Discussion
Raj swims for 6 ½ hours while going 24 km downstream and 36 km upstream. But he takes just 6 hours to swim 36 km downstream and 24 km upstream. At what rate is the river flowing?
Answer: C
Write \( b \) for the boat's speed in still water and \( r \) for the stream's. Then
\[ \text{downstream} = b + r, \qquad \text{upstream} = b - r \]
Set \( x = \dfrac{1}{b+r} \) and \( y = \dfrac{1}{b-r} \):
\[ 24x + 36y = 6.5, \qquad 36x + 24y = 6 \]
Adding: \( 60x + 60y = 12.5 \), so \( x + y = 0.20833 \).
Subtracting: \( -12x + 12y = 0.5 \), so \( y - x = 0.04167 \).
\[ y = 0.125 \Rightarrow b - r = 8, \qquad x = 0.08333 \Rightarrow b + r = 12 \]
\[ r = \frac{12 - 8}{2} = 2 \text{ km/h} \]
Adding and subtracting the two equations is quicker than eliminating one variable at a time, because the coefficients 24 and 36 appear in both.
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