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A sum of money doubles itself on compound interest in 15 years. In what duration will it become 8 times of itself, considering annual compounding?
30 years
40 years
60 years
45 years
45 years
In compound interest, if an amount becomes x times in T years, it becomes xтБ┐ times in (n ├Ч T) years. Here, 8 = 2┬│, so time required is 3 ├Ч 15 = 45 years.
A sum of money doubles in 15 years, meaning the amount A becomes 2 times the principal P in 15 years under compound interest.
A=P(1+r)n
In compound interest, if an amount becomes x times in T years, it becomes xтБ┐ times in (n ├Ч T) years. Here, 8 = 2┬│, so time required is 3 ├Ч 15 = 45 years.
Many students mistake compound interest for simple interest where the time would be 15 * 7 = 105 years, or simply add years (15 + 15 + 15).
Establish the relationship
Given that the sum doubles in 15 years, we use the compound interest formula A=P(1+r)n, where A=2P and n=15. This gives (1+r)15=2.
(1+r)15=2
Set the target condition
We want to find the time t such that the amount becomes 8 times the principal, i.e., A=8P.
P(1+r)t=8PтЗТ(1+r)t=8
Solve for t
Since 8=23, we can substitute 2=(1+r)15 into the equation: (1+r)t=(2)3=((1+r)15)3=(1+r)45. Comparing exponents, we get t=45.
t=15├Ч3=45
D is correct because the sum becomes 8 times (23) in 3├Ч15=45 years.
This concept of geometric progression in time is also used in radioactive decay (half-life) and population growth problems.
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