predicting how much the item will cost after a certain number of years. This prediction can be made using an exponential growth formula, an effective mathematical model when dealing with constant percentage increases over time.
The formula used is: \[ P = P_0 (1 + r)^t \] Where:

• \( P \) is the future price of the item.
• \( P_0 \) is the present price of the item.
• \( r \) is the inflation rate expressed as a decimal.
• \( t \) is the time in years into the future.

By applying this formula, individuals and businesses can make informed predictions about future spending, investment, and pricing strategies.

Percent Increase

A percent increase is a way to express growth in quantities in percentage terms. It quantifies how much a quantity has grown over time relative to its original size.
In the context of inflation and price increases, when we say the price will increase by a certain percentage, we mean that the future price will be a specific percent more than the original price. This is mathematically described by the factor \(1 + r\), where \(r\) is the percent expressed in decimal form (e.g., 4% inflation becomes 0.04).
Generally, percent increase can be calculated as: \[ \text{Percent Increase} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100\% \] This calculation is fundamental in finance, as it helps measure whether and how fast the value of investments, prices, or sales figures are rising.

Algebraic Formulas

Algebraic formulas are powerful tools in mathematics that help solve problems through symbolic expressions and equations. When dealing with financial calculations like inflation, algebraic formulas provide clear and systematic methods to derive unknown values, such as future prices.
The exponential formula used to project future prices, \( P = P_0(1.04)^t \), is a significant example of an algebraic approach. This formula builds on basic algebraic principles to handle multiple operations like multiplication and exponentiation in a single expression.
Using formulas reduces errors and increases efficiency in calculations, whether you're estimating future prices, calculating interest, or determining returns on investments. It's useful not just in academic contexts but in everyday financial decision-making too.