After 30 minutes into a race, Sam has run ⁶₁₃ of the route while Julian has covered only ⁴₁₃ of the distance. Sam runs at the same average speed throughout the race and Julian’s average speed is 112 m/min slower than Sam’s. If Julian wants to finish the race at the same time as Sam, 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 Sam ran more than Julian in first 30 min
= 30 x 112
= 3360 m

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

²₁₃ of the entire distance = 3360 m
¹₁₃ of the entire distance = 3360 ÷ 2 = 1680 m

Distance that Julian ran in the first 30 min
= 4 x 1680
= 6720 m

Julian's average speed
= 6720 ÷ 30
= 224 m/min

Sam's average speed
= 224 + 112
= 336 m/min

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

¹₁₃ of the entire distance = 1680 m
⁷₁₃ of the entire distance = 7 x 1680 = 11760 m

Duration that Sam would take to complete
= 11760 ÷ 336
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1680 = 15120 m  

Speed that Julian needs to take to complete distance at the same time as Sam
= 15120 ÷ 35
= 432 m/min

Increase in Julian's average speed
= 432 - 224
= 208 m/min

Percentage increase in Julian's average speed
= ²⁰⁸₂₂₄ x 100%
≈ 93%

Answer(s): 93%