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

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

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

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

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

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

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

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

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

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

Answer(s): 93%