Answer :
### Solution:
Let's address each part of the question step by step.
#### Part 1: When will Akbar and Birbal meet?
Given:
- Distance apart = 15 km
- Speed of Akbar = 4 km/h
- Speed of Birbal = 6 km/h
To determine the time they will meet, we need to calculate the relative speed when both walk towards each other.
1. Calculate the total relative speed:
[tex]\[ \text{Total speed} = \text{Speed of Akbar} + \text{Speed of Birbal} = 4 \text{ km/h} + 6 \text{ km/h} = 10 \text{ km/h} \][/tex]
2. Calculate the time taken to meet:
[tex]\[ \text{Time to meet} = \frac{\text{Distance apart}}{\text{Total speed}} = \frac{15 \text{ km}}{10 \text{ km/h}} = 1.5 \text{ hours} \][/tex]
3. Convert the time to hours and minutes:
[tex]\[ 1.5 \text{ hours} = 1 \text{ hour} + 0.5 \text{ hour} = 1 \text{ hour} + 30 \text{ minutes} \][/tex]
4. Calculate the meeting time if they start at 10:00 am:
[tex]\[ 10:00 \text{ am} + 1 \text{ hour} + 30 \text{ minutes} = 11:30 \text{ am} \][/tex]
Thus, Akbar and Birbal will meet at 11:30 am.
#### Part 2: At what time will Akbar and Birbal reach home after chatting?
Given:
- They chat for 30 minutes after they meet.
- Both walk back home at their respective speeds.
1. Calculate the time they start walking back home:
[tex]\[ \text{Meeting time} = 11:30 \text{ am} \][/tex]
[tex]\[ \text{Chatting time} = 30 \text{ minutes} \][/tex]
[tex]\[ \text{Start walking back home} = 12:00 \text{ pm} (noon) \][/tex]
2. Calculate the time taken by Birbal to walk home:
- The distance for each to their home from meeting point is half of the initial distance:
[tex]\[ \text{Distance for Birbal} = \frac{15 \text{ km}}{2} = 7.5 \text{ km} \][/tex]
- Birbal's speed = 6 km/h:
[tex]\[ \text{Time to walk home} = \frac{7.5 \text{ km}}{6 \text{ km/h}} = 1.25 \text{ hours} = 1 \text{ hour} + 15 \text{ minutes} \][/tex]
- Time when Birbal reaches home:
[tex]\[ 12:00 \text{ pm} + 1 \text{ hour} + 15 \text{ minutes} = 1:15 \text{ pm} \][/tex]
3. Calculate the time taken by Akbar to walk home:
- The distance for Akbar is the same, 7.5 km:
[tex]\[ \text{Distance for Akbar} = \frac{15 \text{ km}}{2} = 7.5 \text{ km} \][/tex]
- Akbar's speed = 4 km/h:
[tex]\[ \text{Time to walk home} = \frac{7.5 \text{ km}}{4 \text{ km/h}} = 1.875 \text{ hours} = 1 \text{ hour} + 52.5 \text{ minutes} \][/tex]
- Time when Akbar reaches home:
[tex]\[ 12:00 \text{ pm} + 1 \text{ hour} + 52.5 \text{ minutes} = 1:52.5 \text{ pm} \approx 1:53 \text{ pm} \][/tex]
Thus, Birbal will reach home at 1:15 pm and Akbar will reach home at approximately 1:53 pm.
Let's address each part of the question step by step.
#### Part 1: When will Akbar and Birbal meet?
Given:
- Distance apart = 15 km
- Speed of Akbar = 4 km/h
- Speed of Birbal = 6 km/h
To determine the time they will meet, we need to calculate the relative speed when both walk towards each other.
1. Calculate the total relative speed:
[tex]\[ \text{Total speed} = \text{Speed of Akbar} + \text{Speed of Birbal} = 4 \text{ km/h} + 6 \text{ km/h} = 10 \text{ km/h} \][/tex]
2. Calculate the time taken to meet:
[tex]\[ \text{Time to meet} = \frac{\text{Distance apart}}{\text{Total speed}} = \frac{15 \text{ km}}{10 \text{ km/h}} = 1.5 \text{ hours} \][/tex]
3. Convert the time to hours and minutes:
[tex]\[ 1.5 \text{ hours} = 1 \text{ hour} + 0.5 \text{ hour} = 1 \text{ hour} + 30 \text{ minutes} \][/tex]
4. Calculate the meeting time if they start at 10:00 am:
[tex]\[ 10:00 \text{ am} + 1 \text{ hour} + 30 \text{ minutes} = 11:30 \text{ am} \][/tex]
Thus, Akbar and Birbal will meet at 11:30 am.
#### Part 2: At what time will Akbar and Birbal reach home after chatting?
Given:
- They chat for 30 minutes after they meet.
- Both walk back home at their respective speeds.
1. Calculate the time they start walking back home:
[tex]\[ \text{Meeting time} = 11:30 \text{ am} \][/tex]
[tex]\[ \text{Chatting time} = 30 \text{ minutes} \][/tex]
[tex]\[ \text{Start walking back home} = 12:00 \text{ pm} (noon) \][/tex]
2. Calculate the time taken by Birbal to walk home:
- The distance for each to their home from meeting point is half of the initial distance:
[tex]\[ \text{Distance for Birbal} = \frac{15 \text{ km}}{2} = 7.5 \text{ km} \][/tex]
- Birbal's speed = 6 km/h:
[tex]\[ \text{Time to walk home} = \frac{7.5 \text{ km}}{6 \text{ km/h}} = 1.25 \text{ hours} = 1 \text{ hour} + 15 \text{ minutes} \][/tex]
- Time when Birbal reaches home:
[tex]\[ 12:00 \text{ pm} + 1 \text{ hour} + 15 \text{ minutes} = 1:15 \text{ pm} \][/tex]
3. Calculate the time taken by Akbar to walk home:
- The distance for Akbar is the same, 7.5 km:
[tex]\[ \text{Distance for Akbar} = \frac{15 \text{ km}}{2} = 7.5 \text{ km} \][/tex]
- Akbar's speed = 4 km/h:
[tex]\[ \text{Time to walk home} = \frac{7.5 \text{ km}}{4 \text{ km/h}} = 1.875 \text{ hours} = 1 \text{ hour} + 52.5 \text{ minutes} \][/tex]
- Time when Akbar reaches home:
[tex]\[ 12:00 \text{ pm} + 1 \text{ hour} + 52.5 \text{ minutes} = 1:52.5 \text{ pm} \approx 1:53 \text{ pm} \][/tex]
Thus, Birbal will reach home at 1:15 pm and Akbar will reach home at approximately 1:53 pm.
