Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Today a typical family of four spends $$\$ 600 /$$ month for food. If inflation occurs at the rate of \(5 \%\) lyear over the next \(6 \mathrm{yr}\), how much should the typical family of four expect to spend for food 6 yr from now?

Step-by-Step Solution

Answer

The typical family of four should expect to spend approximately \(\$802.58\) per month on food after 6 years if the inflation rate remains at 5% per year.

Step 1: Identify the known variables

We are given: - Present food-cost spending: \(P = \$600\) - Annual inflation rate: \(R = 5\% = 0.05\) - Number of years: \(t = 6 \text{ years}\) We need to find the future food-cost spending, \(F\), after 6 years.

Step 2: Calculate the future value using the Compound Interest formula

We will use the Compound Interest formula to calculate the future food-cost spending: \(F = P(1 + R)^t\) where: - \(F\) is the future value (the food-cost spending after 6 years) - \(P\) is the present value (the current food-cost spending) - \(R\) is the annual inflation rate - \(t\) is the number of years

Step 3: Substitute values into the formula

Now, substitute the given values into the formula: \(F = \$600(1 + 0.05)^6\)

Step 4: Calculate the future value

Now, calculate the future food-cost spending: \(F = \$600(1.05)^6 \approx \$802.58\)

Step 5: Interpret the result

The typical family of four should expect to spend approximately \(\$802.58\) per month on food after 6 years if the inflation rate remains at 5% per year.

Key Concepts

Understanding Inflation Rate

Inflation is a term that describes the increase in prices over time. Think of it as how much pricier things get year after year. When you hear a news report that says inflation is 5% per year, it means that, on average, prices for goods and services are expected to increase by 5% each year.
This affects your buying power. If you hold onto your money without investing it at a rate equal to inflation, you could buy less with the same amount of money in the future. For instance:

• If a loaf of bread costs $2 today, with a 5% inflation rate, you would expect it to cost $2.10 next year.
• This means you'd need extra money to keep up with rising costs.

Understanding inflation helps families plan better for future expenses, especially for fundamental needs like food.

Future Value Calculation and Compound Interest

Future value calculation helps us know how much something will cost or be worth in the future. We used the compound interest formula here, which works for various situations, not just for bank savings. It helps foresee changes in costs or value due to inflation.
The formula for future value using compound interest is:

• \( F = P(1 + R)^t \) where:
• \( F \) is the future value
• \( P \) is the present value
• \( R \) is the rate (expressed as a decimal)
• \( t \) is the time in years

This formula shows how your money grows. Even if we're not saving or investing money, it helps us understand how much more we'll need later. Through this calculation:

• Families can anticipate increases in daily expenses like groceries.
• The formula adapts to any changing rates and periods.

This adaptability makes it a handy tool for almost any financial projection.

The Importance of Financial Planning

Financial planning helps manage your money wisely over time. When people prepare for how much future expenses might be, they're better able to handle the changes inflation brings. By predicting future costs through calculations like the one shown, you can make more informed choices today.
To maintain a stable financial standing:

• Set goals and understand projected incomes and expenses.
• Identify needs versus wants, prioritizing essential expenses.
• Plan your savings or investments with inflation in mind, ensuring money grows at a rate similar to or faster than inflation.

Effective financial planning involves forecasting like we did in our exercise. It leads to better decision-making for today, knowing you’re prepared for tomorrow. Whether it's about grocery bills, education, or housing costs, being proactive helps balance current spending with future savings.