Problem: A boat covers a certain distance between two spots on a river taking 4 hours going downstream and 6 hours going upstream. What time will be taken by the boat to cover the same distance in still water?

Solution:

Understanding the problem:
The problem requires us to find the time taken by a boat to cover a certain distance in still water. We are given the time taken by the boat to cover the same distance going downstream and upstream.

Concepts used:
- Speed of boat in still water
- Speed of current
- Time taken to cover a distance

Steps involved:
1. Let's assume the distance between the two spots is 'd'
2. Let's assume the speed of boat in still water is 'b' and the speed of the current is 'c'
3. Using the formula, speed = distance/time, we can write the following equations:
- (d/t_downstream) = (b+c) - Speed of boat downstream
- (d/t_upstream) = (b-c) - Speed of boat upstream
4. We need to find the time taken by the boat to cover the same distance in still water. Let's assume the time taken is 't'
5. Using the formula, speed = distance/time, we can write the following equation:
- (d/t) = b - Speed of boat in still water
6. We can solve the equations to find the value of 'b'
7. Once we have the value of 'b', we can substitute it in any of the equations to find the value of 't'

Calculations:
Let's solve the equations to find the value of 'b'

d/4 = b+c
d/6 = b-c

Adding the two equations, we get:

d/4 + d/6 = 2b

Simplifying the equation, we get:

b = (5d/24)

Substituting the value of 'b' in the equation (d/t) = b, we get:

t = (d/(5d/24))

Simplifying the equation, we get:

t = 4.8 hours

Answer: The boat will take 4.8 hours to cover the same distance in still water.

Downstream.. t1=d/Vb+Vr.. upstream.. t2=d/Vb-Vr.. time in still water = d/Vb.. where downstream mns same dirxn as of river water.. upstream mns in opposite dirxn of river water.. Vb is speed of boat.. Vr is speed of river water.. to solve the eqn. as both Vr nd Vb is unknwn.. solve both eqns.. 4Vb+4Vr= d.. 6Vb-6Vr=d..multiply scnd eqn with 2/3..now scnd eqn becomes 4Vb-4Vr=2/3 d.. add both eqns.. 8Vb=5/3d..Vb=5d/24..so time comes as 24/5 hrs..