An investment of $1,500 was made in a certain bank account and it earn

Question

An investment of $1,500 was made in a certain bank account and it earned interest that was compounded annually; the annual interest rate was fixed for the entire duration of the investment. If after 12 years the $1,500 increased to $24,000 by earning interest, in how many years after the initial investment was made would the $1,500 have increased to $96,000 by earning interest at the same rate?

A. 15
B. 18
C. 20
D. 21
E. It cannot be determined from the information given

A. 15
B. 18
C. 20
D. 21
E. It cannot be determined from the information given

gmatophobia · Answer

1500(1+\frac{r}{100})^{12} = 24000

(1+\frac{r}{100})^{12} = \frac{24000}{1500}

(1+\frac{r}{100})^{12} = 16

(1+\frac{r}{100})^{12} = 2^4

Taking the fourth root on both sides of the equation

(1+\frac{r}{100})^{3} = 2

Taking the cube root on both sides of the equation

1+\frac{r}{100} = 2^{\frac{1}{3}}

96000 = 24,000 * 4

96000 = 1500(1+\frac{r}{100})^{12} * 4

96000 = 1500(1+\frac{r}{100})^{12} * 2^2

2^2 = (2^{\frac{1}{3}})^6

2^2 = (1+\frac{r}{100})^6

96000 = 1500(1+\frac{r}{100})^{12} * (1+\frac{r}{100})^6

96000 = 1500(1+\frac{r}{100})^{12+6}

96000 = 1500(1+\frac{r}{100})^{18}

Therefore  t = 18  months

Option B

GPD21 · Answer

When compounding is in play, whether it's interest, cells in a petri dish, etc. we can sometimes find an island of nice numbers to make our life easy. Usually doubling works best IMO.

Writing these doublings out we have:

-1,500
\/
-3,000
\/
-6,000
\/
-12,000
\/
-24,000

Now we figure out how many doubles there are (1500 -> 3000 is one, 3000 -> 6000 is two, 6,000 -> 12,000 is three, 12,000 -> 24,000 is four).

We can now see quite quickly it takes 3 years for our number to double (12 years/4 total doubling events).

Knowing 24,000 -> 96,000 is two more doubles (24,000 *2^2 or writing out 24,000 -> 48,000 -> 96,000) we can confidently say it will take 6 more years to grow to 96,000. In total (from initial investment) this would be 18 years.

This can be formalized using the "rule of 72." This rule says that for any growth rate "R" when 72 is divided by R (when R has to be expressed as 4 instead of 4%) the result is the number of years an investment will take to double (approximately).

We could then run the P(1+r/n)^nt = V formula to find our answer, but thats a lot of math

Regardless, 18 years after initial investment (B) is our answer.

Bunuel

An investment of $1,500 was made in a certain bank account and it earned interest that was compounded annually; the annual interest rate was fixed for the entire duration of the investment. If after 12 years the $1,500 increased to $24,000 by earning interest, in how many years after the initial investment was made would the $1,500 have increased to $96,000 by earning interest at the same rate?

A. 15
B. 18
C. 20
D. 21
E. It cannot be determined from the information given

Regardless, 18 years after initial investment (B) is our answer.

An investment of $1,500 was made in a certain bank account and it earn

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