In a 1 km race, A can give B a start of 100 m and C a start of 200 m. If B can give C a 20 m lead in a 200 m race, then time taken by B to run 200 m is:

Given:

A can give B a start of 100 m in a 1 km race.

A can give C a start of 200 m in a 1 km race.

B can give C a lead of 20 m in a 200 m race.

Formula used:

Speed is inversely proportional to the time taken when the distances are equal.

Calculation:

Let the speeds of A, B, and C be S_A, S_B, and S_C respectively.

In a 1 km race:

⇒ A runs 1000 m while B runs 900 m.

⇒ S_A / S_B = 1000 / 900

⇒ S_A / S_B = 10 / 9

A runs 1000 m while C runs 800 m:

⇒ S_A / S_C = 1000 / 800

⇒ S_A / S_C = 5 / 4

From the above ratios:

⇒ S_B / S_C = (S_B / S_A) × (S_A / S_C)

⇒ S_B / S_C = (9 / 10) × (5 / 4)

⇒ S_B / S_C = 45 / 40

⇒ S_B / S_C = 9 / 8

In a 200 m race:

B runs 200 m while C runs 180 m:

⇒ Time taken by B = Time taken by C × (S_C / S_B)

⇒ Time taken by B = Time taken by C × (8 / 9)

Let time taken by B to run 200 m be t:

⇒ t = (Distance run by B / Speed of B)

⇒ t = (200 / S_B)

We know S_B / S_C = 9 / 8:

⇒ S_B = 9 / 8 × S_C

Substitute S:

⇒ t = 200 / (9 / 8 × SC)

⇒ t = (200 × 8) / (9 × SC)

⇒ t = 1600 / (9 × SC)

From 200 m race ratio:

⇒ SC = 180 / Time taken by C

⇒ t = 1600 / (9 × (180 / Time taken by C))

⇒ Time taken by B = 90 / 4

∴ The correct answer is option (3).