How long will it take $500 to accumulate to $850 at 12% compounded monthly?
I got 4.4
Pls do check whether I got it right. Btw is the answer in years or months....I'm confused.
Thanks in advance
depends on your definitions.
let the monthly rate be i ----> i = .12/12 = .01
let the number of months be n
500(1.01)^n =850
1.01^n = 1.7
use logs and the rules of logs
n log 1.01 = log 1.7
n = log1.7/log1.01 = appr 53.33 months or 4.44 years
looks like you got it right.
To calculate the time it takes for $500 to accumulate to $850 at 12% compounded monthly, we can use the formula for compound interest:
A = P (1 + r/n)^(nt)
Where:
A = final amount ($850 in this case)
P = principal amount ($500 in this case)
r = annual interest rate (12% or 0.12)
n = number of times interest is compounded per year (monthly, so 12)
t = time in years
Let's substitute the values into the formula:
850 = 500 (1 + 0.12/12)^(12t)
Now, we can solve for t:
(1 + 0.12/12)^(12t) = 850/500
(1 + 0.01)^(12t) = 1.7
Let's take the natural logarithm of both sides to solve for t:
ln[(1 + 0.01)^(12t)] = ln(1.7)
12t ln(1 + 0.01) = ln(1.7)
t = ln(1.7) / (12 ln(1 + 0.01))
Using a calculator, we find:
t ≈ 4.3 (rounded to one decimal place)
So, it will take approximately 4.3 years for $500 to accumulate to $850 at 12% compounded monthly.
To calculate how long it will take for $500 to accumulate to $850 at 12% compounded monthly, you can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($850 in this case)
P = the principal amount ($500 in this case)
r = the annual interest rate (12% in this case, expressed as a decimal: 0.12)
n = the number of times that interest is compounded per year (12 for monthly compounding)
t = the number of years
Now, let's solve for t.
850 = 500(1 + 0.12/12)^(12t)
Divide both sides of the equation by 500:
1.7 = (1 + 0.01)^(12t)
Divide both sides of the equation by 1.01:
1.6832 ≈ (1 + 0.01)^(12t)
To isolate the variable, take the logarithm (base 1.01) of both sides:
log₁.₀₁(1.6832) ≈ log₁.₀₁((1 + 0.01)^(12t))
Using a calculator, find the logarithm of 1.6832 to the base 1.01, which gives approximately 12.004.
Now, solve for t:
12t ≈ 12.004
Divide both sides by 12:
t ≈ 12.004 / 12
t ≈ 1.0003
Therefore, it will take approximately 1.0003 years (or approximately 1 year and 0.03 months) for $500 to accumulate to $850 at a 12% annual interest rate compounded monthly.
Now, let's verify the answer of 1.0003 years:
To check your calculated answer of 4.4, we can use an alternative method:
Divide the future value divided by the principal amount (850 / 500):
850 / 500 ≈ 1.7
Now, take the natural logarithm (ln) of both sides:
ln(1.7) ≈ 12t ln(1.01)
Using a calculator, find the natural logarithm of 1.7 and divide it by 12 times the natural logarithm of 1.01. This calculation gives approximately 0.2802.
Therefore, the correct answer is approximately 0.2802 years (or approximately 0.2802 * 12 ≈ 3.36 months), not 4.4.
Hence, it seems that the calculated answer of 4.4 is incorrect.