After 15 minutes into a race, Luis has run ⁵₁₁ of the route while Liam has covered only ²₁₁ of the distance. Luis runs at the same average speed throughout the race and Liam’s average speed is 66 m/min slower than Luis’s. If Liam wants to finish the race at the same time as Luis, by how many percent should he increase his average speed by for the remaining part of the race? Round off your answer to the nearest whole number.
Extra distance that Luis ran more than Liam in first 15 min
= 15 x 66
= 990 m

Fraction of the entire distance that was covered by Luis more than Liam in the first 15 min
= ⁵₁₁ - ²₁₁
= ³₁₁

³₁₁ of the entire distance = 990 m
¹₁₁ of the entire distance = 990 ÷ 3 = 330 m

Distance that Liam ran in the first 15 min
= 2 x 330
= 660 m

Liam's average speed
= 660 ÷ 15
= 44 m/min

Luis's average speed
= 44 + 66
= 110 m/min

Fraction of the journey that Luis needed to run to complete the race
= 1 - ⁵₁₁
= ⁶₁₁

¹₁₁ of the entire distance = 330 m
⁶₁₁ of the entire distance = 6 x 330 = 1980 m

Duration that Luis would take to complete
= 1980 ÷ 110
= 18 min

Fraction of the journey that Luis needed to run to complete the race
= 1 - ²₁₁
= ⁹₁₁

⁹₁₁ of the entire distance = 9 x 330 = 2970 m  

Speed that Liam needs to take to complete distance at the same time as Luis
= 2970 ÷ 18
= 165 m/min

Increase in Liam's average speed
= 165 - 44
= 121 m/min

Percentage increase in Liam's average speed
= ¹²¹₄₄ x 100%
≈ 275%

Answer(s): 275%