After 15 minutes into a race, Julian has run ⁵₁₁ of the route while Sam has covered only ³₁₁ of the distance. Julian runs at the same average speed throughout the race and Sam’s average speed is 96 m/min slower than Julian’s. If Sam wants to finish the race at the same time as Julian, 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 Julian ran more than Sam in first 15 min
= 15 x 96
= 1440 m

Fraction of the entire distance that was covered by Julian more than Sam in the first 15 min
= ⁵₁₁ - ³₁₁
= ²₁₁

²₁₁ of the entire distance = 1440 m
¹₁₁ of the entire distance = 1440 ÷ 2 = 720 m

Distance that Sam ran in the first 15 min
= 3 x 720
= 2160 m

Sam's average speed
= 2160 ÷ 15
= 144 m/min

Julian's average speed
= 144 + 96
= 240 m/min

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

¹₁₁ of the entire distance = 720 m
⁶₁₁ of the entire distance = 6 x 720 = 4320 m

Duration that Julian would take to complete
= 4320 ÷ 240
= 18 min

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

⁸₁₁ of the entire distance = 8 x 720 = 5760 m  

Speed that Sam needs to take to complete distance at the same time as Julian
= 5760 ÷ 18
= 320 m/min

Increase in Sam's average speed
= 320 - 144
= 176 m/min

Percentage increase in Sam's average speed
= ¹⁷⁶₁₄₄ x 100%
≈ 122%

Answer(s): 122%