Is my approach accurate to find original position of boat?

A boat goes upstream for $$3$$ hr $$30$$ min and then goes downstream for $$2$$ hr $$30$$ min. If the speed of the current and the speed of the boat in still water are $$\frac{10}{3}$$ kmph and $$\frac{15}{2}$$ kmph respectively, how far from its original position is the boat now?

Speed of boat in still water$$=\frac{15}{2}$$

Speed of stream $$=\frac{10}{3}$$

Upstream speed $$=\frac{15}{2}-\frac{10}{3}$$

Downstream speed $$=\frac{15}{2}+\frac{10}{3}$$

→ Downstream distance - Upstream distance = far from original.

$$=\left(2+\frac{1}{2}\right)\left(\frac{15}{2}+\frac{10}{3}\right) - \left(3+\frac{1}{2}\right)\left(\frac{15}{2}-\frac{10}{3}\right)$$

$$=\left(\frac{5}{2}\right)\left(\frac{65}{6}\right) - \left(\frac{7}{2}\right)\left(\frac{25}{6}\right)$$

$$=\frac{325-175}{12}$$

$$12.5$$ km downstream.

Is my approach accurate to find original position of boat?

• Please remove the gap between ":" and "/" to view the image. – Scott Kooper Nov 19 '19 at 5:48
• I got the same answer, $12.5~\text{km}$. However, I think it would be wiser to set up a coordinate system. It runs so that the downstream distance is positive. In this case, the upstream speed is actually $\frac{10}{3} - \frac{15}{2}$. The answer remains the same, but some of the signs are flipped in this approach. The expression that I have is $$\left( \frac{10}{3}- \frac{15}{2} \right) \frac{7}{2} + \left( \frac{10}{3}+\frac{15}{2} \right) \frac{5}{2} = 12.5$$ – Matti P. Nov 19 '19 at 5:57

With reference to ground, the water travels downstream for $$2.5+3.5=6$$ hours.
With reference to water, the boat went upstream for $$1$$ hour. So the boat is $$\frac{10}{3} \times 6- \frac{15}{2}\times 1 = 12.5$$ $$km$$ downstream from the start.
• What happens if you go north for $3.5$ hours, and then south for $2.5$ hours ? – AgentS Nov 19 '19 at 6:48
• Explain me why subtracting $\frac{15}{2}\times 1$ from total distance i.e $\frac{10}{3} \times 6$? – Scott Kooper Nov 20 '19 at 9:14
• Alright! Imagine you fell into the water and drowning along the stream. Where would you be in $6$ hours? – AgentS Nov 20 '19 at 9:21