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

Fraction of the entire distance that was covered by Caden more than Howard 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 Howard ran in the first 30 min
= 4 x 630
= 2520 m

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

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

Fraction of the journey that Caden 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 Caden would take to complete
= 4410 ÷ 126
= 35 min

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

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

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

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

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

Answer(s): 93%