you deposit 3500 in an account that pays 4.5% annual interest, compounded monthly. How long will it take for the balance to reach 5000? Use formula A= P[1+(r/n)]^(ny) and use knowledge of logarithms.
P = Po(1+r)^n
P = $5000
Po = $3500
r = (4.5%/12)/100% = 0.00375 = Monthly % rate expressed as a decimal.
n = 12Comp/yr * T yrs = 12T = The # of
compounding periods.
P = 3500*(1.00375)^12T = 5000
(1.00375)^12T = 5000/3500 = 1.42857
Take Log of both sides:
12T*Log (1.00375) = Log 1.42857
12T = Log 1.42857/Log (1.00375)=95.29
T = 7.94 or 8 Yrs.
To solve this problem, we can use the formula for compound interest:
A = P[1 + (r/n)]^(ny)
Where:
A is the future value or balance
P is the principal or initial deposit
r is the annual interest rate (expressed as a decimal)
n is the number of times interest is compounded per year
y is the number of years
In this case:
P = $3500
A = $5000
r = 4.5% per year = 0.045 (as a decimal)
n = 12 (compounded monthly)
Let's plug in the values and solve for y:
5000 = 3500[1 + (0.045/12)]^(12y)
Divide both sides by 3500:
5000/3500 = [1 + (0.045/12)]^(12y)
Simplify:
1.428571429 = [1.00375]^(12y)
Apply logarithm to both sides:
log(1.428571429) = log([1.00375]^(12y))
Using logarithm properties, we can bring down the exponent:
log(1.428571429) = 12y * log(1.00375)
Divide both sides by 12 * log(1.00375):
y = log(1.428571429) / (12 * log(1.00375))
Using a calculator, we can find:
y ≈ 4.27 years
So, it will take approximately 4.27 years for the balance to reach $5000.
To find out how long it will take for the balance to reach $5000 using the given formula A = P[1 + (r/n)]^(ny), where:
A = the final amount
P = the initial principle (deposit amount)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
y = the number of years
We can rearrange the formula to solve for y, the time it will take for the balance to reach $5000:
A = P[1 + (r/n)]^(ny)
Divide both sides by P:
A/P = [1 + (r/n)]^(ny)
Take the natural logarithm of both sides:
ln(A/P) = ln([1 + (r/n)]^(ny))
Now let's substitute the given values into the formula:
P = $3500
r = 4.5% or 0.045 (as a decimal)
n = 12 (compounded monthly)
A = $5000
ln(5000/3500) = ln([1 + (0.045/12)]^(12y))
To solve for y, we need to isolate it. Let's simplify the equation step by step:
ln(5/3) = ln((1 + 0.00375)^(12y))
ln(5/3) = ln(1.00375^(12y))
ln(5/3) = 12y * ln(1.00375)
ln(5/3) = 12y * 0.00374165739
Now we can solve for y by dividing both sides of the equation by 12 * 0.00374165739:
ln(5/3) / (12 * 0.00374165739) = y
Using a calculator, we can find the value of ln(5/3) divided by (12 * 0.00374165739):
ln(5/3) / (12 * 0.00374165739) ≈ 7.29
Therefore, it will take approximately 7.29 years for the balance to reach $5000 when $3500 is deposited in an account that pays 4.5% annual interest, compounded monthly.