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 a×10^n where a is single digit.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the ending amount
P = the principal (initial deposit)
r = the interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, P = $100, r = 5% = 0.05, n = 12 (compounded monthly), and t = 10.
Plugging in these values, we have:
A = 100(1 + 0.05/12)^(12*10)
Simplifying:
A = 100(1 + 0.00416666667)^(120)
A = 100(1.00416666667)^(120)
Using a calculator or a computer program, we find that A is approximately equal to $164.6299711.
Rounding to the nearest penny, A is $164.63.
Now, we need to express $164.63 in the form of a×10^n, where a is a single digit.
Since $164.63 is less than $1,000, we can express it in the form a×10^2, where a is the hundredth's place digit.
Thus, $164.63 can be written as 1.6463×10^2.
To find the amount of $15,528.23 after 10 years, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (monthly deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, P = $100, r = 5% (0.05 as a decimal), n = 12 (compounded monthly), and t = 10. Plugging these values into the formula, we get:
A = 100(1 + 0.05/12)^(12*10)
Calculating this expression, we find:
A ≈ $15,528.23
Rounding this amount to the nearest place value, we have $15,528.23.
To express $15,528.23 in the form of a×10^n where a is a single digit, we need to convert it to scientific notation. Start by moving the decimal point until there is only one digit to the left of it:
$15,528.23 = 1.552823 × 10^4
Here, a = 1.552823 and n = 4.
To calculate the final amount in this scenario, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the initial deposit
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, we have:
P = $100 (monthly deposit)
r = 5% (or 0.05 in decimal form)
n = 12 (monthly compounding)
t = 10 years
Substituting these values into the formula, we can calculate the final amount:
A = 100(1 + 0.05/12)^(12*10)
Now, let's solve this equation step by step:
Step 1: Calculate the value inside the brackets first.
(1 + 0.05/12) = 1.004167
Step 2: Raise the value to the power of (12*10) = 120.
(1.004167)^120 = 1.647009
Step 3: Multiply the result by the initial deposit.
A = 100 * 1.647009 = $164.7009
Now, let's round the amount to the nearest place value. Since the hundredth place is the second decimal place, we round it to two decimal places:
A ≈ $164.70
Next, we need to express A = $164.70 in the form a×10^n.
In this case, $164.70 can be written as 1.647 × 10^2.
So, the amount $15,528.23 can be written as 1.553 × 10^4.