Summary of TSD Linear Tracks 1 | CAT Preparation 2024 | Arithmetic | Quantitative Aptitude

Summary of "TSD Linear Tracks 1 | CAT Preparation 2024 | Arithmetic | Quantitative Aptitude"

This video is a detailed tutorial on the concept of linear tracks problems, a common topic in Quantitative Aptitude, especially useful for CAT preparation. The instructor explains the fundamental ideas, introduces key concepts, and demonstrates how to solve problems involving two persons running on a linear track and meeting multiple times.

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Main Ideas and Concepts:

• Setup of Linear Track Problem:
  • Two persons A and B start simultaneously from opposite ends (points P and Q) of a linear track of length \( D \) meters.
  • They run towards each other and meet for the first time at a point \( M_1 \).
  • The distance of \( M_1 \) from one end and the total length \( D \) allows calculating the speed ratio of A and B.
• speed ratio from First Meeting Point:
  • Since time is the same for both when they meet, distance covered is proportional to their speeds.
  • speed ratio \( = \frac{\text{Distance covered by A}}{\text{Distance covered by B}} \).
  • Example: If they meet 40 meters from Q on a 100-meter track, A has traveled 60 meters, B 40 meters, so speed ratio \( = \frac{60}{40} = \frac{3}{2} \).
• Infinite Meetings and Combined Distance:
  • After the first meeting, both continue running back and forth indefinitely between P and Q.
  • Each subsequent meeting occurs after they have together covered multiples of the track length.
  • Combined distance covered between:
    • Start and first meeting \( M_1 \) = \( D \).
    • Between \( M_1 \) and \( M_2 \), \( M_2 \) and \( M_3 \), etc. = \( 2D \) each time.
  • Thus, total combined distance covered till the \( n^{th} \) meeting:
    \[ D + (n-1) \times 2D = (2n - 1) D \]
• Finding Individual Distances for Multiple Meetings:
  • Total combined distance can be split into parts proportional to the speed ratio.
  • If speed ratio is \( 3:2 \), then total distance is divided into 5 parts; A covers 3 parts, B covers 2 parts.
  • Example: For 15 meetings on a 100 m track, total combined distance \(= 29D = 2900 \) meters.
    • Distance by A \(= \frac{3}{5} \times 2900 = 1740\) meters.
    • Distance by B \(= \frac{2}{5} \times 2900 = 1160\) meters.
• Generalization with Larger track length and Number of Meetings:
  • Example problem with track length 1400 m and first meeting 400 m from P.
  • speed ratio calculated similarly (distance/time constant).
  • Total distance till 22nd meeting:
    \[ D + 21 \times 2D = 43D = 43 \times 1400 = 60,200 \text{ meters} \]
  • Distance traveled by A and B can be found by dividing total distance according to speed ratio.
  • To find position of A at the 22nd meeting, relate total distance traveled by A to multiples of track length to determine exact location on the track.
• Key Observations on Position After Multiple Meetings:
  • After even multiples of full track lengths, A or B returns to starting point.
  • Remaining distance after full rounds determines exact position on the track.

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Step-by-Step Methodology for Solving linear track problems:

• Step 1: Identify the length of the track \( D \).
• Step 2: Note the first meeting point distance from one end.
• Step 3: Calculate speed ratio using distance ratio at first meeting:
  \[ \frac{\text{Speed of A}}{\text{Speed of B}} = \frac{\text{Distance covered by A}}{\text{Distance covered by B}} \]
• Step 4: Understand that combined distance covered till first meeting = \( D \).
• Step 5: For subsequent meetings, combined distance covered between meetings = \( 2D \).
• Step 6: Total combined distance till \( n^{th} \) meeting:
  \[ (2n - 1) D \]
• Step 7: Divide total combined distance according to speed ratio to find individual distances traveled by A and B.
• Step 8: Use multiples of \( D \) to determine the exact position of A or B at any meeting point.

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Educational

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