Hugo invests $10,000 with a bank at 0.2% interest. To the nearest dollar, how much money will be in the account after three years if the money is compounded quarterly?
Answer (2)
Apply the compound interest formula: A = P ( 1 + n r ) n t .
Substitute the given values: P = 10000 , r = 0.002 , n = 4 , and t = 3 .
Calculate the future value: A = 10000 ( 1 + 4 0.002 ) ( 4 ) ( 3 ) ≈ 10060.17 .
Round to the nearest dollar: The amount of money in the account after 3 years is 10060 .
Explanation
Understanding the Problem Let's analyze the problem. We are given a principal amount of $10,000 invested at an annual interest rate of 0.2%, compounded quarterly. We need to find the amount of money in the account after 3 years.
Stating the Compound Interest Formula We will use the compound interest formula to solve this problem. The formula is: A = P ( 1 + n r ) n t where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (the initial deposit or loan amount)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years the money is invested or borrowed for
Substituting the Values Now, let's plug in the given values:
P = 10000
r = 0.2% = 0.002
n = 4 (compounded quarterly)
t = 3 years So, the formula becomes: A = 10000 ( 1 + 4 0.002 ) ( 4 ) ( 3 )
Simplifying the Fraction Let's simplify the expression inside the parentheses: 4 0.002 = 0.0005 So we have: A = 10000 ( 1 + 0.0005 ) ( 4 ) ( 3 )
Simplifying the Exponent Now, let's simplify the exponent: ( 4 ) ( 3 ) = 12 So we have: A = 10000 ( 1.0005 ) 12
Calculating the Power Now, we calculate ( 1.0005 ) 12 :
( 1.0005 ) 12 ≈ 1.0060165275 So we have: A = 10000 × 1.0060165275 ≈ 10060.165275
Rounding to the Nearest Dollar Finally, we round the result to the nearest dollar: A ≈ 10060 Therefore, the amount of money in the account after 3 years will be approximately $10,060.
Examples
Compound interest is a powerful concept used in many real-life financial situations. For example, when you deposit money into a savings account, the bank pays you interest, which is often compounded. Similarly, when you take out a loan, the interest you owe is also often compounded. Understanding how compound interest works can help you make informed decisions about saving, investing, and borrowing money. It's also used in calculating the future value of investments, such as retirement accounts, and in determining the cost of loans, like mortgages.
Using the compound interest formula, after three years, the amount in Hugo's account will be approximately $10,060, given a principal of $10,000 and an annual interest rate of 0.2% compounded quarterly.
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