Michael lives 10 km away from where I live. Ahmed lives 5 km away and Susan lives 7 km away from where I live. Arun is farther away than Ahmed but closer than Susan from where I live. From the information provided here, what is one possible distance (in km) at which I live from Arun’s place?
6.02
Understanding the Distances
The problem provides specific distances for Michael, Ahmed, and Susan from my place of residence. We need to determine a possible distance for Arun based on the given relative information.
| Person | Distance from My Place (in km) |
|---|---|
| Michael | 10 |
| Ahmed | 5 |
| Susan | 7 |
Arun's Distance Conditions
The problem states two conditions for Arun's distance from my place:
- Arun is farther away than Ahmed.
- Arun is closer than Susan.
Let D_{Arun} represent the distance of Arun's place from my place.
From the first condition, "Arun is farther away than Ahmed", we know that Arun's distance must be strictly greater than Ahmed's distance. Ahmed lives 5 km away from my place. Therefore, we can write the first inequality as:
$$ D_{Arun} \gt 5 \text{ km} $$
From the second condition, "Arun is closer than Susan", we know that Arun's distance must be strictly less than Susan's distance. Susan lives 7 km away from my place. Therefore, we can write the second inequality as:
$$ D_{Arun} \lt 7 \text{ km} $$
Combining Arun's Distance Inequalities
By combining both of these conditions, we can establish a specific range within which Arun's distance must fall. Arun's distance must be greater than 5 km and less than 7 km simultaneously. This combined condition can be written as a single inequality:
$$ 5 \text{ km} \lt D_{Arun} \lt 7 \text{ km} $$
This means that any valid distance for Arun must be a value strictly between 5 km and 7 km.
Evaluating the Possible Distances
Now, let's examine the given options to see which one satisfies the condition that the distance must be greater than 5 km and less than 7 km.
| Option | Distance (km) | Satisfies 5 \text{ km} \lt D_{Arun} \lt 7 \text{ km}? |
|---|---|---|
| 1 | 3.00 | No ($3.00 \text{ km} \not\gt 5 \text{ km}$) |
| 2 | 4.99 | No ($4.99 \text{ km} \not\gt 5 \text{ km}$) |
| 3 | 6.02 | Yes ($5 \text{ km} \lt 6.02 \text{ km} \lt 7 \text{ km}$) |
| 4 | 7.01 | No ($7.01 \text{ km} \not\lt 7 \text{ km}$) |
Concluding the Possible Distance for Arun
From the analysis of the options, only 6.02 km falls within the required range of (5 km, 7 km). Therefore, 6.02 km is a possible distance at which I live from Arun's place.
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