In how many years will a sum placed on compound interest become 8 times itself if it becomes twice in 6 years?

Question

Accepted Answer

#### Step 1: Understand the Problem
The sum becomes twice in **6 years**. We need to find how long it will take for the sum to become **8 times** its initial value under compound interest.

#### Step 2: Use the Relationship of Compound Interest
For compound interest, the formula for the amount is:  
$$
A = P 	imes (1 + r)^t,
$$  
where $$ A $$ is the final amount, $$ P $$ is the principal, $$ r $$ is the rate of interest, and $$ t $$ is the time.

#### Step 3: Calculate the Rate of Interest
If the sum doubles in **6 years**, then:  
$$
2P = P 	imes (1 + r)^6.
$$  
Simplify:  
$$
(1 + r)^6 = 2.
$$  
Take the 6th root of 2 to find $$ r $$:  
$$
1 + r = 2^{1/6}.
$$

#### Step 4: Find Time to Become 8 Times
To find when the sum becomes **8 times**, use:  
$$
8P = P 	imes (1 + r)^t.
$$  
Simplify:  
$$
(1 + r)^t = 8.
$$

Substitute $$ 1 + r = 2^{1/6} $$:  
$$
(2^{1/6})^t = 8.
$$  
Simplify further:  
$$
2^{t/6} = 2^3.
$$

Equate powers of 2:  
$$
\frac{t}{6} = 3.
$$  
Solve for $$ t $$:  
$$
t = 18.
$$

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Thus, it will take **18 years** for the sum to become **8 times**, making Option **(b)** correct.

Answer

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