Problem 11 - Prealgebra

Question

A These problems are similar to the examples found in this section. They should be set up and solved in the same way. (Problems 1-12 involve simple interest.) Savings Account A woman deposits \(\$ 800\) in a savings account that pays \(5 \% .\) If she withdraws all the money in the account after 120 days, how much does she withdraw?

Step-by-Step Solution

Answer

She withdraws $813.15.

Step 1: Understanding Simple Interest Formula

The formula for simple interest is \( I = P \times r \times t \), where \( I \) is the interest earned, \( P \) is the principal amount (initial amount deposited), \( r \) is the annual interest rate expressed as a decimal, and \( t \) is the time the money is invested or borrowed for, in years.

Step 2: Converting Interest Rate to Decimal

The annual interest rate provided is \( 5\% \). To convert this percentage to a decimal for use in the formula, divide by 100: \( r = \frac{5}{100} = 0.05 \).

Step 3: Calculating Time in Years

The time provided is in days (120 days). To convert this to years, divide by the number of days in a year (assuming 365 days in a year): \( t = \frac{120}{365} \approx 0.3288 \) years.

Step 4: Calculating the Simple Interest Earned

Substitute the known values into the simple interest formula: \( I = 800 \times 0.05 \times 0.3288 \). Calculating this gives the interest: \( I \approx 13.15 \).

Step 5: Calculating Total Withdrawal Amount

Add the interest earned to the initial principal amount to find the total amount withdrawn: \( \text{Total withdrawal} = 800 + 13.15 = 813.15 \).

Key Concepts

Savings Account

A savings account is a type of bank account where you can keep your money safe while earning some interest. When you deposit money into a savings account, the bank pays you interest on this amount. This means that over time, your savings can grow, even if you don't add anything more to them. The interest rate is usually given as a percentage per year. For example, if a bank offers a 5% interest rate, it means you'll earn 5% on your deposited amount each year.

A key benefit of a savings account is its liquidity. This means you can access your money relatively easily, unlike some other investments where your money might be tied up for a longer time. Nevertheless, a savings account is not meant for daily transactions but for storing funds you plan to save over the medium to long term.

Interest Rate Calculation

Interest rate calculation is crucial for determining how much interest your savings will generate over a specific period. It involves converting percentage rates into a usable format for calculations. Once the interest rate is given as a percentage, you must convert this into a decimal for use in financial formulas.

Here's how you do it:

• Take the percentage rate.
• Divide it by 100 to convert it into a decimal format. For example, a 5% interest rate becomes 0.05 when divided by 100.

This conversion lets you accurately calculate the interest earned over time using the simple interest formula.

Time Conversion to Years

To use the simple interest formula, time needs to be in terms of years, even if the problem gives you time in days. This conversion is important for accuracy in calculations.

To convert time from days to years, follow these steps:

• Take the number of days.
• Divide by 365, which is the number of days in a year. For example, if you have 120 days, divide 120 by 365.
• The result is that time in years. In this case, it would be approximately 0.3288 years.

This conversion ensures that the simple interest formula reflects the true period for which the interest applies, yielding accurate financial results.