It takes 10 years for a $100 monthly bank deposit to amount to $15,528.23 when invested at 5% interest compounded monthly. Round to the nearest place value, then express the amount $15,528.23 in the form of a * 10 ^ n where a is single digit.
To find the interest rate, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (monthly deposit)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = number of years
In this case, we know the future value (A = $15,528.23), the principal amount (P = $100 monthly), the number of times compounded per year (n = 12) and the number of years (t = 10).
We need to solve for the interest rate (r):
15,528.23 = 100(1 + r/12)^(12*10)
Simplifying the equation, we get:
15,528.23/100 = (1 + r/12)^(120)
155.2823 = (1 + r/12)^(120)
Taking the 120th root of both sides:
(1 + r/12) = (155.2823)^(1/120)
1 + r/12 = 1.0039806
r/12 = 1.0039806 - 1
r/12 = 0.0039806
r = 12 * 0.0039806
r ≈ 0.0478
Therefore, the interest rate is approximately 0.0478 or 4.78%.
Next, let's express $15,528.23 in the form of a * 10 ^ n where a is a single digit:
$15,528.23 can be written as 1.552823 * 10^4
To find the amount $15,528.23 in the form of a * 10 ^ n, where a is a single digit, we need to round the number to the nearest place value and determine the exponent.
Given:
Principal (P) = $100
Interest rate (r) = 5% or 0.05 (as a decimal)
Time (t) = 10 years
The formula to calculate the future value (A) of an investment with compound interest is:
A = P(1 + r/n)^(nt)
Where:
n is the number of times the interest is compounded per year.
Since the interest is compounded monthly, n = 12 (12 months in a year).
Plugging in the values, we have:
A = $100(1 + 0.05/12)^(12*10)
A = $100(1.00416666667)^(120)
A ≈ $100(1.6487212707)
A ≈ $164.87212707
Now, we need to round $164.87212707 to the nearest place value:
$164.87212707 rounded to the nearest hundredth = $164.87
Finally, expressing $164.87 in the form of a * 10 ^ n:
$164.87 = 1.6487 * 10 ^ 2
Therefore, the amount $15,528.23 can be expressed as 1.6487 * 10 ^ 2 where a is the single-digit value 1.