After 30 minutes into a race, Luis has run ⁶₁₃ of the route while Ken has covered only ⁴₁₃ of the distance. Luis runs at the same average speed throughout the race and Ken’s average speed is 112 m/min slower than Luis’s. If Ken 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 Ken in first 30 min
= 30 x 112
= 3360 m

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

²₁₃ of the entire distance = 3360 m
¹₁₃ of the entire distance = 3360 ÷ 2 = 1680 m

Distance that Ken ran in the first 30 min
= 4 x 1680
= 6720 m

Ken's average speed
= 6720 ÷ 30
= 224 m/min

Luis's average speed
= 224 + 112
= 336 m/min

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

¹₁₃ of the entire distance = 1680 m
⁷₁₃ of the entire distance = 7 x 1680 = 11760 m

Duration that Luis would take to complete
= 11760 ÷ 336
= 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 1680 = 15120 m  

Speed that Ken needs to take to complete distance at the same time as Luis
= 15120 ÷ 35
= 432 m/min

Increase in Ken's average speed
= 432 - 224
= 208 m/min

Percentage increase in Ken's average speed
= ²⁰⁸₂₂₄ x 100%
≈ 93%

Answer(s): 93%