Question

A sum of money is invested at 12% compounded quarterly. About how long will it take for the amount of money to double?

Compound interest formula: (image uploaded V(t) )

t = years since initial deposit
n = number of times compounded per year
r = annual interest rate (as a decimal)
P = initial (principal) investment
V(t) = value of investment after t years

A. 5.9 years
B. 6.1 years
C. 23.4 years
D. 24.5 years

Answer 1

Follow the given formula.  The initial amount of money invested, P, becomes 2P (same thing as "doubles) after t years.  Since compounding is quarterly, n=4.  The annual interest rate is 12%.  That is, r=0.12.

Then we have 2P = P (1 + 0.12/4)^(4t) and need only solve for time, t.

Simplifying the above equation:  2 = (1.03)^(4t)

We must isolate 4t, and then isolate t.  To do this, take the common log of both sides of the above equation.  We get:

log 2 = (4t) log 1.03.  This gives us 4t = [log 2] / [log 1.03], or

4t =  23.4498

Dividing both sides by 4, we get     t = 5.86 (years).

Answer 2

Compound interest is the interest generated overtime on a sum of money invested. The time it will take for the amount of money to double is 5.9 years

How to calculate the compound interest?

Compound interest is the interest generated overtime on a sum of money invested. The formula for calculating compound interest is expressed as:
V(t) = P(1+r/n)^nt

where:

• t = years since initial deposit
• n = number of times compounded per year
• r = annual interest rate (as a decimal)
• P = initial (principal) investment
• V(t) = value of investment after t years

Given the following

V(t) = 2P
r = 0.12
n = 4

Substitute
2P = P(1+0.12/4)^4t
2 =  (1.03)^4t
ln2 = 4t ln(1.03)
0.6931 = 0.1182t
t = 0.6931/0.1182
t = 5.9 years

Hence the time it will take for the amount of money to double is 5.9 years

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