After 35 minutes into a race, Archie has run ⁷₁₃ of the route while Justin has covered only ⁴₁₃ of the distance. Archie runs at the same average speed throughout the race and Justin’s average speed is 54 m/min slower than Archie’s. If Justin wants to finish the race at the same time as Archie, 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 Archie ran more than Justin in first 35 min
= 35 x 54
= 1890 m

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

³₁₃ of the entire distance = 1890 m
¹₁₃ of the entire distance = 1890 ÷ 3 = 630 m

Distance that Justin ran in the first 35 min
= 4 x 630
= 2520 m

Justin's average speed
= 2520 ÷ 35
= 72 m/min

Archie's average speed
= 72 + 54
= 126 m/min

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

¹₁₃ of the entire distance = 630 m
⁶₁₃ of the entire distance = 6 x 630 = 3780 m

Duration that Archie would take to complete
= 3780 ÷ 126
= 30 min

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

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

Speed that Justin needs to take to complete distance at the same time as Archie
= 5670 ÷ 30
= 189 m/min

Increase in Justin's average speed
= 189 - 72
= 117 m/min

Percentage increase in Justin's average speed
= ¹¹⁷₇₂ x 100%
≈ 163%

Answer(s): 163%