Question

How long compounded annually triples in 3 years, how long 9 times itself?

A. 6 years
B. 9 years
C. 12 years
D. 18 years

Answer 1

It would take approximately 3^(2/3) years for an amount to increase 9 times itself when compounded annually.

Step-by-step explanation:

The question asks how long it would take for an amount to increase 9 times itself when compounded annually. To answer this, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. In this case, the final amount is 9 times the principal amount, so we can set up the equation 9P = P(1 + r/n)^(nt). We are given that when compounded annually, an amount triples in 3 years, so we can use this information to find the interest rate and substitute it into our equation.

Let's solve the equation 9P = P(1 + r/1)^(1*3) for r:

1. 
   
3. 9P = P(1 + r)^(3)
4. 
   
6. 9 = (1 + r)^(3)
7. 
   
9. Using the cube root of both sides, we find (1 + r) = 3^(1/3)
10. 
    
12. Subtracting 1 from both sides, we get r = 3^(1/3) - 1

Now, let's substitute this interest rate into the equation A = P(1 + r/n)^(nt) and solve for t:

1. 
   
3. 9P = P(1 + (3^(1/3) - 1)/1)^(1*t)
4. 
   
6. 9 = (1 + 3^(1/3) - 1)^(t)
7. 
   
9. 9 = 3^(1/3)^(t)
10. 
    
12. Using the cube root of both sides, we find t = (3^(1/3))^(-1)
13. 
    
15. Simplifying, t = 3^(2/3)

Therefore, it would take approximately 3^(2/3) years for an amount to increase 9 times itself when compounded annually.