After 30 minutes into a race, Liam has run ⁶₁₃ of the route while Tim has covered only ⁴₁₃ of the distance. Liam runs at the same average speed throughout the race and Tim’s average speed is 35 m/min slower than Liam’s. If Tim wants to finish the race at the same time as Liam, 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 Liam ran more than Tim in first 30 min
= 30 x 35
= 1050 m

Fraction of the entire distance that was covered by Liam more than Tim in the first 30 min
= ⁶₁₃ - ⁴₁₃
= ²₁₃

²₁₃ of the entire distance = 1050 m
¹₁₃ of the entire distance = 1050 ÷ 2 = 525 m

Distance that Tim ran in the first 30 min
= 4 x 525
= 2100 m

Tim's average speed
= 2100 ÷ 30
= 70 m/min

Liam's average speed
= 70 + 35
= 105 m/min

Fraction of the journey that Liam needed to run to complete the race
= 1 - ⁶₁₃
= ⁷₁₃

¹₁₃ of the entire distance = 525 m
⁷₁₃ of the entire distance = 7 x 525 = 3675 m

Duration that Liam would take to complete
= 3675 ÷ 105
= 35 min

Fraction of the journey that Liam needed to run to complete the race
= 1 - ⁴₁₃
= ⁹₁₃

⁹₁₃ of the entire distance = 9 x 525 = 4725 m  

Speed that Tim needs to take to complete distance at the same time as Liam
= 4725 ÷ 35
= 135 m/min

Increase in Tim's average speed
= 135 - 70
= 65 m/min

Percentage increase in Tim's average speed
= ⁶⁵₇₀ x 100%
≈ 93%

Answer(s): 93%