Introduction:
In this problem, we are given two bodies with masses 4 kg and 16 kg, initially at rest. We need to find the ratio of times for which the same force must act on them to produce the same kinetic energy in both bodies.

Explanation:
To solve this problem, we need to understand the relationship between force, mass, and kinetic energy.

1. Kinetic Energy:
The kinetic energy of an object is given by the formula:
K.E. = 1/2 * m * v^2
where K.E. is the kinetic energy, m is the mass of the object, and v is the velocity of the object.

2. Force and Kinetic Energy:
According to Newton's second law of motion, the force acting on an object is given by:
F = m * a
where F is the force, m is the mass of the object, and a is the acceleration produced by the force.

3. Acceleration and Velocity:
We know that acceleration is the rate of change of velocity. So, we can write:
a = (v - u) / t
where a is the acceleration, v is the final velocity, u is the initial velocity, and t is the time taken.

4. Relationship between Force, Mass, and Acceleration:
Substituting the value of acceleration in the equation of force, we get:
F = m * (v - u) / t
Rearranging the equation, we get:
F * t = m * (v - u)

5. Relationship between Force, Mass, and Kinetic Energy:
Substituting the value of force in the equation of kinetic energy, we get:
K.E. = 1/2 * m * v^2 = F * t * v

6. Ratio of Times:
To find the ratio of times, we can equate the kinetic energy equations for both bodies:
(1/2 * m1 * v1^2) / (1/2 * m2 * v2^2) = (F * t1 * v1) / (F * t2 * v2)
Simplifying the equation, we get:
m1 / m2 = t1 / t2

Conclusion:
Therefore, the ratio of times for which the same force must act on the two bodies to produce the same kinetic energy is equal to the ratio of their masses. In this case, the ratio is 4:16, which simplifies to 1:4.

1:4