After 20 minutes into a race, Valen has run ⁵₁₁ of the route while Nick has covered only ³₁₁ of the distance. Valen runs at the same average speed throughout the race and Nick’s average speed is 90 m/min slower than Valen’s. If Nick 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 Nick in first 20 min
= 20 x 90
= 1800 m

Fraction of the entire distance that was covered by Valen more than Nick in the first 20 min
= ⁵₁₁ - ³₁₁
= ²₁₁

²₁₁ of the entire distance = 1800 m
¹₁₁ of the entire distance = 1800 ÷ 2 = 900 m

Distance that Nick ran in the first 20 min
= 3 x 900
= 2700 m

Nick's average speed
= 2700 ÷ 20
= 135 m/min

Valen's average speed
= 135 + 90
= 225 m/min

Fraction of the journey that Valen needed to run to complete the race
= 1 - ⁵₁₁
= ⁶₁₁

¹₁₁ of the entire distance = 900 m
⁶₁₁ of the entire distance = 6 x 900 = 5400 m

Duration that Valen would take to complete
= 5400 ÷ 225
= 24 min

Fraction of the journey that Valen needed to run to complete the race
= 1 - ³₁₁
= ⁸₁₁

⁸₁₁ of the entire distance = 8 x 900 = 7200 m  

Speed that Nick needs to take to complete distance at the same time as Valen
= 7200 ÷ 24
= 300 m/min

Increase in Nick's average speed
= 300 - 135
= 165 m/min

Percentage increase in Nick's average speed
= ¹⁶⁵₁₃₅ x 100%
≈ 122%

Answer(s): 122%