After 35 minutes into a race, Vincent has run ⁷₁₃ of the route while Caden has covered only ⁴₁₃ of the distance. Vincent runs at the same average speed throughout the race and Caden’s average speed is 120 m/min slower than Vincent’s. If Caden wants to finish the race at the same time as Vincent, 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 Vincent ran more than Caden in first 35 min
= 35 x 120
= 4200 m

Fraction of the entire distance that was covered by Vincent more than Caden in the first 35 min
= ⁷₁₃ - ⁴₁₃
= ³₁₃

³₁₃ of the entire distance = 4200 m
¹₁₃ of the entire distance = 4200 ÷ 3 = 1400 m

Distance that Caden ran in the first 35 min
= 4 x 1400
= 5600 m

Caden's average speed
= 5600 ÷ 35
= 160 m/min

Vincent's average speed
= 160 + 120
= 280 m/min

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

¹₁₃ of the entire distance = 1400 m
⁶₁₃ of the entire distance = 6 x 1400 = 8400 m

Duration that Vincent would take to complete
= 8400 ÷ 280
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1400 = 12600 m  

Speed that Caden needs to take to complete distance at the same time as Vincent
= 12600 ÷ 30
= 420 m/min

Increase in Caden's average speed
= 420 - 160
= 260 m/min

Percentage increase in Caden's average speed
= ²⁶⁰₁₆₀ x 100%
≈ 163%

Answer(s): 163%