Question 1

We need to find the starting points for two runners so they both run 350 m to a common finish line.

Step 1 — Find track perimeters

First, let us find the total length of each track. This is called the perimeter. A square track has 4 equal sides.

For the inner track: Each side is 100 m. The perimeter is 4 times the side length. $4 \times 100 \text{ m}$ $= 400 \text{ m}$

$\boxed{\text{Inner track perimeter} = 400 \text{ m}}$

For the outer track: Each side is 150 m. The perimeter is 4 times the side length. $4 \times 150 \text{ m}$ $= 600 \text{ m}$

$\boxed{\text{Outer track perimeter} = 600 \text{ m}}$

Step 2 — Locate the finish line

Look at the figure. The common finishing line is in the center of the bottom side of both tracks. The arrows show that runners move in a clockwise direction.

For the inner track: The bottom side is 100 m long. The finish line is at the center, so it is 100 m divided by 2 from either corner. $100 \text{ m} \div 2$ $= 50 \text{ m}$ So, the finish line is 50 m from the bottom-left corner, moving right. It is also 50 m from the bottom-right corner, moving left.

For the outer track: The bottom side is 150 m long. The finish line is at the center, so it is 150 m divided by 2 from either corner. $150 \text{ m} \div 2$ $= 75 \text{ m}$ So, the finish line is 75 m from the bottom-left corner, moving right. It is also 75 m from the bottom-right corner, moving left.

Step 3 — Find starting point 'A' for the inner track

The runner on the inner track must run 350 m to reach the finish line. The starting point 'A' must be 350 m away from the finish line, measured in the direction of running (clockwise). To find 'A', we can go 350 m backwards from the finish line. Going backwards means moving counter-clockwise.

Let us trace 350 m counter-clockwise from the finish line:

1. The finish line is 50 m from the bottom-left corner (moving right). Moving counter-clockwise from the finish line, we first go 50 m to the bottom-left corner. Distance covered = 50 m. Remaining distance to find 'A' = 350 m - 50 m = 300 m.
2. From the bottom-left corner, we go up the left side. This side is 100 m long. Distance covered = 100 m. Remaining distance to find 'A' = 300 m - 100 m = 200 m.
3. From the top-left corner, we go right along the top side. This side is 100 m long. Distance covered = 100 m. Remaining distance to find 'A' = 200 m - 100 m = 100 m.
4. From the top-right corner, we go down the right side. This side is 100 m long. Distance covered = 100 m. Remaining distance to find 'A' = 100 m - 100 m = 0 m. We have covered 350 m counter-clockwise. This brings us to the bottom-right corner. So, the starting point 'A' is at the bottom-right corner of the inner track.

Let us check this: If runner A starts at the bottom-right corner and runs clockwise:

• Runs 100 m along the right side (to top-right corner).
• Runs 100 m along the top side (to top-left corner).
• Runs 100 m along the left side (to bottom-left corner).
• Runs 50 m along the bottom side (to the finish line). Total distance = 100 m + 100 m + 100 m + 50 m = 350 m. This is correct.

Step 4 — Find starting point 'B' for the outer track

The runner on the outer track must run 350 m to reach the finish line. The starting point 'B' must be 350 m away from the finish line, measured in the direction of running (clockwise). To find 'B', we can go 350 m backwards from the finish line. Going backwards means moving counter-clockwise.

Let us trace 350 m counter-clockwise from the finish line:

1. The finish line is 75 m from the bottom-left corner (moving right). Moving counter-clockwise from the finish line, we first go 75 m to the bottom-left corner. Distance covered = 75 m. Remaining distance to find 'B' = 350 m - 75 m = 275 m.
2. From the bottom-left corner, we go up the left side. This side is 150 m long. Distance covered = 150 m. Remaining distance to find 'B' = 275 m - 150 m = 125 m.
3. From the top-left corner, we go right along the top side. We need to cover 125 m. This point is our starting point 'B'. So, the starting point 'B' is 125 m from the top-left corner, along the top side (moving right).

Let us check this: If runner B starts 125 m from the top-left corner (along the top side, moving right) and runs clockwise:

• Runs (150 m - 125 m) = 25 m along the top side (to top-right corner).
• Runs 150 m along the right side (to bottom-right corner).
• Runs 75 m along the bottom side (to the finish line). Total distance = 25 m + 150 m + 75 m = 250 m. This is not 350 m.

Let us re-calculate for 'B'. The starting point 'B' is such that running 350 m clockwise leads to the finish line. This means the starting point 'B' is 350 m before the finish line, measured clockwise. This is the same as saying the starting point 'B' is (Perimeter - 350 m) after the finish line, measured clockwise. Perimeter of outer track = 600 m. Distance 'after' finish line = 600 m - 350 m = 250 m. So, 'B' is 250 m clockwise from the finish line.

Let us trace 250 m clockwise from the finish line:

1. The finish line is 75 m from the bottom-right corner (moving left). Moving clockwise from the finish line, we first go 75 m to the bottom-right corner. Distance covered = 75 m. Remaining distance to find 'B' = 250 m - 75 m = 175 m.
2. From the bottom-right corner, we go up the right side. This side is 150 m long. Distance covered = 150 m. Remaining distance to find 'B' = 175 m - 150 m = 25 m.
3. From the top-right corner, we go left along the top side. We need to cover 25 m. This point is our starting point 'B'. So, the starting point 'B' is 25 m from the top-right corner, along the top side (moving left).

Let us check this: If runner B starts 25 m from the top-right corner (along the top side, moving left) and runs clockwise:

• Runs (150 m - 25 m) = 125 m along the top side (to top-left corner).
• Runs 150 m along the left side (to bottom-left corner).
• Runs 75 m along the bottom side (to the finish line). Total distance = 125 m + 150 m + 75 m = 350 m. This is correct.

Answer

(i) For the inner track, the starting point 'A' is at the bottom-right corner. (ii) For the outer track, the starting point 'B' is 25 m from the top-right corner, along the top side (moving left).