Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Minimum wage In 1971 the minimum wage in the United States was \(\$ 1.60\) per hour. Assuming that the rate of inflation is \(5 \%\) per year, find the equivalent minimum wage in the year 2010 .

Step-by-Step Solution

Answer

In 2010, the equivalent minimum wage is approximately $12.69.

Step 1: Understand the Inflation Formula

To solve for the equivalent minimum wage in a future year with an annual inflation rate, we use the formula for compounded inflation: \[ P = P_0 (1 + r)^t \] where \( P \) is the future value, \( P_0 \) is the initial value, \( r \) is the annual inflation rate, and \( t \) is the number of years.

Step 2: Identify Given Values

From the problem statement, we have: - Initial minimum wage, \( P_0 = 1.60 \) dollars- Inflation rate, \( r = 5\% = 0.05 \)- Time period, \( t = 2010 - 1971 = 39 \) years.

Step 3: Calculate the Equivalent Minimum Wage

Plug the values into the compound interest formula: \[ P = 1.60 \times (1 + 0.05)^{39} \] Calculating this gives: \[ P = 1.60 \times (1.05)^{39} \] We need to calculate \((1.05)^{39}\) first, then multiply by 1.60.

Step 4: Compute the Future Value

Calculate \((1.05)^{39}\). This is approximately 7.93. Now compute:\[ P = 1.60 \times 7.93 \approx 12.69 \]So, the equivalent minimum wage in 2010, considering 5% inflation, would be approximately 12.69 dollars.

Key Concepts

Understanding the Compound Interest Formula

The compound interest formula is not just for banking and investments but is useful in calculating inflation as well. When prices increase due to inflation, it's as if our money in the past grows "interest" that reduces its purchasing power over time. To estimate the future value of money, we use the formula:\[ P = P_0 (1 + r)^t \]Here:

• \( P \) stands for the future value we want to estimate.
• \( P_0 \) is the present or initial value, such as the starting wage or price.
• \( r \) is the annual inflation rate expressed as a decimal.
• \( t \) is the number of years that have passed.

By applying this formula, you essentially allow the value of money to "grow" as if being compounded annually by the rate \( r \). This method effectively captures how inflation compounds and increases over time.

Exploring the Annual Inflation Rate

The annual inflation rate represents how much prices overall are expected to increase each year. It is an average rate of price changes typically defined by government reports or economic studies. To use it effectively in calculations:

• First, convert the percentage to a decimal for the formula. For instance, 5% becomes 0.05.
• The rate compounds, meaning each year's increase is based on the new total of the previous year.

If the inflation rate is constant over many years, you can simplify calculations using the compound inflation formula. This gives a more realistic estimate than just multiplying each year's rate. Thus, understanding how an annual inflation rate compounds lets you anticipate future changes in costs or wages. This is crucial for financial planning and historical price comparisons.

Calculating Future Value Using Inflation

Future value calculations help us understand what a certain amount of money today will be worth in the future after considering inflation. It reflects purchasing power in economic terms. To calculate the future value:

• Start with the initial value, such as a minimum wage figure.
• Use the yearly inflation rate and the number of years to find the compounded value.
• Plug these numbers into the compound interest formula \[ P = P_0 (1 + r)^t \].

For instance, starting with an initial minimum wage from 1971, using an inflation rate and knowing the future year, you can calculate the future equivalent wage. This involves raising the inflation factor to the power of the number of years, multiplying it with the initial amount. Calculating the future value helps in assessing the real value of money across time periods, crucial for planning and adjustment of salaries or budgets in response to inflation.