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Chapter 10: Problem 24

What will be the price of a $$\$ 20,000$$ car in 5 years if the inflation rate is \(6 \% ?\)

Short Answer

Expert verified
The price of the car in 5 years would be approximately \$26,801.32 assuming an inflation rate of 6% per annum

Step by step solution

01

Understand the problem and write down the values

The student must interpret the problem correctly and write down the given values: \n- The present value of the car: \(\$ 20,000) \n- The annual inflation rate: \(6%\) \n- The number of years: \(5\)
02

Convert the inflation rate into decimal

We need to convert the percentage value of the inflation rate into a decimal for the calculations. To do this, divide the inflation rate by \(100\), so \(6\%\) becomes \(0.06\)
03

Calculate the future value

The formula to calculate the future value based on inflation is: \(Future Value = Present Value * (1 + inflation rate) ^ {number of years}\) \nBy substituting the known values into the formula, Future Value of the car would be \(Future Value = $20,000*(1+0.06)^5\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Future Value
To determine what the price of an item, like a car, will be after a certain period considering inflation, we need to calculate its future value. The future value represents how much money will be worth in the future given a specific interest or inflation rate. For example, if you have a car worth $20,000 today and you want to know its price in 5 years with a 6% annual inflation rate, you need to use the future value formula.
The formula is:
  • \[ \text{Future Value} = \text{Present Value} \times (1 + \text{inflation rate}) ^ \text{number of years} \]
This formula helps you understand how money grows over time with inflation or interest. By using it, you can plan better for financial goals that depend on predicting future prices or values.
Present Value
Understanding present value is crucial when trying to determine the future cost of goods and services. Present value refers to the current worth of an asset or sum of money. In our example, the present value is the current price of the car: $20,000.
Present value allows us to think about how much money we need today to reach a financial goal in the future, considering inflation or interest. It is the starting point for any future value calculation and is pivotal in financial planning. By knowing the present value and applied rate, one can move to further calculations with confidence.
Inflation Rate
The inflation rate is a measure of how prices increase over a specific period, often expressed annually. It shows the percentage change in prices, affecting purchasing power. In the exercise, a 6% inflation rate means that prices are expected to rise by 6% each year.
To use this rate in mathematical calculations, it must first be converted from a percentage to a decimal. Hence, 6% is represented as 0.06. Recognizing how inflation impacts your finances allows you to better predict future expenses, adjust budgets, and plan investments to ensure that you can afford future costs of living.
Mathematical Problem Solving
Mathematical problem-solving involves understanding how to apply formulas and logical reasoning to real-life problems. In solving our exercise, we followed specific steps to calculate the future value of the car.
Here's a summary of the process:
  • Identify given values: present value, inflation rate, and time period.
  • Convert the inflation rate from a percentage to a decimal.
  • Use the future value formula to make the calculation.
  • Interpret the result to understand its meaning in context.
This approach makes math problems more manageable, helping you make informed predictions that are vital for planning and decision-making.

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Most popular questions from this chapter

Mr. Tong puts aw ay $$\$ 500$$ per month for 10 years in an account that earns \(9.3 \%\). After 10 years, he decides to withdraw $$\$ 1,000$$ per month. If the interest rate stays the same, how long will it take Mr. Tong to deplete the account?

Tasha signs a note for a discounted loan agreeing to pay $$\$ 1200$$ in 8 months at an \(18 \%\) discount rate. Determine the amount of the discount and the proceeds to her.

At an interest rate of \(8 \%\) compounded continuously, how many years will it take to double your money? Hint: You may do this on your calculator by trial and error.

The City Library has ordered a new computer system costing $$\$ 158,000$$. The system will be delivered in 6 months, and the full amount will be due 30 days after delivery. How much should be deposited today into an account paying \(7.5 \%\) compounded monthly to have $$\$ 158,000$$ in 7 months?

Find the effective interest rate for an account paying \(7.2 \%\) compounded quarterly.
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