After 30 minutes into a race, Seth has run ⁶₁₃ of the route while Ken has covered only ⁴₁₃ of the distance. Seth runs at the same average speed throughout the race and Ken’s average speed is 35 m/min slower than Seth’s. If Ken wants to finish the race at the same time as Seth, 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 Seth ran more than Ken in first 30 min
= 30 x 35
= 1050 m

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

²₁₃ of the entire distance = 1050 m
¹₁₃ of the entire distance = 1050 ÷ 2 = 525 m

Distance that Ken ran in the first 30 min
= 4 x 525
= 2100 m

Ken's average speed
= 2100 ÷ 30
= 70 m/min

Seth's average speed
= 70 + 35
= 105 m/min

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

¹₁₃ of the entire distance = 525 m
⁷₁₃ of the entire distance = 7 x 525 = 3675 m

Duration that Seth would take to complete
= 3675 ÷ 105
= 35 min

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

⁹₁₃ of the entire distance = 9 x 525 = 4725 m  

Speed that Ken needs to take to complete distance at the same time as Seth
= 4725 ÷ 35
= 135 m/min

Increase in Ken's average speed
= 135 - 70
= 65 m/min

Percentage increase in Ken's average speed
= ⁶⁵₇₀ x 100%
≈ 93%

Answer(s): 93%