The principal represents an amount of money deposited in a savings account subject to compound interest at the given rate. Answer parts (a) and (b).
| Principal | Rate | Compounded | Time |
| --------- | ---- | ---------- | ---- |
| $7500 | 4.5% | monthly | 5 years |
(i) Click the icon to view some finance formulas.
a. Find how much money there will be in the account after the given number of years.
The amount of money in the account after 5 years is $ [ ]. (Round to the nearest hundredth as needed.)
Answer (1)
Identify the principal P = $7500 , annual interest rate r = 0.045 , number of times interest is compounded per year n = 12 , and the number of years t = 5 .
Use the compound interest formula: A = P ( 1 + n r ) n t .
Substitute the values into the formula: A = 7500 ( 1 + 12 0.045 ) ( 12 × 5 ) .
Calculate the final amount: A ≈ 9388.47 .
Explanation
Understanding the Problem We are given a principal amount of $7500 deposited in a savings account with an annual interest rate of 4.5% compounded monthly for 5 years. We need to find the amount of money in the account after 5 years.
Identifying the Formula To solve this, we will use the compound interest formula: A = P ( 1 + r / n ) n t , where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
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.
Assigning Values In this case, we have:
P = $7500
r = 4.5% = 0.045
n = 12 (compounded monthly)
t = 5 years
Plugging in the Values Now, we plug these values into the formula: A = 7500 ( 1 + 0.045/12 ) ( 12 × 5 )
Calculating the Amount First, calculate the monthly interest rate: r / n = 0.045/12 = 0.00375 Next, calculate the number of compounding periods: n t = 12 × 5 = 60 Now, substitute these values back into the formula: A = 7500 ( 1 + 0.00375 ) 60 A = 7500 ( 1.00375 ) 60
Final Calculation Calculating ( 1.00375 ) 60 gives approximately 1.251796. Therefore, A = 7500 × 1.251796 ≈ 9388.47
Final Answer The amount of money in the account after 5 years is approximately $9388.47.
Examples
Compound interest is a powerful concept that applies to many real-life financial situations. For example, when you deposit money into a savings account, the bank pays you interest, which is often compounded. Understanding compound interest can help you make informed decisions about saving and investing. Another example is when you take out a loan; the interest you pay on the loan is also often compounded. Knowing how compound interest works can help you understand the true cost of borrowing money and manage your debts effectively. It's also used in retirement planning to project the growth of investments over time.
