Question

A boat covers 24 km upstream and 18 km downstream in 6 hours, while it covers 36 km upstream and 36 km downstream in 10 hours. The speed of the current is:

• A. 0.75 km/h
• B. 1.5 km/h
• C. 1 km/h
• D. 2 km/h

Answer 1

The correct option is B 1.5 km/h

Method 1:
Let the speed of the boat be = x km/h
Let the speed of the stream be = y km/h
Speed of the boat against the stream will be = (x - y) km/h
Speed of the boat in the direction of the stream will be = (x + y) km/h
$\frac{24}{(x-y)}+\frac{18}{(x+y)}=6h---\left(1\right)\frac{36}{(x-y)}+\frac{36}{(x+y)}=10h---\left(2\right)$
For simplification,
Let us substitute, $\frac{1}{(x-y)}$= m and $\frac{1}{(x-y)}$ = n.
Such that the equations will get reduced as:
24m + 18n = 6 --- (3)
36m + 36n =10 --- (4)
Equation (3) can further be simplified as: 4m+3n = 1 ----- (5)
Equation (4) can further be simplified as : 18m+18n = 5 ---- (6)
Multiplying Eq. (5) with 18 and Eq. (6) with 4 so as to solve the equations;
72m + 54n = 18 ---- (7)
72m + 72n = 20 ----- (8)
(8) - (7) = 18 n = 2
Thus, n = $\frac{2}{18}=\frac{1}{9}$
m = $\frac{(1-3n)}{4}=\frac{(1-3\times \frac{1}{9})}{4}=\frac{1}{6}$
Thus, $\frac{(1-3n)}{4}=\frac{(1-3\times (\frac{1}{9})}{4}$
Similarly, m = $\frac{1}{(x+y)}=\frac{1}{9}\Rightarrow$ x-y = 9
Adding the two equations, 2x = 6+ 9 =15, x = 7.5 km/hr
y = 1.5 km/hr

Method 2:
Downstream speed = x + y = $\frac{(24\times 36)-(36\times 18)}{(24\times 10)-(36\times 6)}=\frac{216}{24}$ = 9 km/hr ---> (1)
Upstream speed = x-y = $\frac{\left(\right(18\times 36)-(24\times 36\left)\right)}{\left(\right(18\times 10)-(36\times 6\left)\right)}=\frac{216}{36}$ = 6 km/hr -----> (2)

Solving, y = 1.5 km/hr