In this article we solve Question 1 of the compound interest quiz.

Question

One dollar growing at a rate of 10% annually will become two dollars after ________.
A. about 2 years
B. about 5 years
C. about 7 years
D. about 10 years
E. none of the above

Solution

Answer: C.

Let’s look at how the one dollar is going to grow over time.
At the end of year 1, it will be $1.00 × 1.1 = $1.10
At the end of year 2, it will be $1.10 × 1.1 = $1.21
At the end of year 3, it will be $1.21 × 1.1 = $1.331
At the end of year 4, it will be $1.331 × 1.1 = $1.4641
At the end of year 5, it will be $1.4641 × 1.1 = $1.61051
At the end of year 6, it will be $1.61051 × 1.1 = $1.771561
At the end of year 7, it will be $1.771561 × 1.1 = $1.9487171
At the end of year 8, it will be $1.9487171 × 1.1 = $2.14358881

From the calculations above, we can see that at the end of year 7, the one dollar will have grown to $1.9487171, which is almost two dollars while at the end of year 8 it will have grown to $2.14358881. This means that somewhere between year 7 and year 8 it will be exactly $2.00. To find the exact time where it will be two dollars, we would have to solve for x in the exponential equation 1.1^x = 2. Solving for x gives x = (log 2) / (log 1.1) = 7.27254089734.

The rule of 72

This problem illustrates the rule of 72 which gives the approximate number of years it would take for an amount of money growing at a given rate to double. By this rule, an amount of money growing at a rate of x% per annum will double in 72/x years. Thus, by that rule, it would take 72/10 = 7.2 years for money growing at 10% per annum to double.