After 30 minutes into a race, Ryan has run ⁵₁₁ of the route while Warren has covered only ³₁₁ of the distance. Ryan runs at the same average speed throughout the race and Warren’s average speed is 120 m/min slower than Ryan’s. If Warren wants to finish the race at the same time as Ryan, 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 Ryan ran more than Warren in first 30 min
= 30 x 120
= 3600 m

Fraction of the entire distance that was covered by Ryan more than Warren in the first 30 min
= ⁵₁₁ - ³₁₁
= ²₁₁

²₁₁ of the entire distance = 3600 m
¹₁₁ of the entire distance = 3600 ÷ 2 = 1800 m

Distance that Warren ran in the first 30 min
= 3 x 1800
= 5400 m

Warren's average speed
= 5400 ÷ 30
= 180 m/min

Ryan's average speed
= 180 + 120
= 300 m/min

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

¹₁₁ of the entire distance = 1800 m
⁶₁₁ of the entire distance = 6 x 1800 = 10800 m

Duration that Ryan would take to complete
= 10800 ÷ 300
= 36 min

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

⁸₁₁ of the entire distance = 8 x 1800 = 14400 m  

Speed that Warren needs to take to complete distance at the same time as Ryan
= 14400 ÷ 36
= 400 m/min

Increase in Warren's average speed
= 400 - 180
= 220 m/min

Percentage increase in Warren's average speed
= ²²⁰₁₈₀ x 100%
≈ 122%

Answer(s): 122%