Understanding Meeting Points in Opposite Directions

When two individuals, Siddharth and Yash, run on a circular race track in opposite directions, they will meet at various points along the circumference. The key is to determine how many unique locations these meetings occur at. The number of these distinct meeting points depends on their relative speeds, which can be derived from the time they take to complete one round.

Calculating Speed Ratios

To find the number of meeting points, we first need to understand the relationship between their speeds. Speed is defined as distance covered per unit of time.

Let the length of the circular race track be represented by '$C$'.

• Siddharth's time for one round ($T_S$) = 85 seconds.
• Yash's time for one round ($T_Y$) = 45 seconds.

We can express their speeds:

• Siddharth's speed ($v_S$) = $\frac{C}{T_S} = \frac{C}{85}$ meters per second.
• Yash's speed ($v_Y$) = $\frac{C}{T_Y} = \frac{C}{45}$ meters per second.

The ratio of their speeds is:

$$ \frac{v_S}{v_Y} = \frac{C/T_S}{C/T_Y} = \frac{T_Y}{T_S} $$

Substituting the given times:

$$ \frac{v_S}{v_Y} = \frac{45}{85} $$

To simplify this ratio, we find the greatest common divisor (GCD) of 45 and 85, which is 5:

$$ \frac{v_S}{v_Y} = \frac{45 \div 5}{85 \div 5} = \frac{9}{17} $$

This implies that their speeds are in the ratio 9:17. For every 9 units of distance Siddharth covers, Yash covers 17 units.

Determining Distinct Meeting Points

When two people run in opposite directions on a circular track, they meet whenever the sum of the distances they have covered equals an integer multiple of the track's circumference. Let '$t$' be the time elapsed since they started, and let '$n$' be a positive integer representing the number of times the sum of their distances equals the circumference.

Distance covered by Siddharth = $d_S = v_S \times t$

Distance covered by Yash = $d_Y = v_Y \times t$

Meeting condition: $d_S + d_Y = n \times C$

Substituting the speeds ($v_S = 9k$ and $v_Y = 17k$ for some constant $k$ representing the speed unit):

$$ (9k)t + (17k)t = nC $$

$$ 26kt = nC $$

We can express the time '$t$' as: $$ t = \frac{nC}{26k} $$

Now, let's find the position of Siddharth at the time of meeting. The distance Siddharth covers is:

$$ d_S = v_S \times t = (9k) \times \frac{nC}{26k} = \frac{9nC}{26} $$

The position of the meeting point on the circumference can be represented as a fraction of the total circumference $C$. This fraction is:

$$ \text{Position Fraction} = \frac{d_S}{C} = \frac{9n}{26} $$

We are interested in the number of *distinct* meeting points. This means we need to find how many unique values the expression $\frac{9n}{26} \pmod{1}$ can take as '$n$' increases ($n = 1, 2, 3, \dots$).

The number of distinct values for an expression of the form $\frac{an}{m} \pmod{1}$ is given by the formula $\frac{m}{\text{gcd}(a, m)}$.

In this case, $a=9$ and $m=26$. We need to calculate the greatest common divisor of 9 and 26.

• Factors of 9 are 1, 3, 9.
• Factors of 26 are 1, 2, 13, 26.

The greatest common divisor is $\text{gcd}(9, 26) = 1$.

Final Answer Calculation

Using the formula for distinct meeting points:

$$ \text{Number of distinct meeting points} = \frac{m}{\text{gcd}(a, m)} = \frac{26}{\text{gcd}(9, 26)} $$

$$ \text{Number of distinct meeting points} = \frac{26}{1} = 26 $$

Therefore, there are 26 different meeting points on the circumference when Siddharth and Yash run in opposite directions.