Question

You put \(\$ 1,000\) into a savings account that pays an interest rate of 6 percent annually. How many years before your savings doubles?

Short answer

It will take approximately 12 years for the savings to double.

Step-by-step solution

Set Up The Equation

Using the formula for compound interest A = P(1 + r/n)^(nt), and knowing that we want A to be double the initial deposit, we get the following equation: 2 * \$1000 = \$1000 (1 + 0.06/1)^(1t). We simplify this to 2 = (1 + 0.06)^t.

Solve the Equation

To solve for t, we will take the natural logarithm (ln) of both sides of the equation, which allows us to use the property of logs that brings down the exponent. It becomes ln(2) = t*ln(1.06). Divide both sides by ln(1.06) to find t. The calculation gives t ≈ 11.9.

Conclude

The answer means it will take approximately 12 years for the investment to double, rounding up as we can't have partial years.

Key concepts

Savings Accounts

Savings accounts are a secure and convenient way to store money while earning interest over time. When you deposit money into a savings account, the bank pays you for keeping your money with them. This payment comes in the form of interest.

One of the main advantages of savings accounts is that they help your money grow. This growth occurs because the bank adds interest to your account balance, usually on an annual or monthly basis. Over time, this interest compounds, meaning you earn interest on your initial deposit and on any accumulated interest. It acts as a financial growth engine, allowing your savings to increase without additional deposits.

• Savings accounts provide safety since the funds are usually insured by the government up to a certain limit.
• They also offer liquidity, allowing you to easily withdraw or transfer funds when needed.

Choosing a savings account with a favorable interest rate is essential to maximize the benefit from your saved funds. Every percentage point counts in boosting your savings over time.

Interest Rates

Interest rates are critical in determining how much your savings will grow over time. The interest rate is the percentage of the principal, or initial amount, paid to the account holder regularly. For savings accounts, these rates can be influenced by several factors.

An understanding of interest rates will help you make better financial decisions. The rate at which your money grows through interest is usually expressed annually, known as the Annual Percentage Rate (APR).

• Higher interest rates lead to faster growth of your savings through compounding.
• Even small differences in rates can significantly affect the amount of interest earned over a long period.

When considering interest rates, always look for savings accounts with competitive rates that align with your financial goals. Understanding the types of interest—simple and compound—also aids in selecting an account. Compound interest, where the interest is calculated on the initial principal and also on the accumulated interest, accelerates the growth of savings, unlike simple interest which only pays on the initial principal.

Financial Mathematics

Financial mathematics is the backbone of understanding how money grows, especially through savings accounts and investments. It involves using mathematical formulas to calculate the future value of money based on variables like interest rates and time.

The compound interest formula, \( A = P(1 + r/n)^{nt} \), helps calculate the total amount (A) accumulated over time, given an initial deposit (P), interest rate (r), number of times the interest is compounded per year (n), and number of years (t). Solving for any of these variables requires careful understanding of mathematical principles.

• Use logarithms to solve equations where the variable is in the exponent, as seen in the original exercise.
• Understanding exponential growth through time, where the value of savings can increase significantly, especially with higher interest rates.

Grasping the mathematical principles behind financial decisions ensures that you can plan and manage your savings effectively. Financial mathematics allows you to predict the impact of different interest rates and timeframes, thereby helping you achieve your financial goals.