Find the compound amount for the deposit and the amount of interest earned.
$5800 at 5% compounded quarterly for 2 years
The compound amount after 2 years is $
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (1)
Calculate the quarterly interest rate: i = 4 0.05 = 0.0125 .
Calculate the total number of compounding periods: n = 4 × 2 = 8 .
Calculate the compound amount: A = 5800 ( 1 + 0.0125 ) 8 = 6406.02 .
Calculate the interest earned: I n t eres t = 6406.02 − 5800 = 606.02 . The compound amount is $\boxed{ 6406.02} and the interest earned is $\boxed{ 606.02} .
Explanation
Problem Overview We are given a principal amount of $5800 that is deposited at an annual interest rate of 5%, compounded quarterly for 2 years. We need to find the compound amount after 2 years and the amount of interest earned.
Calculating Quarterly Interest Rate First, we need to determine the quarterly interest rate. Since the annual interest rate is 5% or 0.05, and it is compounded quarterly (4 times a year), the quarterly interest rate is calculated as follows: i = 4 0.05 = 0.0125
Calculating Total Compounding Periods Next, we calculate the total number of compounding periods. Since the investment is for 2 years and it is compounded quarterly, the total number of compounding periods is: n = 4 × 2 = 8
Calculating Compound Amount Now, we can calculate the compound amount (A) using the formula: A = P ( 1 + i ) n where P is the principal amount ( 5800 ) , ii s t h e q u a r t er l y in t eres t r a t e ( 0.0125 ) , an d ni s t h e t o t a l n u mb ero f co m p o u n d in g p er i o d s ( 8 ) . A = 5800 ( 1 + 0.0125 ) 8 A = 5800 ( 1.0125 ) 8 A = 5800 × 1.104486 A = 6406.019386852189 $
Rounding to the nearest cent, the compound amount is $6406.02.
Calculating Interest Earned Finally, we calculate the amount of interest earned by subtracting the principal amount from the compound amount: I n t eres t = A − P I n t eres t = 6406.02 − 5800 I n t eres t = 606.02 Therefore, the amount of interest earned is $606.02.
Final Answer The compound amount after 2 years is $6406.02, and the amount of interest earned is $606.02.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your investments grow over time with compound interest helps you estimate the potential returns and adjust your savings strategy accordingly. Similarly, when taking out a loan, understanding the compound interest can help you assess the total cost of borrowing and compare different loan options to choose the most favorable one. This knowledge empowers you to make sound financial choices and achieve your long-term financial goals.
