L can run 1 km in 10 min while M can run it in 12 min . How many metres start can L give M in a km race , so that the race may end in a dead heat ?

Options

A. ` 166 2/3 m`
B. `166 1/3 m`
C. `166 3/5 m`
D. `200 m`

Answer

To solve the problem of how many meters start L can give M in a 1 km race so that the race ends in a dead heat, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the speeds of L and M:**
- L can run 1 km in 10 minutes.
- M can run 1 km in 12 minutes.
- Convert these times into seconds:
- L's time = 10 minutes = 10 × 60 = 600 seconds.
- M's time = 12 minutes = 12 × 60 = 720 seconds.

2. **Calculate the speeds:**
- Speed is calculated as distance divided by time.
- Speed of L = Distance / Time = 1000 meters / 600 seconds = 5/3 meters per second.
- Speed of M = Distance / Time = 1000 meters / 720 seconds = 25/18 meters per second.

3. **Determine the time difference:**
- To finish at the same time, we need to find out how much time L has over M.
- The time difference is 720 seconds (M's time) - 600 seconds (L's time) = 120 seconds.

4. **Calculate the distance M can cover in that time:**
- We need to find out how far M can run in the 120 seconds that L has over M.
- Distance covered by M in 120 seconds = Speed of M × Time = (25/18) meters/second × 120 seconds.

5. **Perform the calculation:**
- Distance = (25/18) × 120 = (25 × 120) / 18 = 3000 / 18 = 500 / 3 meters.

6. **Convert to mixed fraction:**
- 500 / 3 = 166 and 2/3 meters.

### Final Answer:
L can give M a start of **166 and 2/3 meters** in a 1 km race for it to end in a dead heat.

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