After 35 minutes into a race, Dylan has run ⁵₁₁ of the route while Ken has covered only ³₁₁ of the distance. Dylan runs at the same average speed throughout the race and Ken’s average speed is 30 m/min slower than Dylan’s. If Ken wants to finish the race at the same time as Dylan, 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 Dylan ran more than Ken in first 35 min
= 35 x 30
= 1050 m

Fraction of the entire distance that was covered by Dylan more than Ken in the first 35 min
= ⁵₁₁ - ³₁₁
= ²₁₁

²₁₁ of the entire distance = 1050 m
¹₁₁ of the entire distance = 1050 ÷ 2 = 525 m

Distance that Ken ran in the first 35 min
= 3 x 525
= 1575 m

Ken's average speed
= 1575 ÷ 35
= 45 m/min

Dylan's average speed
= 45 + 30
= 75 m/min

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

¹₁₁ of the entire distance = 525 m
⁶₁₁ of the entire distance = 6 x 525 = 3150 m

Duration that Dylan would take to complete
= 3150 ÷ 75
= 42 min

Fraction of the journey that Dylan needed to run to complete the race
= 1 - ³₁₁
= ⁸₁₁

⁸₁₁ of the entire distance = 8 x 525 = 4200 m  

Speed that Ken needs to take to complete distance at the same time as Dylan
= 4200 ÷ 42
= 100 m/min

Increase in Ken's average speed
= 100 - 45
= 55 m/min

Percentage increase in Ken's average speed
= ⁵⁵₄₅ x 100%
≈ 122%

Answer(s): 122%