Sohan and Lallan leave from their house at same time. After 36 min, Sohan meet his uncle going towards house and Lallan meet his uncle 12 min later than that of Sohan. After meeting with Lallan and after 24 min., Uncle rings doorbell of Sohan and Lallan's house. Assume that each persons is walking at equal speed. Find the ratio of speeds of Sohan, Lallan and their uncle-

Options

A. A - `1 : 2 : 2`
B. B - `1 : 3 : 2`
C. C - `3 : 1 : 3`
D. D - `2 : 1 : 2`

Answer

To solve the problem step by step, let's analyze the information given and derive the required speeds and their ratio.

### Step 1: Understand the scenario
- Sohan and Lallan leave their house at the same time.
- Sohan meets his uncle after 36 minutes.
- Lallan meets his uncle 12 minutes after Sohan, which means Lallan meets his uncle after 48 minutes.
- After meeting Lallan, the uncle takes an additional 24 minutes to reach Sohan and Lallan's house.

### Step 2: Determine the time taken by the uncle
- The total time taken by the uncle to reach Sohan and Lallan's house can be calculated as follows:
- Time taken to meet Sohan = 36 minutes
- Time taken to meet Lallan = 48 minutes
- Time taken after meeting Lallan to reach home = 24 minutes
- Therefore, the total time taken by the uncle = 48 minutes (to meet Lallan) + 24 minutes = 72 minutes.

### Step 3: Calculate the distance covered
- Let's assume the speed of Sohan, Lallan, and their uncle is the same (denote it as 'v').
- The distance covered by Sohan when he meets his uncle is:
- Distance = Speed × Time = v × (36/60) hours = 0.6v (since 36 minutes = 36/60 hours).
- The distance covered by Lallan when he meets his uncle is:
- Distance = Speed × Time = v × (48/60) hours = 0.8v (since 48 minutes = 48/60 hours).
- The distance covered by the uncle to reach Sohan and Lallan's house is:
- Distance = Speed × Time = v × (72/60) hours = 1.2v (since 72 minutes = 72/60 hours).

### Step 4: Establish the relationship between distances
- Since Sohan and Lallan meet their uncle at different times but are walking towards the same point (their house), the distances they cover can be equated based on the time taken.
- The distance covered by the uncle when he meets Lallan is equal to the distance covered by Lallan:
- Distance covered by uncle = Distance covered by Lallan
- 1.2v = 0.8v + Distance from Lallan to home.
- The distance from Lallan to home can be calculated as:
- Distance = Speed × Time = v × (24/60) hours = 0.4v.

### Step 5: Solve for the speeds
- Now we can set up the equation:
- 1.2v = 0.8v + 0.4v
- This simplifies to:
- 1.2v = 1.2v, which confirms our assumption that the speeds are equal.

### Step 6: Calculate the ratio of speeds
- Since Sohan, Lallan, and their uncle are all walking at the same speed, the ratio of their speeds is:
- Speed of Sohan : Speed of Lallan : Speed of Uncle = v : v : v = 1 : 1 : 1.

### Final Answer
The ratio of speeds of Sohan, Lallan, and their uncle is **1 : 1 : 1**.