After 15 minutes into a race, Jack has run ⁵₁₁ of the route while Liam has covered only ³₁₁ of the distance. Jack runs at the same average speed throughout the race and Liam’s average speed is 108 m/min slower than Jack’s. If Liam wants to finish the race at the same time as Jack, 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 Jack ran more than Liam in first 15 min
= 15 x 108
= 1620 m

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

²₁₁ of the entire distance = 1620 m
¹₁₁ of the entire distance = 1620 ÷ 2 = 810 m

Distance that Liam ran in the first 15 min
= 3 x 810
= 2430 m

Liam's average speed
= 2430 ÷ 15
= 162 m/min

Jack's average speed
= 162 + 108
= 270 m/min

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

¹₁₁ of the entire distance = 810 m
⁶₁₁ of the entire distance = 6 x 810 = 4860 m

Duration that Jack would take to complete
= 4860 ÷ 270
= 18 min

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

⁸₁₁ of the entire distance = 8 x 810 = 6480 m  

Speed that Liam needs to take to complete distance at the same time as Jack
= 6480 ÷ 18
= 360 m/min

Increase in Liam's average speed
= 360 - 162
= 198 m/min

Percentage increase in Liam's average speed
= ¹⁹⁸₁₆₂ x 100%
≈ 122%

Answer(s): 122%