Upstream Downstream Boat Math Problem

Problem Statement

If a man travel's by his motor boat down a river to his office and back. With the speed of the river unchanged, if he doubles the speed of the boat, then his total travel time gets reduced by 75%. The ratio of the original speed fo the boat to speed of the river is ?

Strategy

The boat problem usually has 3 components to it

Downstream speed would be

S down = S b + S r

And upstream speed would be

S up = S b - S r

The time taken to complete the round trip would be

T = D S up + D S down

Condition given in the problem

If speed of the boat is doubled, time reduces by 75% (i.e its 25% of original time)

1 4 T = D 2S b - S r + D 2S b + S r

Let's make T 100% in the equation

T = 4D 2S b - S r + 4D 2S b + S r

Finding the ratio now

D S b - S r + D S b + S r = 4D 2S b - S r + 4D 2S b + S r

The distance can be removed from both LHS and RHS

1 S b - S r + 1 S b + S r = 4 1 2S b - S r + 1 2S b + S r

Let's try to reduce the equations.

2S b S b 2 - S r 2 = 4 4S b 4S b 2 - S r 2

And reduce it even further

8 S b 2 - S r 2 = 4S b 2 - S r 2

Moving the sides around we get

8S b 2 - 4S b 2 = 8S r 2 - S r 2

The ratio of the squares would be.

S b 2 S r 2 = 7 4

Answer

S b S r = 7 2
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