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Chapter 5: Problem 3

Find the accumulated amount at the end of \(9 \mathrm{mo}\) on an \(\$ 800\) deposit in a bank paying simple interest at a rate of \(6 \% /\) year.

Short Answer

Expert verified
The accumulated amount at the end of 9 months on an $800 deposit in a bank paying simple interest at a rate of 6% per year is $836.

Step by step solution

01

Convert the given time to years

As the interest rate is given in per annum (per year), we need to convert the given time (9 months) to years. To do this, we will divide the given time by the total number of months in a year (12 months). Time in years = \(\frac{9 \, \text{months}}{12 \, \text{months/year}} = \frac{3}{4}\) years. Now we have the time in years, which is \(\frac{3}{4}\) years.
02

Calculate the Simple Interest

We can now calculate the simple interest using the formula: Simple Interest = Principal Amount 脳 Interest Rate 脳 Time in years By plugging in the values, we get, Simple Interest = \(800\, \$ \times \,0.06\,\times \,\frac{3}{4}\, \text{years} \) Simple Interest = \(800\, \$ \times \,0.06\,\times \,\frac{3}{4} = 36 \, \$ \) So, the simple interest accumulated in 9 months is \(36\$.\)
03

Calculate the Accumulated Amount

Now, we will find the accumulated amount which is the sum of the principal amount and the simple interest. We can calculate this using the formula: Accumulated Amount = Principal Amount + Simple Interest By plugging in the values, we get, Accumulated Amount = \(800 \$ + 36 \$ \) Accumulated Amount = $836 \$$ Therefore, the accumulated amount at the end of 9 months on an 800\( deposit in a bank paying simple interest at a rate of 6% per year is \)836.

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Key Concepts

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

Interest Rate
When we talk about the interest rate in the context of simple interest calculations, we're looking at the percentage of the principal amount that the bank or financial institution will pay over time. This rate is typically expressed on an annual basis, referred to as the 'annual percentage rate' or APR. For example, if a bank offers a 6% interest rate, this means for every \(100 deposited, the bank will pay \)6 in interest per year.

In our exercise, the interest rate is 6% per year. It's crucial for students to understand that the interest rate directly influences how much extra money they will earn (or owe, if they are the ones borrowing) over the life of the investment or loan. The higher the rate, the more money the deposited principal amount will generate over time. However, it's important to note that the given rate often needs to be adjusted based on the actual time the money is invested, which is commonly less than a full year.
Accumulated Amount
The accumulated amount refers to the total value of an investment after the interest has been added to the initial principal amount. It's what the depositor will receive at the end of the investment period. To calculate the accumulated amount for simple interest, the formula is quite straightforward: Add the initial deposit (the principal) to the interest earned over the time period specified.

The key takeaway for students here is that the accumulated amount will always be greater than the principal amount, assuming the interest rate is positive. In our exercise, the accumulated amount came to \(836 after a 9-month period on an \)800 initial deposit with a 6% interest rate. This is the sum of the original principal and the interest earned, demonstrating the growth of the investment over time.
Principal Amount
The principal amount, often simply called 'the principal,' is the initial sum of money that is either invested or borrowed. It is the base figure on which interest is calculated. If you deposit \(800 into a savings account, for instance, that \)800 is the principal. It is essential for students to recognize that the principal amount remains unchanged during the investment period when it comes to simple interest; only the interest accrues or compounds over time.

In our problem, we start with the principal amount of $800. It is this amount which, when multiplied by the interest rate and the time factor, enables us to find the interest earned. Understanding what constitutes the principal helps students to identify the starting point of any financial calculation involving interest.

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Most popular questions from this chapter

Having received a large inheritance, Jing-mei's parents wish to establish a trust for her college education. If 7 yr from now they need an estimated \(\$ 70,000\), how much should they set aside in trust now, if they invest the money at \(10.5 \%\) compounded quarterly? Continuously?

Auro FiNANCING Paula is considering the purchase of a new car. She has narrowed her search to two cars that are equally appealing to her. Car A costs \(\$ 28,000\), and car \(B\) costs \(\$ 28,200\). The manufacturer of car A is offering \(0 \%\) financing for 48 months with zero down, while the manufacturer of car \(\mathrm{B}\) is offering a rebate of \(\$ 2000\) at the time of purchase plus financing at the rate of \(3 \%\) year compounded monthly over 48 mo with zero down. If Paula has decided to buy the car with the lower net cost to her, which car should she purchase?

Jodie invested \(\$ 15,000\) in a mutual fund 4 yr ago. If the fund grew at the rate of \(9.8 \%\) lyear compounded monthly, what would Jodie's account be worth today?

Use logarithms to solve each problem. Find the interest rate needed for an investment of \(\$ 5000\) to grow to an amount of \(\$ 6000\) in 3 yr if interest is compounded continuously.

Use logarithms to solve each problem. How long will it take an investment of \(\$ 8000\) to double if the investment earns interest at the rate of \(8 \%\) compounded continuously?
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