Compound Interest

[goyalsau]

A certain investment grows at an annual interest rate of 8%, compounded quarterly. Which of the following equations can be solved to find the number of years, x, that it would take for the investment to increase by a factor of 16?

16 = (1.02)^x/4

2 = (1.02)^x

16 = (1.08)^4x

2 = (1.02)^x/4

1/16 = (1.02)^4x

Source: Manhattan Gmat

[Rahul@gurome]

A certain investment grows at an annual interest rate of 8%, compounded quarterly. Which of the following equations can be solved to find the number of years, x, that it would take for the investment to increase by a factor of 16?

16 = (1.02)^x/4
2 = (1.02)^x
16 = (1.08)^4x
2 = (1.02)^x/4
1/16 = (1.02)^4x

Say the investment = a and the number of years required = x

After 1 quarter it grows to [a + a*(0.08/4)] = a*(1.02)
After 2 quarters it grows to = a*(1.02)*(1.02)
...
After 4 quarters (= 1 year) it grows to = a*(1.02)^4
...
After 4x quarters (= x years) it grows to = a*(1.02)^4x

Now this must be equal to 16 times the investment i.e. 16a.
=> 16a = a*(1.02)^4x
=> 16 = (1.02)^4x

As both quantities are positive, we can take the fourth root of both sides.
Thus, 2 = a*(1.02)^x

The correct answer is B.