After 15 minutes into a race, Cole has run ⁵₁₁ of the route while Luke has covered only ³₁₁ of the distance. Cole runs at the same average speed throughout the race and Luke’s average speed is 48 m/min slower than Cole’s. If Luke wants to finish the race at the same time as Cole, 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 Cole ran more than Luke in first 15 min
= 15 x 48
= 720 m

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

²₁₁ of the entire distance = 720 m
¹₁₁ of the entire distance = 720 ÷ 2 = 360 m

Distance that Luke ran in the first 15 min
= 3 x 360
= 1080 m

Luke's average speed
= 1080 ÷ 15
= 72 m/min

Cole's average speed
= 72 + 48
= 120 m/min

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

¹₁₁ of the entire distance = 360 m
⁶₁₁ of the entire distance = 6 x 360 = 2160 m

Duration that Cole would take to complete
= 2160 ÷ 120
= 18 min

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

⁸₁₁ of the entire distance = 8 x 360 = 2880 m  

Speed that Luke needs to take to complete distance at the same time as Cole
= 2880 ÷ 18
= 160 m/min

Increase in Luke's average speed
= 160 - 72
= 88 m/min

Percentage increase in Luke's average speed
= ⁸⁸₇₂ x 100%
≈ 122%

Answer(s): 122%