Problem Statement
A motor boat can travel 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can travel 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.
Verified Solution & Marking Scheme
Formulate System of Equations
Let the speed of the boat in still water be $x$ km/h and speed of the stream be $y$ km/h ($x > y$).
- Speed upstream $= x - y$ km/h
- Speed downstream $= x + y$ km/h
Given conditions:
$\frac{30}{x - y} + \frac{44}{x + y} = 10$ ...(1)
$\frac{40}{x - y} + \frac{55}{x + y} = 13$ ...(2)
Substitute u = 1/(x - y) and v = 1/(x + y)
$30u + 44v = 10 \implies 15u + 22v = 5$ ...(3)
$40u + 55v = 13$ ...(4)
Multiply (3) by 8 and (4) by 3:
$120u + 176v = 40$
$120u + 165v = 39$
Subtracting: $11v = 1 \implies v = \frac{1}{11}$.
Substitute $v = \frac{1}{11}$ into (3):
$15u + 22\left(\frac{1}{11}\right) = 5 \implies 15u + 2 = 5 \implies 15u = 3 \implies u = \frac{1}{5}$.
Solve for x and y
$x - y = \frac{1}{u} = 5$ ...(5)
$x + y = \frac{1}{v} = 11$ ...(6)
Adding (5) and (6):
$2x = 16 \implies x = 8 \text{ km/h}$
Subtracting (5) from (6):
$2y = 6 \implies y = 3 \text{ km/h}$
Thus, speed of the boat in still water is $8$ km/h, and speed of the stream is $3$ km/h.