Problem:
A sum becomes two times in 7 years at compound interest. In how many years will the same sum become 16 times?

Solution:

To solve this problem, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the final amount
P = the principal sum (initial amount)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years

Given:
The sum becomes two times in 7 years at compound interest. This means that the final amount (A) is twice the initial amount (P) after 7 years.

We can set up the equation as follows:

2P = P(1 + r/n)^(nt)

Step 1: Simplify the equation

Divide both sides of the equation by P:

2 = (1 + r/n)^(nt)

Step 2: Find the value of (1 + r/n)^(nt)

To find the value of (1 + r/n)^(nt), we need to know the value of r (interest rate) and n (number of times compounded per year). However, these values are not given in the problem statement. Therefore, we need to make an assumption about these values in order to solve the problem.

Assumption:
Let's assume that the interest is compounded annually (n = 1) and the interest rate is constant over time (r remains the same). This assumption allows us to solve the problem using the given information.

Step 3: Solve for t

Now we can substitute the assumed values into the equation:

2 = (1 + r/1)^(1*t)
2 = (1 + r)^t

Since the left side of the equation is a constant (2), and (1 + r) is also a constant, the only variable in the equation is t. Therefore, we need to solve for t.

Step 4: Solve the exponential equation

To solve the exponential equation, we can take the logarithm (base 2) of both sides:

log2(2) = log2((1 + r)^t)
1 = t * log2(1 + r)

Now we can solve for t:

t = 1 / log2(1 + r)

Step 5: Calculate the value of t

Since the value of r is not given in the problem, we cannot calculate the exact value of t. However, we can determine the relationship between t and r based on the given options.

Option a) 21 years: t = 1 / log2(1 + r1)
Option b) 28 years: t = 1 / log2(1 + r2)
Option c) 35 years: t = 1 / log2(1 + r3)
Option d) 19 years: t = 1 / log2(1 + r4)

Since t is in the denominator of the equation, a smaller value of t corresponds to a larger value of r. Therefore, we need to find the option that gives the smallest value of t.

From the given options, we can see that option b) 28 years gives the largest value of t.