After 30 minutes into a race, Xavier has run ⁶₁₃ of the route while Ken has covered only ⁴₁₃ of the distance. Xavier runs at the same average speed throughout the race and Ken’s average speed is 42 m/min slower than Xavier’s. If Ken wants to finish the race at the same time as Xavier, 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 Xavier ran more than Ken in first 30 min
= 30 x 42
= 1260 m

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

²₁₃ of the entire distance = 1260 m
¹₁₃ of the entire distance = 1260 ÷ 2 = 630 m

Distance that Ken ran in the first 30 min
= 4 x 630
= 2520 m

Ken's average speed
= 2520 ÷ 30
= 84 m/min

Xavier's average speed
= 84 + 42
= 126 m/min

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

¹₁₃ of the entire distance = 630 m
⁷₁₃ of the entire distance = 7 x 630 = 4410 m

Duration that Xavier would take to complete
= 4410 ÷ 126
= 35 min

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

⁹₁₃ of the entire distance = 9 x 630 = 5670 m  

Speed that Ken needs to take to complete distance at the same time as Xavier
= 5670 ÷ 35
= 162 m/min

Increase in Ken's average speed
= 162 - 84
= 78 m/min

Percentage increase in Ken's average speed
= ⁷⁸₈₄ x 100%
≈ 93%

Answer(s): 93%