You have $2500 to invest at 6% interest compounded quarterly. For how many years will the money need to be invested for that amount to triple?
correct answer is in decimal form rounded to the nearest half year
.06/4 = .015
1.015^(n) = 3
n log 1.015 = log 3
n = 73.8 quarterly periods
n/4 = 18.4 or 18 1/2 years
To determine the number of years it will take for the initial investment to triple, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, triple the initial investment)
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
Given:
P = $2500
r = 6% or 0.06
n = 4 (compounded quarterly)
We want to find t.
Using the formula, we substitute the values:
3P = P(1 + r/n)^(nt)
3(2500) = 2500(1 + 0.06/4)^(4t)
7500 = 2500(1 + 0.015)^(4t)
Dividing both sides by 2500:
3 = (1.015)^(4t)
To solve for t, we can take the logarithm of both sides (base 10 or natural log), but since we want the answer rounded to the nearest half-year, it's easier to use trial and error.
We can start by trying different values of t until we reach an approximate value of 3 on the left side.
When t = 1, (1.015)^(4*1) = 1.061
When t = 2, (1.015)^(4*2) = 1.125
When t = 3, (1.015)^(4*3) = 1.192
When t = 4, (1.015)^(4*4) = 1.263
When t = 5, (1.015)^(4*5) = 1.339
When t = 6, (1.015)^(4*6) = 1.418
When t = 6.5, (1.015)^(4*6.5) = 1.462
Based on the calculations above, when t is approximately 6.5 years (or halfway between 6 and 7 years), the investment will triple.
Therefore, the correct answer to the nearest half-year is 6.5 years.