Logarithm question using compound interest formula

I.e., 12% compounded annually is more than 12% compounded monthly, because the compounded annual rate is really 1.1268% per month.) But for mortgage purposes, the difference is not significant.
  • #1
Schaus
118
5

Homework Statement


A colony of ants will grow by 12% per month. If the colony originally contains 2000 ants how long will it take for the colony to double in size?

Answer - 6.12 months

Homework Equations


A = P(1+r/n)nt

The Attempt at a Solution


r = 12% = 0.12
n = 12
P = 2000
A = 4000
t = ?

A = P(1+r/n)nt
4000 = 2000(1+0.12/12)12t
2 = (1.01)12t
log2 = log(1.01)12t
log2 = 12t log(1.01)
log2/12log(1.01) = t
t = 5.8 months

I don't know what I'm doing wrong. I've followed another compound interest question that is very similar to this one but my answer is still 0.3 months out. Can anyone point out what I'm doing wrong?
Thanks for your help
 
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  • #2
You seem to be mixing up the formula.
Your base for the exponent should be 1 + the percentage growth for each time increment...in this case months.
If you were given a yearly growth rate and asked to figure out monthly growth, then n would be 12, but in this case, n is 1, and you can let t represent months.
This gives you
## A = P(1 + .12)^t. ##
 
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Likes Schaus
  • #3
Yes that was definitely my problem. Physics Forum for the win again! Thanks a lot for your help!
 
  • #4
I'd actually suggest that you are solving for n, not t. Why? you have periodic model that is fixed at months, and you want to know how many iterations of your model occur before population size doubles. Furthermore, having t in the exponent is troubling / not interpretable from a units perspective. Exponents are always dimensionless. n iterations is per se dimensionless. Time always has units. So if you want the equation to be transparent/ dimensionally correct, the below is more appropriate

## A = P(1 + .12)^n.##
 
  • #5
Schaus said:
Yes that was definitely my problem. Physics Forum for the win again! Thanks a lot for your help!

Just to clarify: the formula you wrote initially would be appropriate to a mortgage situation with an annual interest rate (or growth rate) of 12%, but compounded monthly. Mortgages are typically computed that way, so an annual rate of 12% is regarded as a rate of 1% per month in a normal mortgage contract.

Mathematically that is not quite true, because a monthly interest rate of r (0 < r < 1, not a percentage) compounds monthly to ##(1+r)^{12}## in one year. So, mathematically, to get 12% = 0.12 in one year we need ##r = 1.12^{1/12}-1 \doteq 0.00948879293##, about 0.9489%. Or, to put it another way, 1% compounded monthly produces ##1.01^{12} \doteq 1.1268,## so a bit more than 12% per year.
 

FAQ: Logarithm question using compound interest formula

1. What is the compound interest formula?

The compound interest formula is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

2. How do logarithms relate to compound interest?

Logarithms are used to solve for unknown variables in the compound interest formula. By taking the logarithm of both sides of the equation, we can isolate the variable we are solving for.

3. Can you explain the concept of compounding in compound interest?

Compounding refers to the process of adding earned interest back to the principal amount, so that interest is earned on the new total amount. This results in earning interest on interest, which leads to faster growth of the initial investment.

4. How do you calculate the annual interest rate using the compound interest formula?

The annual interest rate can be calculated by rearranging the compound interest formula to r = (A/P)^(1/nt) - 1. This formula can be used to find the interest rate when all other variables are known.

5. Is the compound interest formula applicable to all types of investments?

The compound interest formula is commonly used for investments that earn interest, such as savings accounts, bonds, and certain types of annuities. However, it may not be applicable to investments that have variable or non-guaranteed returns.

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