7. A woman wishes to invest $5,800. She can buy savings bonds which pay simple interest at the rate of 8.5% per annum or she can start a savings account which pays compound interest at the same rate. Calculate the difference in the amounts of the two investments at the end of five years.
To calculate the difference in the amounts of the two investments at the end of five years, we need to calculate the amounts separately for each investment.
For the savings bonds:
Simple interest formula: A = P(1 + rt)
Where:
A = final amount
P = principal amount = $5,800
r = interest rate per period = 8.5% = 0.085
t = number of periods = 5 years
A = 5800(1 + 0.085 * 5)
A = 5800(1 + 0.425)
A = 5800(1.425)
A = $8,215
For the savings account:
Compound interest formula: A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount = $5,800
r = interest rate per period = 8.5% = 0.085
n = number of compounding periods per year (assuming it's compounded annually)
t = number of years = 5
A = 5800(1 + 0.085)^5
A = 5800(1.085)^5
A = 5800(1.498357)
A = $8,684.76
The difference in the amounts of the two investments at the end of five years is:
$8,684.76 - $8,215 = $469.76
To calculate the difference in the amounts of the two investments at the end of five years, we'll need to calculate the amount for each investment separately.
Let's start with the savings bonds:
The formula to calculate simple interest is:
Interest = Principal * Rate * Time
Given:
Principal = $5,800
Rate = 8.5% per annum
Time = 5 years
Using the formula, we can calculate the interest earned on savings bonds:
Interest = $5,800 * 8.5% * 5 = $2,467.50
Adding the interest to the principal gives us the total amount at the end of five years:
Total amount with savings bonds = $5,800 + $2,467.50 = $8,267.50
Now let's calculate the amount in the savings account with compound interest:
The formula to calculate compound interest is:
Amount = Principal * (1 + Rate)^Time
Given:
Principal = $5,800
Rate = 8.5% per annum
Time = 5 years
Using the formula, we can calculate the amount in the savings account:
Amount = $5,800 * (1 + 8.5%)^5
Amount = $5,800 * (1 + 0.085)^5
Amount ≈ $8,719.46 (rounded to 2 decimal places)
Now, let's calculate the difference in the amounts of the two investments at the end of five years:
Difference = Amount in savings account - Total amount with savings bonds
Difference = $8,719.46 - $8,267.50
Difference ≈ $451.96 (rounded to 2 decimal places)
Therefore, the difference in the amounts of the two investments at the end of five years is approximately $451.96.