When dealing with problems involving speed, time, and distance, it’s important to set up equations based on the formula \( \text{Distance} = \text{Speed} \times \text{Time} \). For problems involving upstream and downstream travel, remember that the effective speed is altered by the speed of the stream. In this case, the upstream speed is reduced and the downstream speed is increased due to the current. Simplify the equation step by step to find the unknown distance, ensuring the units of time and speed match up correctly.
Given:
Speed in still water \(v = 5 \text{ km/hr}\),
Speed of stream \(u = 2 \text{ km/hr}\),
Additional time taken upstream = 20 minutes = \(\frac{1}{3}\) hours.
Effective speeds:
Speed Upstream = \(v - u = 5 - 2 = 3 \text{ km/hr}\),
Speed Downstream = \(v + u = 5 + 2 = 7 \text{ km/hr}\).
Let the distance between the two points be \(d \text{ km}\). Time taken for upstream and downstream travel:
Time Upstream = \(\frac{d}{3}\),
Time Downstream = \(\frac{d}{7}\).
The time difference between upstream and downstream travel is:
\(\frac{d}{3} - \frac{d}{7} = \frac{1}{3}\).
Equation simplification:
\(\frac{7d - 3d}{21} = \frac{1}{3}\),
\(\frac{4d}{21} = \frac{1}{3}\).
Multiplying by 21:
\(4d = 7\), \(d = \frac{7}{4} = 1.75 \text{ km}\).
Therefore, the distance between the two points is 1.75 km.