A(t) = Aₒ(1 + r)^t
Used when compounding
Aₒ = starting amount
r = rate
t = amount of time compound in given timespan
A(t) = Aₒb^(t/k)
Used when things are doubling, tripling, halfing, etc.
Aₒ = starting amount
b = doubling, tripling, etc.
t = amount of time
k = how long it takes to double, triple, etc.
Rule of 72 → 72 ÷ r
Used to find how long it takes for something to double
r is the rate. Not used as a decimal, but a percent.
P(t) = Pₒe^(rt)
only used when compounding continuously
Pₒ = starting amount
e = on ln button
r = rate
t = time
Ex 1. A bank advertises that if you open a savings account, you can double your money in 10 years. If you invest $2,000, how much money will you have after 3 years?
It gives you how long it takes to double, so you use formula 2.
Aₒ = $2,000
b = 2
t = 3
k = 10
A(t) = 2,000(2)^3/10
= $2462.29
Ex 2. If $1,000 is invested in a savings account that is compounding monthly at .8% for 6 years how much money did you gain? What if it was compounding continuously?'
First, we will use the compounding formula, then the compounding continuously formula for the second question. It is compounding monthly so we have to convert the years into months.
Aₒ = $1,000
r = .008
t = 6*12 = 72
A(t) = 1,000(1 + .008)^72
= $1,774.84
They asked for how much gain so we will subtract the original amount from the answer we got. Amount gained was $774.84
Second part:
Compounding continuously
Pₒ = $1,000
e = ...e
r = .008
t = 72 months
P(t) = 1000(e^(.008*72))
= $1,778.91 – 1,000 = $778.91
Ex 3. How long does it take for any given amount to double at a rate of 8% increase per year?
This is a rule of 72 problem.
72 ÷ 8 = 9 years
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