After 30 minutes into a race, Archie has run ⁶₁₃ of the route while Xavier has covered only ⁴₁₃ of the distance. Archie runs at the same average speed throughout the race and Xavier’s average speed is 49 m/min slower than Archie’s. If Xavier 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 Xavier in first 30 min
= 30 x 49
= 1470 m

Fraction of the entire distance that was covered by Archie more than Xavier in the first 30 min
= ⁶₁₃ - ⁴₁₃
= ²₁₃

²₁₃ of the entire distance = 1470 m
¹₁₃ of the entire distance = 1470 ÷ 2 = 735 m

Distance that Xavier ran in the first 30 min
= 4 x 735
= 2940 m

Xavier's average speed
= 2940 ÷ 30
= 98 m/min

Archie's average speed
= 98 + 49
= 147 m/min

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

¹₁₃ of the entire distance = 735 m
⁷₁₃ of the entire distance = 7 x 735 = 5145 m

Duration that Archie would take to complete
= 5145 ÷ 147
= 35 min

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

⁹₁₃ of the entire distance = 9 x 735 = 6615 m  

Speed that Xavier needs to take to complete distance at the same time as Archie
= 6615 ÷ 35
= 189 m/min

Increase in Xavier's average speed
= 189 - 98
= 91 m/min

Percentage increase in Xavier's average speed
= ⁹¹₉₈ x 100%
≈ 93%

Answer(s): 93%