After 35 minutes into a race, Ryan has run ⁷₁₃ of the route while Tim has covered only ⁴₁₃ of the distance. Ryan runs at the same average speed throughout the race and Tim’s average speed is 96 m/min slower than Ryan’s. If Tim 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 Tim in first 35 min
= 35 x 96
= 3360 m

Fraction of the entire distance that was covered by Ryan more than Tim in the first 35 min
= ⁷₁₃ - ⁴₁₃
= ³₁₃

³₁₃ of the entire distance = 3360 m
¹₁₃ of the entire distance = 3360 ÷ 3 = 1120 m

Distance that Tim ran in the first 35 min
= 4 x 1120
= 4480 m

Tim's average speed
= 4480 ÷ 35
= 128 m/min

Ryan's average speed
= 128 + 96
= 224 m/min

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

¹₁₃ of the entire distance = 1120 m
⁶₁₃ of the entire distance = 6 x 1120 = 6720 m

Duration that Ryan would take to complete
= 6720 ÷ 224
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1120 = 10080 m  

Speed that Tim needs to take to complete distance at the same time as Ryan
= 10080 ÷ 30
= 336 m/min

Increase in Tim's average speed
= 336 - 128
= 208 m/min

Percentage increase in Tim's average speed
= ²⁰⁸₁₂₈ x 100%
≈ 163%

Answer(s): 163%