MBA Nexus - CAT & OMETS 2024, [29-10-2024 09:13]
Q6 A and B started running simultaneously from a point around a circular track of 900 m. The ratio of their speeds is 3 : 7. For how many times will A and B be at a distance of 10 m from each other, if they are running in opposite directions and stops only when they meet for the first time at the starting point again?

MBA Nexus - CAT & OMETS 2024, [29-10-2024 09:13]
[ Poll : Ans ]
- 20
- 10
- 16
- Can't be determined

To solve this, let's analyze the given conditions:

1. **Track Length**: 900 meters.
2. **Ratio of Speeds of A and B**: 3 : 7.
3. **They are running in opposite directions**.

Let's break down the solution:

### Step 1: Determine Individual Speeds
Since the ratio of their speeds is 3:7, we can represent A's speed as \(3x\) and B's speed as \(7x\), where \(x\) is a common multiplier.

### Step 2: Relative Speed
Since they are running in opposite directions, their relative speed will be the sum of their speeds:
\[
3x + 7x = 10x
\]

### Step 3: Time to Complete One Lap
Since they start at the same point and will meet for the first time at the starting point after both completing a full lap, we calculate the time it takes them to meet at the starting point again.

Since B's speed is faster, we’ll calculate based on the time A takes to complete the lap:
- Total distance = 900 meters.
- A’s speed = \(3x\), so \( \text{Time} = \frac{900}{3x} = 300/x \).

Thus, both A and B will meet at the starting point after \(300/x\) seconds.

### Step 4: Calculate Instances of 10m Distance
Since their relative speed is \(10x\), the time taken to have a separation of 10 meters is:
\[
\text{Time} = \frac{10}{10x} = \frac{1}{x} \text{ seconds}
\]

Within the total time of \(300/x\) seconds, the number of times A and B are at a distance of 10 meters from each other is:
\[
\frac{300/x}{1/x} = 300
\]

Thus, A and B will be at a distance of 10 meters from each other **300 times**.

Since this answer is not in the options, let's verify if an adjustment is required based on problem constraints or typical errors in interpreting "distance times." Given the current structure, if an answer adjustment arises through simplification or clarifying meeting points only:

### Conclusion: Final Answer Not Match

Q6 A and B started running simultaneously from a point around a circular track of 900 m. The ratio of their speeds is 3 : 7. For how many times will A and B be at a distance of 10 m from each other, if they are running in opposite directions and stops only when they meet for the first time at the starting point again?

Explore a detailed solution to calculating how many times runners A and B are at a 10m distance from each other on a 900m circular track, given their speed ratio is 3:7 and they run in opposite directions. This problem covers concepts of relative speed, distance intervals, and circular motion, suitable for students in Grades 10-12.

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