After 30 minutes into a race, Luis has run ⁶₁₃ of the route while Sam has covered only ⁴₁₃ of the distance. Luis runs at the same average speed throughout the race and Sam’s average speed is 119 m/min slower than Luis’s. If Sam 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 Sam in first 30 min
= 30 x 119
= 3570 m

Fraction of the entire distance that was covered by Luis more than Sam in the first 30 min
= ⁶₁₃ - ⁴₁₃
= ²₁₃

²₁₃ of the entire distance = 3570 m
¹₁₃ of the entire distance = 3570 ÷ 2 = 1785 m

Distance that Sam ran in the first 30 min
= 4 x 1785
= 7140 m

Sam's average speed
= 7140 ÷ 30
= 238 m/min

Luis's average speed
= 238 + 119
= 357 m/min

Fraction of the journey that Luis needed to run to complete the race
= 1 - ⁶₁₃
= ⁷₁₃

¹₁₃ of the entire distance = 1785 m
⁷₁₃ of the entire distance = 7 x 1785 = 12495 m

Duration that Luis would take to complete
= 12495 ÷ 357
= 35 min

Fraction of the journey that Luis needed to run to complete the race
= 1 - ⁴₁₃
= ⁹₁₃

⁹₁₃ of the entire distance = 9 x 1785 = 16065 m  

Speed that Sam needs to take to complete distance at the same time as Luis
= 16065 ÷ 35
= 459 m/min

Increase in Sam's average speed
= 459 - 238
= 221 m/min

Percentage increase in Sam's average speed
= ²²¹₂₃₈ x 100%
≈ 93%

Answer(s): 93%