In a 1000 metres race Ravi gives Vinod a start of 40 m and beats him by 19 seconds. If Ravi gives a start of 30 seconds then Vinod beats Ravi by 40 m. What is the ratio of speed of Ravi to that of Vinod?

Options

A. `4:5`
B. `6:5`
C. `3:8`
D. `5:4`

Answer

To solve the problem, we will analyze the two scenarios given in the question step by step.

### Step 1: Analyze the first scenario
In the first scenario, Ravi gives Vinod a start of 40 meters in a 1000-meter race. This means:
- Ravi runs 1000 meters.
- Vinod runs \(1000 - 40 = 960\) meters.

Let \(T_1\) be the time taken by Ravi to finish the race. Since Ravi beats Vinod by 19 seconds, Vinod takes \(T_1 + 19\) seconds to finish his race.

### Step 2: Establish equations for speeds
We can express the speeds of Ravi and Vinod as follows:
- Speed of Ravi, \(S_R = \frac{1000}{T_1}\)
- Speed of Vinod, \(S_V = \frac{960}{T_1 + 19}\)

### Step 3: Analyze the second scenario
In the second scenario, Ravi gives Vinod a start of 30 seconds. This means:
- When Vinod finishes the race (1000 meters), Ravi has run \(1000 - 40 = 960\) meters.
- Let \(T_2\) be the time taken by Ravi to cover 960 meters.

Since Vinod starts 30 seconds earlier, he takes \(T_2 + 30\) seconds to finish his race.

### Step 4: Establish equations for the second scenario
For this scenario, we can express the speeds as:
- Speed of Ravi, \(S_R = \frac{960}{T_2}\)
- Speed of Vinod, \(S_V = \frac{1000}{T_2 + 30}\)

### Step 5: Set the equations equal
Since both expressions equal the speed of Ravi, we can set them equal to each other:
\[ \frac{1000}{T_1} = \frac{960}{T_2} \]
Cross-multiplying gives us:
\[ 1000 T_2 = 960 T_1 \quad \Rightarrow \quad T_1 = \frac{1000}{960} T_2 = \frac{25}{24} T_2 \]

### Step 6: Substitute \(T_1\) into Vinod's speed equation from the first scenario
Now, substituting \(T_1\) into Vinod's speed equation from the first scenario:
\[ S_V = \frac{960}{T_1 + 19} = \frac{960}{\frac{25}{24} T_2 + 19} \]

### Step 7: Set the second scenario's speed equations equal
From the second scenario, we have:
\[ S_V = \frac{1000}{T_2 + 30} \]
Setting these two expressions for \(S_V\) equal gives:
\[ \frac{960}{\frac{25}{24} T_2 + 19} = \frac{1000}{T_2 + 30} \]

### Step 8: Cross-multiply and simplify
Cross-multiplying:
\[ 960 (T_2 + 30) = 1000 \left(\frac{25}{24} T_2 + 19\right) \]
Expanding both sides:
\[ 960 T_2 + 28800 = \frac{25000}{24} T_2 + 19000 \]

### Step 9: Combine like terms
To eliminate the fraction, multiply the entire equation by 24:
\[ 23040 T_2 + 691200 = 25000 T_2 + 456000 \]
Rearranging gives:
\[ 691200 - 456000 = 25000 T_2 - 23040 T_2 \]
\[ 235200 = 1960 T_2 \quad \Rightarrow \quad T_2 = \frac{235200}{1960} = 120 \text{ seconds} \]

### Step 10: Find \(T_1\)
Now substituting \(T_2\) back to find \(T_1\):
\[ T_1 = \frac{25}{24} \times 120 = 125 \text{ seconds} \]

### Step 11: Calculate the speeds
Now we can calculate the speeds:
- Speed of Ravi:
\[ S_R = \frac{1000}{T_1} = \frac{1000}{125} = 8 \text{ m/s} \]
- Speed of Vinod:
\[ S_V = \frac{1000}{T_2 + 30} = \frac{1000}{120 + 30} = \frac{1000}{150} = \frac{20}{3} \text{ m/s} \]

### Step 12: Find the ratio of speeds
Finally, the ratio of the speeds of Ravi to Vinod is:
\[ \text{Ratio} = \frac{S_R}{S_V} = \frac{8}{\frac{20}{3}} = 8 \times \frac{3}{20} = \frac{24}{20} = \frac{6}{5} \]

### Final Answer
The ratio of the speed of Ravi to that of Vinod is \(6:5\).