A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

# A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

1. A
4 km/h and 8 km/h
2. B
3 km/h and 8 km/h
3. C
4 km/h and 6 km/h
4. D
4 m/h and 5 km/h

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### Solution:

Given that a boat goes 30 km upstream and 44 km downstream in 10 hours.
In 13 hours, it can go 40 km upstream and 55 km downstream.
We have to find the speed of stream and that of the boat in still water.
Let speed of boat in still water be x km/h and speed of stream be y km/h.
Time taken to cover 30 km upstream is  .
Time taken to cover 44 km downstream is  .
When the boat takes 10 hours, the following equation can be written,
Simplifying this equation,

Similarly,
Time taken to cover 40 km upstream is $\frac{40}{x-y}$.
Time taken to cover 55 km downstream is $\frac{55}{x+y}$.
When the boat takes 13 hours, the following equation can be written,
….(2)
Let   and  .
Then, the equations become,
55p+40q-13=0
22p+15q-5=0
The solution of the equations by cross multiplication method is given by the formula,
$\frac{p}{{b}_{1}{c}_{2}-{b}_{2}{c}_{1}}=\frac{q}{{a}_{2}{c}_{1}-{a}_{1}{c}_{2}}=\frac{1}{{a}_{1}{b}_{2}-{a}_{2}{b}_{1}}$
Here,
Using the cross-multiplication method to find the value of p and q,
$⇒\frac{p}{40\left(-5\right)-15\left(-13\right)}=\frac{q}{22\left(-13\right)-55\left(-5\right)}=\frac{1}{55\left(15\right)-40\left(22\right)}$
$⇒\frac{p}{-200+195}=\frac{q}{-286+275}=\frac{1}{825-880}$
$⇒\frac{p}{-5}=\frac{q}{-11}=\frac{1}{-55}$
$⇒p=\frac{1}{11}$
$⇒q=\frac{1}{5}$
This gives us,
Now,
$x+y+x-y=11+5$
$⇒2x=16$
$⇒x=8$
Putting value of x in equation 4 and solving for y,
$8-y=5$
$⇒y=3$
Therefore, the values are x=8 and y=3.
Hence, the speed of the stream and that of the boat in still water are 3 km/h and 8 km/h respectively.
Hence, option (2) is correct.

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