A sum of money invested at compound interest doubles itself in $4\text{ years}$. The number of years it will take for the same sum to become $8$ times its original value at the same annual interest rate compounded annually is ________ (answer in integer).
Answer Key:
12
Step-by-Step Solution & Explanation
Under compound interest compounding annually:
$A = P(1 + r)^t$
Given that the sum doubles in 4 years:
$2P = P(1 + r)^4 \implies (1 + r)^4 = 2$
We want to find $t$ such that the amount becomes $8P$:
$8P = P(1 + r)^t \implies (1 + r)^t = 8 = 2^3$
Substituting $2 = (1 + r)^4$:
$(1 + r)^t = \left[ (1 + r)^4 \right]^3 = (1 + r)^{12}$
Equating exponents gives $t = 12\text{ years}$.