Perimeter and Area | FIO

Question 9

Think and mark the positions as directed—

a. Mark 'A' at the point where Akshi will be after she ran 250 m. b. Mark 'B' at the point where Akshi will be after she ran 500 m. c. Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'C'. d. Mark 'X' at the point where Toshi will be after she ran 250 m. e. Mark 'Y' at the point where Toshi will be after she ran 500 m. f. Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'Z'.

Solution

Understand the Question

Perimeter of Akshi's outer track ( $70\text{ m} \times 40\text{ m}$ ): $\text{Perimeter} = 2 \times (70\text{ m} + 40\text{ m}) = 2 \times 110\text{ m} = 220\text{ m}$ So, $1$ full round for Akshi $= 220\text{ m}$ .
Perimeter of Toshi's inner track ( $60\text{ m} \times 30\text{ m}$ ): $\text{Perimeter} = 2 \times (60\text{ m} + 30\text{ m}) = 2 \times 90\text{ m} = 180\text{ m}$ So, $1$ full round for Toshi $= 180\text{ m}$ .
Both runners start at the bottom-right corner and run clockwise along the perimeter of their respective rectangular tracks.
To find each runner's position, find how many complete rounds they ran and how many metres remain from the starting point.

(a) Mark 'A' at the point where Akshi will be after she ran 250 m.

Step 1 · Find Akshi's Position after 250 m (Point A)

Akshi runs $250\text{ m}$ on a track of perimeter $220\text{ m}$ . Diagram 1

$250 \div 220 = 1 \text{ round with a remainder}$

Distance covered in $1$ full round: $1 \times 220\text{ m} = 220\text{ m}$

Remaining distance: $250\text{ m} - 220\text{ m} = 30\text{ m}$

Starting from the bottom-right corner and moving clockwise, point 'A' is on the bottom side, $30\text{ m}$ away from the starting corner.

Answer

(a) Point 'A' is on the bottom side, $30\text{ m}$ from the starting point.

(b) Mark 'B' at the point where Akshi will be after she ran 500 m.

Step 1 · Find Akshi's Position after 500 m (Point B)

Akshi runs $500\text{ m}$ on a track of perimeter $220\text{ m}$ . Diagram 2

$500 \div 220 = 2 \text{ rounds with a remainder}$

Distance covered in $2$ full rounds: $2 \times 220\text{ m} = 440\text{ m}$

Remaining distance: $500\text{ m} - 440\text{ m} = 60\text{ m}$

Point 'B' is on the bottom side, $60\text{ m}$ from the starting corner.

Answer

(b) Point 'B' is on the bottom side, $60\text{ m}$ from the starting point.

(c) Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'C'.

Step 1 · Find Akshi's Full Rounds and Position after 1000 m (Point C)

Akshi runs $1000\text{ m}$ on a track of perimeter $220\text{ m}$ . Diagram 3

$1000 \div 220 = 4 \text{ with a remainder of } 120$

Akshi finishes $4$ full rounds.

Distance in $4$ rounds: $4 \times 220\text{ m} = 880\text{ m}$

Remaining distance: $1000\text{ m} - 880\text{ m} = 120\text{ m}$

Tracing $120\text{ m}$ clockwise from start:

Bottom side $= 70\text{ m}$
Left side $= 40\text{ m}$
Distance covered along bottom and left sides $= 70\text{ m} + 40\text{ m} = 110\text{ m}$
Distance on the top side $= 120\text{ m} - 110\text{ m} = 10\text{ m}$

Point 'C' is on the top side, $10\text{ m}$ from the top-left corner.

Answer

(c) Akshi finishes $4$ full rounds. Position 'C' is on the top side, $10\text{ m}$ from the top-left corner ( $120\text{ m}$ from the start).

(d) Mark 'X' at the point where Toshi will be after she ran 250 m.

Step 1 · Find Toshi's Position after 250 m (Point X)

Toshi runs $250\text{ m}$ on the inner track of perimeter $180\text{ m}$ . Diagram 4

$250 \div 180 = 1 \text{ round with a remainder}$

Distance covered in $1$ round: $1 \times 180\text{ m} = 180\text{ m}$

Remaining distance: $250\text{ m} - 180\text{ m} = 70\text{ m}$

Tracing clockwise from the bottom-right corner:

Bottom side $= 60\text{ m}$
Distance along the left side $= 70\text{ m} - 60\text{ m} = 10\text{ m}$

Point 'X' is on the left side, $10\text{ m}$ from the bottom-left corner.

Answer

(d) Point 'X' is on the left side, $10\text{ m}$ from the bottom-left corner ( $70\text{ m}$ from the start).

(e) Mark 'Y' at the point where Toshi will be after she ran 500 m.

Step 1 · Find Toshi's Position after 500 m (Point Y)

Toshi runs $500\text{ m}$ on the inner track of perimeter $180\text{ m}$ . Diagram 5

$500 \div 180 = 2 \text{ rounds with a remainder}$

Distance covered in $2$ rounds: $2 \times 180\text{ m} = 360\text{ m}$

Remaining distance: $500\text{ m} - 360\text{ m} = 140\text{ m}$

Tracing clockwise from the bottom-right corner:

Bottom side $= 60\text{ m}$
Left side $= 30\text{ m}$
Distance along bottom and left sides $= 60\text{ m} + 30\text{ m} = 90\text{ m}$
Distance along the top side $= 140\text{ m} - 90\text{ m} = 50\text{ m}$

Point 'Y' is on the top side, $50\text{ m}$ from the top-left corner.

Answer

(e) Point 'Y' is on the top side, $50\text{ m}$ from the top-left corner ( $140\text{ m}$ from the start).

(f) Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'Z'.

Step 1 · Find Toshi's Full Rounds and Position after 1000 m (Point Z)

Toshi runs $1000\text{ m}$ on the inner track of perimeter $180\text{ m}$ . Diagram 6

$1000 \div 180 = 5 \text{ with a remainder of } 100$

Toshi finishes $5$ full rounds.

Distance in $5$ rounds: $5 \times 180\text{ m} = 900\text{ m}$

Remaining distance: $1000\text{ m} - 900\text{ m} = 100\text{ m}$

Tracing clockwise from the bottom-right corner:

Bottom side $= 60\text{ m}$
Left side $= 30\text{ m}$
Distance along bottom and left sides $= 60\text{ m} + 30\text{ m} = 90\text{ m}$
Distance along the top side $= 100\text{ m} - 90\text{ m} = 10\text{ m}$

Point 'Z' is on the top side, $10\text{ m}$ from the top-left corner.

Answer

(f) Toshi finishes $5$ full rounds. Position 'Z' is on the top side, $10\text{ m}$ from the top-left corner ( $100\text{ m}$ from the start).

Common Mistakes

Mixing Track Perimeters: Akshi runs on the outer track ( $220\text{ m}$ ), while Toshi runs on the inner track ( $180\text{ m}$ ). Using the wrong perimeter leads to incorrect round counts and positions.
Not Subtracting Side Lengths: For remaining distances greater than the first side, remember to subtract each side successively (bottom side, then left side) to locate the exact position on the correct edge.
Direction Error: Ensure points are measured in the clockwise direction from the starting bottom-right corner.

Question 9

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