Problem Solving

ru2008 wrote:An investment doubles in value apprx every 69/r years, where r is the annual continuously compounding rate of interest. In approximately how many years will an initial $1000 investment grow to $8000, at an interest rate of 5%?

a. 24
b. 28
c. 36
d. 42
e. 56
Source: Veritas Prep
Can someone show me appropriate formula for this? I dont understand their reasoning

The formula is A = P (1 + r/n)^ (nt), where A is the amount after time t, P is the principal amount, r is the annual interest rate, t is the no. of years, n number of times the interest is compounded per year.

A = 1000(1 + 0.05/(69/5))^(69 * t)/5 = 1000[1 + 0.0036]^(13.8t) = 1000 * (1.0036) ^(13.8t)
So, 8000 = 1000 * (1.0036) ^(13.8t)
8 = (1.0036) ^(13.8t) and when you solve it, you get t = 42 approximately.

The correct answer is (D).

Rahul@gurome wrote:

ru2008 wrote:An investment doubles in value apprx every 69/r years, where r is the annual continuously compounding rate of interest. In approximately how many years will an initial $1000 investment grow to $8000, at an interest rate of 5%?

a. 24
b. 28
c. 36
d. 42
e. 56
Source: Veritas Prep
Can someone show me appropriate formula for this? I dont understand their reasoning

The formula is A = P (1 + r/n)^ (nt), where A is the amount after time t, P is the principal amount, r is the annual interest rate, t is the no. of years, n number of times the interest is compounded per year.

A = 1000(1 + 0.05/(69/5))^(69 * t)/5 = 1000[1 + 0.0036]^(13.8t) = 1000 * (1.0036) ^(13.8t)
So, 8000 = 1000 * (1.0036) ^(13.8t)
8 = (1.0036) ^(13.8t) and when you solve it, you get t = 42 approximately.

The correct answer is (D).

That would be the correct calculation if we were compounding 69/5 times per year (and it would be completely impractical to complete that calculation with pen and paper, but that's a separate issue). We are certainly not compounding 69/5 times per year in this question; as highlighted in red above, the compounding is occurring continuously. That is, this question concerns an account which is earning a minuscule amount of interest every instant, and this interest is continually compounded. The formula in this situation is one you will absolutely never need on the GMAT, but it is Final Value = P*e^(rt), where P is your initial investment, r the interest rate, t the time in years, and e a number roughly equal to 2.7. You'll certainly encounter that in any finance course, but you won't need to understand any of this for the GMAT.

As an aside, the answer obtained above is close to the correct answer only because continuous compounding doesn't make much more of a difference than 'frequent' compounding, at least for small interest rates. If you solve as if you're compounding monthly or quarterly, say, you'll also find that 42 is the best answer.

In any case, the whole point of the question is that you do not need any formula. The question tells you the value of the investment will double every 69/r years, and that r=5. That's all the information you need here. Our investment doubles every 69/5 = 13.8 years. We need it to double three times (1000 --> 2000 --> 4000 --> 8000), so it will take 3*13.8 = 41.4 years. The first solution given by 4GMATMumbai above is perfect.