Question

$3500 is deposited into a bank account earning 6.75% interest compounded monthly. How long will it take to double in the account?
Suppose the account was compounding interest continuously. How much more would the account earn in the time it would have originally taken to double?

(I am confused, please help by showing work so I can understand)

Answer 1

Answer:

Compounded monthly:  It will take 10 years, 4 months to double

Compounded continuously:  They will make $13.65 more

Step-by-step explanation:

Set up the first part using the equation for compounding interest that is not continuous...

A = P(1 + r/n)^(nt)    where r is the rate as a decimal, n is the number of times the interest is compounded, A is the final amount, t is number if years it is compounded for, and P is the initial principle (or deposit)

Plug in what w have.  We want our $3,500 to double, so P = 3500, A = 7000,

r = 0,0675 and n = 12

7000 = 3500(1 + 0.0675/12)^(12t)

We need to solve for t.  First divide both sides by 3500

2 = (1 + 0.0675/12)^(12t)

use natual log rules to solve for t...

ln (2) = (12t)ln(1.005625)

[ln (2)]/[ln (1.005625)] = 12t

123.5724150 = 12t

 

   10.29770125 = t

 

      Or 10 years 4 months  (technically 10 years, 3.57 months, but the interest is compounded by the full month, so we need to round up)

When the interest is compounded continuously, we use the fromula

A = Pe^(rt)        plug in the info we are given to find out how much more we can make...

A = 3500e^(0.0675*10.29771025)       { here we use the actual value we found for t in the last part since the interest is compounded continuously}

A = 7,013.65

That's $13.65 more than when it was compounded monthly for the same time