A and B are two points in a river. Point X and Y divide the line AB in three equal parts. Let the flow of the river be from A to B. The ratio of the time taken to row from point A to Y and from B to Y is 3 : 4. Find the ratio of the speed of the boat upstream to the flow of the river.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6 : 5
Going with the flow (downstream) the boat’s speed is \(b+r\); going against it (upstream) it is \(b-r\), where \(b\) is the speed of the boat in still water and \(r\) the speed of the river. The river flows from A to B, so A to Y is downstream and B to Y is upstream.
The two distances (X and Y cut AB into three equal parts; take each part as 1):
A to Y \(=2\) parts, downstream
B to Y \(=1\) part, upstream
The ratio of the times (time \(=\) distance \(\div\) speed):
\(\displaystyle \frac{2}{b+r}:\frac{1}{b-r}=3:4\)
\(\displaystyle \frac{2(b-r)}{b+r}=\frac{3}{4}\)
\(8b-8r=3b+3r\)
\(5b=11r\), so \(\displaystyle b=\frac{11r}{5}\)
The ratio asked for:
Upstream speed \(\displaystyle =b-r=\frac{11r}{5}-r=\frac{6r}{5}\)
Upstream speed : speed of the river \(\displaystyle =\frac{6r}{5}:r=6:5\)