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×10n
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 final amount
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, we have:
A = $15,528.23
P = $100
r = 5% = 0.05
n = 12 (compounded monthly)
t = 10 years
Substituting these values into the formula:
$15,528.23 = $100(1 + 0.05/12)^(12*10)
Dividing both sides by $100:
155.2823 = (1 + 0.05/12)^(12*10)
Taking the logarithm of both sides (base 10):
log(155.2823) = log[(1 + 0.05/12)^(12*10)]
Using the logarithmic property: log(a^b) = b*log(a)
log(155.2823) = (12*10) * log(1 + 0.05/12)
Simplifying:
log(155.2823) = 120 * log(1 + 0.05/12)
To solve for log(1 + 0.05/12), we need to isolate it:
log(1 + 0.05/12) = log(155.2823) / 120
Using a calculator:
log(1 + 0.05/12) ≈ 0.012573
Substituting this value back into the equation:
log(155.2823) = 120 * 0.012573
Using a calculator:
log(155.2823) ≈ 1.699702
Finally, we can solve for 155.2823:
155.2823 = 10^1.699702
Using a calculator:
155.2823 ≈ 46.39
Therefore, in the form of a×10n, 15528.23 is approximately equal to 46.39×10^2.
To find the future value of a monthly deposit of $100 compounded monthly at an interest rate of 5% over 10 years, we can use the formula for compound interest:
\(A = P \times \left(1 + \frac{r}{n}\right)^{nt}\),
where:
A = the future value of the investment
P = the principal amount (the initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, we have:
P = $100 (monthly deposit)
r = 5% = 0.05
n = 12 (compounded monthly)
t = 10 years
Substituting these values into the formula, we get:
\(A = 100 \times \left(1 + \frac{0.05}{12}\right)^{(12 \times 10)}\)
Simplifying the expression inside the brackets:
\(A = 100 \times \left(1 + \frac{0.00416667}{12}\right)^{120}\)
Calculating the value inside the brackets:
\(A = 100 \times \left(1.00416667\right)^{120}\)
Using a calculator, we find:
\(A \approx 15,528.23\)
Rounding this value to the nearest place value, we get:
A ≈ $15,528.23
Now, let's express $15,528.23 in the form of a×10^n, where a is a single digit. To do this, we need to move the decimal point to the left until there is only one non-zero digit to the left of the decimal point. We will count how many places we moved the decimal point and set n equal to that number. Starting with $15,528.23, we move the decimal point three places to the left to get $15.52823. Therefore, we have:
A = 1.552823 × 10^4
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (the initial deposit)
r = annual interest rate (written as a decimal)
n = number of times interest is compounded per year
t = number of years the money is invested for
In this case, the principal amount (P) is $100, the interest rate (r) is 5% or 0.05, the number of times interest is compounded per year (n) is 12 (monthly compounding), and the number of years (t) is 10.
So, the formula becomes:
A = 100(1 + 0.05/12)^(12*10)
To find the future value of the investment (A), we can calculate it:
A = 100(1 + 0.00416667)^(120)
Now, you can use a calculator or a software program to evaluate this expression:
A ≈ $16470.09
Rounding this value to the nearest cent gives: $16470.10
To express this amount in the form of a × 10^n, we need to determine the values of a and n.
In this case, a single digit will be the first digit of the rounded amount, which is 1. Therefore, a is 1.
To find n, we need to determine the exponent of 10 that can be multiplied by a to obtain the original value. In this case, we can move the decimal point from its current position at the end of the number to its original position at the end of $15,470.10 by multiplying it by 10^3.
So, the value $15,470.10 can be expressed as 1.54701 × 10^4.