Understanding the Market Meeting Problem

The problem asks us to find the next day Ramu and Ravi will meet in the market, given their meeting pattern and the date of their previous meeting.

Here's what we know:

• Ramu goes to the market every 4th day.
• Ravi goes to the market every 5th day.
• They last met on the 5th day of the month.

We need to find the earliest day after the 5th day when both Ramu and Ravi will be in the market again.

Applying the Concept of Least Common Multiple (LCM)

Ramu's market visits occur on days that are multiples of 4 (relative to his first visit after the 5th). Ravi's visits occur on days that are multiples of 5 (relative to his first visit after the 5th).

For them to meet, the number of days passed since their last meeting must be a multiple of both 4 and 5. The soonest they will meet again is after a number of days equal to the Least Common Multiple (LCM) of 4 and 5.

Calculating the LCM of 4 and 5

To find the LCM of 4 and 5, we can list the multiples of each number:

• Multiples of 4: 4, 8, 12, 16, 20, 24, ...
• Multiples of 5: 5, 10, 15, 20, 25, ...

The smallest common multiple is 20.

Alternatively, since 4 and 5 are prime to each other (they have no common factors other than 1), their LCM is simply their product.

LCM(4, 5) $= 4 \times 5 = 20$

This means they will meet again 20 days after their last meeting.

Calculating the Next Market Meeting Day

They last met on the 5th day of the month.

The number of days until they meet again is the LCM of 4 and 5, which is 20 days.

To find the date of their next meeting, we add the number of days until the next meeting to the date of their last meeting:

Next Meeting Day = Day of Last Meeting + LCM

Next Meeting Day $= 5 + 20 = 25$

So, they will meet again on the 25th day of the month.

Let's verify:

• Ramu visits every 4 days after the 5th: Day 5, Day $5+4=9$, Day $9+4=13$, Day $13+4=17$, Day $17+4=21$, Day $21+4=\textbf{25}$, ...
• Ravi visits every 5 days after the 5th: Day 5, Day $5+5=10$, Day $10+5=15$, Day $15+5=20$, Day $20+5=\textbf{25}$, ...

The next day they both visit is the 25th day.

Final Answer

Based on the calculation, Ramu and Ravi will meet again on the 25th day of the month.

Revision Table: Key Concepts

Additional Information: LCM and Real-World Problems

The concept of the Least Common Multiple (LCM) is very useful for solving problems involving events that happen at regular intervals, just like Ramu and Ravi's market visits. Here are a few other examples:

• Scheduling: Finding when two buses will arrive at the same stop at the same time again, given their respective schedules.
• Cycling: Determining when two or more cyclists, riding on a track at different speeds, will cross the starting line together again.
• Machine Cycles: In manufacturing, finding when two machines operating on different cycles will complete a process step simultaneously.

In all these cases, you find the LCM of the time intervals to determine when the events will coincide again. Understanding LCM helps solve various problems related to cycles and synchronization.