After 30 minutes into a race, Jack has run ⁶₁₃ of the route while Rael has covered only ⁴₁₃ of the distance. Jack runs at the same average speed throughout the race and Rael’s average speed is 112 m/min slower than Jack’s. If Rael 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 Rael in first 30 min
= 30 x 112
= 3360 m

Fraction of the entire distance that was covered by Jack more than Rael 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 Rael ran in the first 30 min
= 4 x 1680
= 6720 m

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

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

Fraction of the journey that Jack 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 Jack would take to complete
= 11760 ÷ 336
= 35 min

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

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

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

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

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

Answer(s): 93%