**Boats and Streams :**

In this section, we are going to learn, how to solve problems on boats and streams step by step.

1. The direction along the stream in water is called downstream.

2. The direction against the stream in water is called upstream.

3. When the speed of a boat in still water is **a** miles/hr and the speed of the stream is **b** miles/hr,

Speed downstream = **(a + b)** miles/hr

Speed upstream = ** (a - b)** miles/hr

4. When the speed downstream is **u** miles/hr and the speed upstream stream is **v** miles/hr,

Speed in still water = **1/2 ****⋅ ****(u + v)** miles/hr

Rate of stream or current = **1/2 ****⋅ ****(u - v)** miles/hr

5. Meaning :

Speed of boat in still water = Actual speed of the boat.

**Problem 1 :**

The speed of a boat in still water is 10 miles per hours. If it can travel 26 miles downstream and 14 miles upstream in the same time, find the speed of the current.

**Solution :**

Let "x" be the speed of the current

Speed downstream = (10 + x) mph

Speed upstream = (10 - x) mph

The formula to find the time :

Time = Distance / Speed

We can use the above formula and find time downstream and time upstream as given below.

Time downstream = 26 / (10 + x)

Time upstream = 14 / (10 - x)

**Given : **It can travel 26 miles downstream and 14 miles upstream in the same time.

Then, we have

26 / (10 + x) = 14 / (10 - x)

Take reciprocal on both sides.

(10 + x) / 26 = (10 - x) / 14

14(10 + x) = 26(10 - x)

140 + 14x = 260 - 26x

40x = 120

x = 3

Hence, the speed of the current is 3 mph.

**Problem 2 :**

A man can row 18 kmph in still water. It takes him thrice as long as to row up as to row down the river. Find the rate of current in kmph.

**Solution :**

Let "x" be the rate of current.

Then,

Speed downstream = 18 - x

Speed upstream = 18 + x

Let us assume that the man travels 100 kms.

Time downstream = 100 / (18 + x) km/hr

Time upstream = 100 / (18 - x) km/hr

**Given : **It takes him thrice as long as to row up as to row down the river.

So, we have

Time upstream = 3 ⋅ Time downstream

100 / (18 - x) = 3 ⋅ 100 / (18 + x)

1 / (18 - x) = 3 / (18 + x)

1(18 + x) = 3(18 - x)

18 + x = 54 - 3x

4x = 36

x = 9

Hence, the rate of current is 9 kmph.

**Problem 3 :**

A boat takes 19 hours for traveling downstream from point A to point B and coming back to a point C midway between A and B. If the speed of the current is 4 kmph and the speed of the boat in still water is 14 kmph, what is the distance between A and B ?

**Solution :**

From the given information, we have

Speed downstream = 14 + 4 = 18 kmph

Speed upstream = 14 - 4 = 10 kmph

Let "x" be the distance between A and B.

Because C is the midpoint of A and B, the distance from B to C is x / 2.

The formula to find the time :

Time = Distance / Speed

We can use the above formula and find time downstream and time upstream as given below.

Time downstream (A to B) = x / 18

Time upstream (B to C) = (x/2) / 10 = x / 20

It is clear that,

Time downstream + Time upstream = Total time

(x / 18) + (x / 20) = 19

L.C.M of (18, 20) is 180.

Then, we have

(10x / 180) + (9x / 180) = 19

(10x + 9x) / 180 = 19

Multiply both sides by 180.

19x = 19 ⋅ 180

Divide both sides by 19.

x = 180

Hence the distance between A and B is 180 km.

**Problem 4 :**

A boat covers 24 miles upstream and 36 miles downstream in 6 hours while it covers 36 miles upstream and 24 miles down stream in 6.5 hours. Find the speed of the stream.

**Solution :**

Let "x" and "y" be speed upstream and downstream respectively.

The formula to find the time :

Time = Distance / Speed

We can use the above formula and find the following times.

Time taken for 24 miles upstream = 24 / x

Time taken for 36 miles downstream = 36 / y

**Given : **The boat covers 24 miles upstream and 36 miles downstream in 6 hours.

So, we have

(24 / x) + (36 / y) = 6 ------(1)

**Given : **The boat covers 36 miles upstream and 24 miles downstream in 6.5

So, we have

(36 / x) + (24 / y) = 6.5 ------(1)

Adding (1) and (2), we get,

(60 / x) + (60 / y) = 12.5

60(1/x + 1/y) = 12.5

1/x + 1/y = 12.5/60

1/x + 1/y = 5 / 24 ------(3)

Subtracting (1) from (2), we get

(12 / x) - (12 / y) = 1 / 2

12(1/x - 1/y) = 1 / 2

1/x - 1/y = 1 / 24 ------(4)

Adding (3) and (4), we get

2 / x = 6 / 24

x / 2 = 24 / 6

x / 2 = 4

x = 8

Plug x = 8 in (3).

(3)-----> 1/8 + 1/y = 5 / 24

1 / y = (5 / 24) - (1 / 8)

1 / y = 1 / 12

y / 1 = 12 / 1

y = 12

Speed of the stream is

= 1/2 ⋅ (y - x)

= 1/2 ⋅ (12 - 8)

= 1/2 ⋅ 4

= 2

Hence, the speed of the stream is 2 mph..

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