A bank offers 7% compunded continously. How soon will a deposit: triple and how soon will it increase by 25%?

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The formula involved for continuous compound is as follows:

`A = Pe^(rt)`

where:

- A is the initial amount
- P is the principal
- r is the interest rate
- t is the time in years

We know that the interset rate is 7%:

`A = Pe^(0.07t)`

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We want to know how long it takes for the amount to triple. Tripling the principal means that `A = 3P` . We substitute this to the equation and solve for `t` .

`A = 3P = Pe^(0.07t)`

`3P = Pe^(0.07t)`

`3 = e^(0.07t)`

`ln(3) = ln(e^(0.07t))`

`ln(3) = 0.07t`

`t = (ln(3))/(0.07)`

`t = 15.69`

Hence,** the amount will triple in 15.69 years.**

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We want to know how long it takes before the amount increases by 25%. This means that `A = (1.25)P` . Solving this uses the same technique as the first one:

`A = 1.25P = Pe^(0.07t)`

`1.25P = Pe^(0.07t)`

`1.25 = e^(0.07t)`

`ln(1.25) = ln(e^(0.07t))`

`ln(1.25) = 0.07t`

`t = (ln(1.25))/(0.07)`

`t = 3.19`

Hence, it takes 3.19 years for the amount to increase by 25%.

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