After 30 minutes into a race, Valen has run ⁶₁₃ of the route while Bobby has covered only ⁴₁₃ of the distance. Valen runs at the same average speed throughout the race and Bobby’s average speed is 84 m/min slower than Valen’s. If Bobby wants to finish the race at the same time as Valen, 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 Valen ran more than Bobby in first 30 min
= 30 x 84
= 2520 m

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

²₁₃ of the entire distance = 2520 m
¹₁₃ of the entire distance = 2520 ÷ 2 = 1260 m

Distance that Bobby ran in the first 30 min
= 4 x 1260
= 5040 m

Bobby's average speed
= 5040 ÷ 30
= 168 m/min

Valen's average speed
= 168 + 84
= 252 m/min

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

¹₁₃ of the entire distance = 1260 m
⁷₁₃ of the entire distance = 7 x 1260 = 8820 m

Duration that Valen would take to complete
= 8820 ÷ 252
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1260 = 11340 m  

Speed that Bobby needs to take to complete distance at the same time as Valen
= 11340 ÷ 35
= 324 m/min

Increase in Bobby's average speed
= 324 - 168
= 156 m/min

Percentage increase in Bobby's average speed
= ¹⁵⁶₁₆₈ x 100%
≈ 93%

Answer(s): 93%