Total distance between A and B is d kms . If the distance travelled along the stream is three time of the total distance and the distance travelled against the stream is two times of the total distance . If the time taken to cover the distance along the stream is 10% less then the time taken to cover the distance against the stream .if a person cover a distance of 21 km in 1 hr 24 min along the stream , then find the rate of current ?

Options

A. 2 km /hr
B. 3 km/hr
C. 1 km/hr
D. 4 km/hr

Explanation

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Define the Variables
Let:
- \( D \) = total distance between A and B (in km)
- \( X \) = speed of the person in still water (in km/hr)
- \( Y \) = speed of the current (in km/hr)

### Step 2: Distances Along and Against the Stream
According to the problem:
- Distance travelled along the stream = \( 3D \)
- Distance travelled against the stream = \( 2D \)

### Step 3: Time Taken to Travel Distances
The time taken to cover the distance along the stream can be expressed as:
\[ \text{Time along the stream} = \frac{3D}{X + Y} \]
The time taken to cover the distance against the stream can be expressed as:
\[ \text{Time against the stream} = \frac{2D}{X - Y} \]

### Step 4: Relationship Between the Times
We know that the time taken along the stream is 10% less than the time taken against the stream:
\[ \frac{3D}{X + Y} = 0.9 \times \frac{2D}{X - Y} \]

### Step 5: Simplifying the Equation
We can simplify this equation by eliminating \( D \) (since it is common in both terms):
\[ \frac{3}{X + Y} = 0.9 \times \frac{2}{X - Y} \]
Cross-multiplying gives:
\[ 3(X - Y) = 1.8(X + Y) \]
Expanding both sides:
\[ 3X - 3Y = 1.8X + 1.8Y \]
Rearranging the terms:
\[ 3X - 1.8X = 3Y + 1.8Y \]
This simplifies to:
\[ 1.2X = 4.8Y \]
Dividing both sides by 1.2:
\[ X = 4Y \]

### Step 6: Using the Given Information
We know that the person covers a distance of 21 km in 1 hour 24 minutes (which is 84 minutes or \( \frac{84}{60} = 1.4 \) hours) along the stream:
\[ X + Y = \frac{21}{1.4} = 15 \text{ km/hr} \]

### Step 7: Setting Up the Equations
Now we have two equations:
1. \( X = 4Y \)
2. \( X + Y = 15 \)

### Step 8: Substitute and Solve
Substituting \( X = 4Y \) into the second equation:
\[ 4Y + Y = 15 \]
This simplifies to:
\[ 5Y = 15 \]
Thus:
\[ Y = 3 \text{ km/hr} \]

### Step 9: Find the Rate of Current
The rate of current \( Y \) is therefore:
\[ \text{Rate of current} = 3 \text{ km/hr} \]

### Final Answer
The rate of current is **3 km/hr**.

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