Understanding the Circular Track Crossing Problem

This problem involves two individuals, P and Q, moving in opposite directions on a circular track from the same starting point. To find out how many times they cross each other, we first need to determine their relative speed. When objects move in opposite directions, their speeds add up to give the relative speed at which the distance between them changes.

Calculating Relative Speed on the Track

P walks at a speed of 5 rounds per hour.

Q walks at a speed of 3 rounds per hour.

Since they are moving in opposite directions, their relative speed is the sum of their individual speeds:

Relative Speed = Speed of P + Speed of Q

Relative Speed = 5 rounds/hour + 3 rounds/hour = 8 rounds/hour.

This means that the distance between them effectively covers 8 full rounds of the track every hour. They cross each other every time their combined distance covers exactly one full round. Therefore, they will cross each other 8 times every hour.

Determining the Time Interval

The problem asks for the number of crossings between 5:20 a.m. and 7:00 a.m.

The starting time is 5:00 a.m.

We need to consider the interval from 5:20 a.m. up to, but not including, 7:00 a.m.

Calculating the Time of Each Crossing

They cross each other when their relative distance covered is 1 round, 2 rounds, 3 rounds, and so on.

The time it takes for them to cover one relative round is:

Time per crossing = $\frac{1 \text{ round}}{\text{Relative Speed}} = \frac{1 \text{ round}}{8 \text{ rounds/hour}} = \frac{1}{8}$ hour.

Converting this to minutes: $\frac{1}{8} \text{ hour} \times 60 \text{ minutes/hour} = 7.5$ minutes.

So, the crossings occur at intervals of 7.5 minutes starting from after 5:00 a.m.

The times of crossing relative to the start time (5:00 a.m.) are at $n \times 7.5$ minutes, where $n$ is the crossing number (1st, 2nd, 3rd, ...).

• 1st crossing: $1 \times 7.5 = 7.5$ minutes past 5:00 a.m. (at 5:07:30 a.m.)
• 2nd crossing: $2 \times 7.5 = 15$ minutes past 5:00 a.m. (at 5:15:00 a.m.)
• 3rd crossing: $3 \times 7.5 = 22.5$ minutes past 5:00 a.m. (at 5:22:30 a.m.)
• and so on...

Counting Crossings Between 5:20 a.m. and 7:00 a.m.

The time interval is from 5:20 a.m. to 7:00 a.m. We need to find the crossings that occur strictly within this open interval (5:20 a.m., 7:00 a.m.).

Let's look at the crossing times:

• Crossings up to 5:20 a.m.: 5:07:30 a.m. (1st), 5:15:00 a.m. (2nd). The 3rd crossing is at 5:22:30 a.m., which is after 5:20 a.m. So, 2 crossings occurred before or at 5:20 a.m.
• Crossings up to 7:00 a.m.: The total time from 5:00 a.m. to 7:00 a.m. is 2 hours, which is $2 \times 60 = 120$ minutes. The number of crossings up to 7:00 a.m. can be found by dividing the total time by the time per crossing: $\frac{120 \text{ minutes}}{7.5 \text{ minutes/crossing}} = 16$ crossings. The 16th crossing occurs exactly at 7:00:00 a.m.

We want the number of crossings that happen strictly after 5:20 a.m. and strictly before 7:00 a.m.

The first crossing after 5:20 a.m. is the 3rd crossing (at 5:22:30 a.m.).

The last crossing before 7:00 a.m. is the one just before the 16th crossing. The 16th crossing is at 7:00 a.m. ($16 \times 7.5 = 120$ minutes past 5:00 a.m.). So, the 15th crossing is at $15 \times 7.5 = 112.5$ minutes past 5:00 a.m., which is at 5:00 a.m. + 112.5 minutes = 6:52:30 a.m.

So, the crossings between 5:20 a.m. and 7:00 a.m. are the 3rd, 4th, 5th, ..., up to the 15th crossing.

The number of crossings from the 3rd to the 15th is $15 - 3 + 1 = 13$.

Alternatively, we can think about the time elapsed from 5:00 a.m.

• 5:20 a.m. is 20 minutes from 5:00 a.m. ($20/60 = 1/3$ hour). Number of crossings completed exactly at or before 5:20 a.m. = $\lfloor \text{Relative Speed} \times \text{Time} \rfloor = \lfloor 8 \times \frac{1}{3} \rfloor = \lfloor 2.66... \rfloor = 2$.
• 7:00 a.m. is 120 minutes from 5:00 a.m. (2 hours). Number of crossings completed exactly at or before 7:00 a.m. = $\lfloor \text{Relative Speed} \times \text{Time} \rfloor = \lfloor 8 \times 2 \rfloor = \lfloor 16 \rfloor = 16$.

The crossings occur at times $t_n = n/8$ hours from 5:00 a.m. We want the number of values of $n$ such that $1/3 < n/8 < 2$.

Multiplying by 8, we get $8/3 < n < 16$.

Which simplifies to $2.66... < n < 16$.

Since $n$ must be an integer representing the crossing number (starting from 1), the possible values for $n$ are 3, 4, 5, ..., 15.

The number of integers in this range is $15 - 3 + 1 = 13$.

The crossings are the 3rd through the 15th, inclusive. The total count is 13.

Final Answer Summary

P and Q walk on a circular track in opposite directions. Their relative speed is 8 rounds per hour. This means they cross each other every $\frac{1}{8}$ hour, or 7.5 minutes.

The crossings occur at 5:07:30, 5:15:00, 5:22:30, ..., up to 7:00:00.

We need the crossings between 5:20 a.m. and 7:00 a.m. (exclusive of the endpoints).

The first crossing after 5:20 a.m. is at 5:22:30 a.m. (the 3rd crossing).

The last crossing before 7:00 a.m. is at 6:52:30 a.m. (the 15th crossing).

The crossing at 7:00:00 a.m. (the 16th crossing) is not strictly between 5:20 a.m. and 7:00 a.m.

So, the crossings are the 3rd, 4th, ..., 15th crossings.

Number of crossings = 15 - 3 + 1 = 13.

Revision Table: Key Concepts

Additional Information: Circular Track Problems

Circular track problems often involve relative speed. The concept changes slightly depending on whether the people move in the same or opposite directions.

• Same Direction: Relative speed is the difference between the faster and slower speeds. They meet when the faster person has gained one round on the slower person.
• Opposite Directions: Relative speed is the sum of the speeds. They meet/cross when their combined distance covers one round.

The number of times they meet/cross in a given time 'T' (in hours) is $\lfloor \text{Relative Speed} \times T \rfloor$ or related to it, depending on whether the start point is counted as a meeting and whether the endpoints of the time interval are inclusive or exclusive.

In problems where they start at the same point and move in opposite directions, the first crossing happens after some time, not at the start. Subsequent crossings happen at regular intervals based on the relative speed.