Awnser: Here’s how to calculate the total amount and interest earned on Miguel’s CD:
(a) Total Amount After 5 Years
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
Let’s plug in Miguel’s information:
P = $1100
r = 1.49% = 0.0149 (convert percentage to decimal)
n = 365 (compounded daily)
t = 5 years
A = 1100(1 + 0.0149/365)^(365*5) A = 1100(1.0000408219)^1825 A ≈ 1184.74
Therefore, Miguel will have approximately $1184.74 in his account after 5 years.
(b) Total Interest Earned
To find the total interest earned, subtract the initial principal from the final amount:
Interest = A - P Interest = $1184.74 - $1100 Interest ≈ $84.74
Therefore, Miguel will earn approximately $84.74 in interest after 5 years.
Step-by-step explanation:
Let’s break down how we calculate the total amount and interest earned on Miguel’s CD step-by-step:
(a) Total Amount After 5 Years
We’ll use the compound interest formula: A = P(1 + r/n)^(nt)
Step 1: Understand the Formula
A: This is what we want to find – the total amount in the account after 5 years.
P: The principal amount Miguel invests, which is $1100.
r: The annual interest rate, but expressed as a decimal. So, 1.49% becomes 0.0149 (divide by 100).
n: The number of times interest is compounded per year. Since it’s compounded daily, n = 365.
t: The time in years the money is invested, which is 5 years.
Step 2: Plug in the Values
A = 1100 (1 + 0.0149 / 365)^(365 * 5)
Step 3: Simplify Inside the Parentheses
A = 1100 (1 + 0.0000408219)^(365 * 5)
Step 4: Calculate the Exponent
A = 1100 (1.0000408219)^1825
Step 5: Calculate the Final Amount
A ≈ 1184.74
Therefore, Miguel will have approximately $1184.74 in his account after 5 years.
(b) Total Interest Earned
Step 1: Understand the Calculation
To find the interest earned, we subtract the initial principal from the final amount we just calculated.
Step 2: Calculate the Interest
Interest = A - P Interest = $1184.74 - $1100 Interest ≈ $84.74
Therefore, Miguel will earn approximately $84.74 in interest after 5 years.