After 30 minutes into a race, Bryan has run ⁶₁₃ of the route while Caden has covered only ⁴₁₃ of the distance. Bryan runs at the same average speed throughout the race and Caden’s average speed is 119 m/min slower than Bryan’s. If Caden wants to finish the race at the same time as Bryan, 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 Bryan ran more than Caden in first 30 min
= 30 x 119
= 3570 m

Fraction of the entire distance that was covered by Bryan more than Caden 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 Caden ran in the first 30 min
= 4 x 1785
= 7140 m

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

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

Fraction of the journey that Bryan 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 Bryan would take to complete
= 12495 ÷ 357
= 35 min

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

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

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

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

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

Answer(s): 93%