Answer: It will take about 7.75 years for the allowance to reach $15, starting from an initial allowance of $3 with a 15% annual increase.
Explanation:
We can solve this problem by using a simple formula for calculating compound interest:
A = P(1 + r)^n
Where:
A = the final amount (in this case, $15)
P = the initial amount (in this case, $3)
r = the annual interest rate (in this case, 15% or 0.15)
n = the number of years
We want to find out how many years (n) it will take for the allowance to reach $15, given an initial allowance of $3 and an annual increase of 15%.
Plugging in the values we have:
$15 = $3(1 + 0.15)^n
Dividing both sides by $3, we get:
5 = (1 + 0.15)^n
Taking the logarithm of both sides (to isolate n), we get:
log(5) = n*log(1.15)
Solving for n, we get:
n = log(5) / log(1.15)
Using a calculator, we find that n is approximately 7.75 years.
Therefore, it will take about 7.75 years for the allowance to reach $15, starting from an initial allowance of $3 with a 15% annual increase.