MATH SOLVE

4 months ago

Q:
# 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 depositn = number of times compounded per yearr = annual interest rate (as a decimal)P = initial (principal) investmentV(t) = value of investment after t yearsA. 5.9 yearsB. 6.1 yearsC. 23.4 yearsD. 24.5 years

Accepted Solution

A:

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).

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).