Veena and Karmesh Meeting Time Calculation

This problem involves calculating when two people traveling towards each other will meet. We are given their starting points, starting times, and arrival times over the same route.

Understanding Travel Details

First, let's break down the information given for Veena and Karmesh:

Note: Karmesh started 2 hours later than Veena (8:00 a.m. + 2 hours = 10:00 a.m.).

Calculating Speeds and Initial Distance

To find out when they meet, we need to know their speeds. Let's assume the distance between Place A and Place B is $D$.

• Veena's total travel time = 8 hours. Her speed ($V_V$) is:

  $$ V_V = \frac{\text{Distance}}{\text{Time}} = \frac{D}{8} \text{ units per hour} $$

• Karmesh's total travel time = 7 hours. His speed ($V_K$) is:

  $$ V_K = \frac{\text{Distance}}{\text{Time}} = \frac{D}{7} \text{ units per hour} $$

Karmesh starts at 10:00 a.m. By this time, Veena has already been traveling for 2 hours (from 8:00 a.m. to 10:00 a.m.).

Distance covered by Veena in these 2 hours:

$$ \text{Distance covered by Veena} = V_V \times 2 \text{ hours} = \left( \frac{D}{8} \right) \times 2 = \frac{2D}{8} = \frac{D}{4} $$

So, at 10:00 a.m., when Karmesh starts from Place B, Veena has already covered $D/4$ of the total distance. The remaining distance between them is:

$$ \text{Remaining Distance} = D - \frac{D}{4} = \frac{3D}{4} $$

Determining the Meeting Time

From 10:00 a.m. onwards, both Veena and Karmesh are traveling towards each other. To find the time it takes for them to meet, we use the concept of relative speed.

Their relative speed is the sum of their individual speeds because they are moving towards each other:

$$ \text{Relative Speed} = V_V + V_K = \frac{D}{8} + \frac{D}{7} $$

To add these fractions, we find a common denominator, which is 56:

$$ \text{Relative Speed} = \frac{7D}{56} + \frac{8D}{56} = \frac{15D}{56} \text{ units per hour} $$

Now, we can calculate the time it takes for them to cover the remaining distance ($3D/4$) at their relative speed:

$$ \text{Time to meet} = \frac{\text{Remaining Distance}}{\text{Relative Speed}} = \frac{3D/4}{15D/56} $$

$$ \text{Time to meet} = \frac{3D}{4} \times \frac{56}{15D} $$

The $D$ cancels out:

$$ \text{Time to meet} = \frac{3}{4} \times \frac{56}{15} = \frac{3 \times 56}{4 \times 15} = \frac{168}{60} \text{ hours} $$

Simplify the fraction:

$$ \text{Time to meet} = \frac{168 \div 12}{60 \div 12} = \frac{14}{5} \text{ hours} $$

Converting this to hours and minutes:

$$ \frac{14}{5} \text{ hours} = 2 \frac{4}{5} \text{ hours} $$

$$ \frac{4}{5} \text{ hours} = \frac{4}{5} \times 60 \text{ minutes} = 4 \times 12 \text{ minutes} = 48 \text{ minutes} $$

So, they will meet 2 hours and 48 minutes after Karmesh started (which was at 10:00 a.m.).

Meeting Time = 10:00 a.m. + 2 hours 48 minutes = 12:48 p.m.