Question

A certain account pays 1.5 percent compound interest every 3 months. A person invested an initial amount and did not invest any more money in the account after that. If after exactly 5 years, the amount of money in the account was \(T\) dollars, which of the following is an expression for the original number of dollars invested in the account? A $$(1.015)^{4} T$$ B $$(1.015)^{15} T$$ C $$(1.015)^{20} T$$ D $$\frac{T}{(1.015)^{15}}$$ E $$\frac{T}{(1.015)^{20}}$$

Short answer

Option E: \(\frac{T}{(1.015)^{20}}\).

Step-by-step solution

- Determine the number of compounding periods

Interest is compounded every 3 months. In 1 year, there are 4 quarters (3 months each). Therefore, in 5 years, there are a total of \[5 \times 4 = 20\] compounding periods.

- Identify the interest rate per compounding period

The interest rate for each 3-month period is 1.5 percent. Convert this percentage to a decimal to use in further calculations: \[1.5\% = 0.015\]

- Use the compound interest formula

The amount in the account after compounding can be described by the compound interest formula: \[A = P (1 + r)^n\] where \(A\) is the amount after 5 years, \(P\) is the initial principal, \(r\) is the interest rate per period, and \(n\) is the number of periods.

- Substitute the given values

Given \(r = 0.015\) (interest rate per period), \(n = 20\) (number of periods), and \(A = T\) (final amount), the equation becomes \[T = P (1.015)^{20}\]

- Solve for the initial principal (P)

Rearrange the equation to solve for \(P\): \[P = \frac{T}{(1.015)^{20}}\]

- Select the correct answer

Comparing with the given options, the expression for the initial amount invested is: \(\frac{T}{(1.015)^{20}}\). Therefore, the correct answer is option E.

Key concepts

compound interest

Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is only computed on the principal amount, compound interest allows wealth to grow faster over time. This growth is often described using the formula: \( A = P (1 + r)^n \) , where:

• \(A\) is the amount of money accumulated after n periods, including interest.
• \(P\) is the principal amount (the initial sum of money).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of compounding periods.

In the given exercise, the interest rate is compounded quarterly (every 3 months), which affects how the formula is applied. This frequency means the principal amount grows more quickly compared to annual compounding.

financial mathematics

Financial mathematics deals with the application of mathematical methods to financial problems. It includes various topics such as compound interest, growth rates, annuities, and loan amortization. The key to solving financial mathematics problems is understanding the formulas and knowing how to manipulate variables.
In this exercise, the problem-focused on the calculation of the initial investment using compound interest. The steps involved are typical of financial math problems: identifying the number of compounding periods, determining the interest rate per period, and applying the compound interest formula. Financial mathematics is essential for various applications, including investments, savings, and understanding the cost of borrowing money.

exponential growth

Exponential growth refers to growth that increases at a consistent rate over time. This type of growth is represented mathematically by an exponential function, where the amount grows by a constant percentage each period.

• The equation \( A = P (1 + r)^n \) embodies exponential growth, where each compounding period adds a constant percentage to the accumulated total.
• For example, in the exercise provided, the amount in the account grows by 1.5% every three months.

With every compounding period, the interest is calculated on a progressively larger amount. Therefore, exponential growth is significantly faster than linear growth, and understanding this concept is crucial for accurate financial planning and forecasting.

educational test preparation

Adequate educational test preparation is crucial for performing well in standardized tests like the GMAT. To prepare effectively:

• Understand key concepts: Ensure you grasp fundamental mathematical principles, such as compound interest and financial mathematics.


• Practice problems: Regularly solve practice exercises to become familiar with various problem types and their solutions.


• Review step-by-step solutions: Go through detailed solutions to understand where you might have made mistakes and learn how to correct them.


• Time management: Practice under timed conditions to manage your time efficiently during the actual test.

By breaking down complex concepts into simpler parts and consistently practicing, you can improve your problem-solving skills and boost your confidence for test day.