A dog after travelling 50 km meets a swami who counsels him to go slower. He then proceeds at 3/4 of his former speed and arrives at his destination 35 minutes late. Had the meeting occurred 24 km further the dog would have reached its destination 25 minutes late. The speed of the dog is

Options

A. 48 km/h
B. 36 km/h
C. 54 km/h
D. 58 km/h

Answer

To solve the problem, we need to determine the speed of the dog based on the information provided. Let's break it down step by step.

### Step 1: Define Variables
Let the original speed of the dog be \( v \) km/h.

### Step 2: Determine Distances and Times
The dog travels 50 km before meeting the swami. After the meeting, he travels at \( \frac{3}{4}v \) km/h.

Let the total distance to the destination be \( D \) km. After the meeting, the remaining distance to the destination is \( D - 50 \) km.

### Step 3: Calculate Time Taken
1. **Time taken to travel the remaining distance at original speed:**
\[ \text{Time} = \frac{D - 50}{v} \]

2. **Time taken to travel the remaining distance at reduced speed:**
\[ \text{Time} = \frac{D - 50}{\frac{3}{4}v} = \frac{4(D - 50)}{3v} \]

### Step 4: Set Up the First Equation
According to the problem, after meeting the swami, the dog arrives 35 minutes late. This means:
\[ \frac{4(D - 50)}{3v} = \frac{D - 50}{v} + \frac{35}{60} \]

### Step 5: Simplify the First Equation
Multiply through by \( 3v \) to eliminate the denominators:
\[ 4(D - 50) = 3(D - 50) + \frac{35}{20}v \]
\[ 4D - 200 = 3D - 150 + \frac{35}{20}v \]
\[ D - 50 = \frac{35}{20}v \]
\[ D = \frac{35}{20}v + 50 \]

### Step 6: Set Up the Second Equation
If the meeting occurred 24 km further (i.e., after traveling 74 km), the remaining distance would be \( D - 74 \) km. The dog would be 25 minutes late:
\[ \frac{4(D - 74)}{3v} = \frac{D - 74}{v} + \frac{25}{60} \]

### Step 7: Simplify the Second Equation
Multiply through by \( 3v \):
\[ 4(D - 74) = 3(D - 74) + \frac{25}{20}v \]
\[ 4D - 296 = 3D - 222 + \frac{25}{20}v \]
\[ D - 74 = \frac{25}{20}v \]
\[ D = \frac{25}{20}v + 74 \]

### Step 8: Set the Two Equations for D Equal
Now we have two expressions for \( D \):
1. \( D = \frac{35}{20}v + 50 \)
2. \( D = \frac{25}{20}v + 74 \)

Setting them equal:
\[ \frac{35}{20}v + 50 = \frac{25}{20}v + 74 \]

### Step 9: Solve for v
Rearranging gives:
\[ \frac{35}{20}v - \frac{25}{20}v = 74 - 50 \]
\[ \frac{10}{20}v = 24 \]
\[ v = 24 \times 2 = 48 \text{ km/h} \]

### Final Answer
The speed of the dog is **48 km/h**.

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