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

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

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

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

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

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

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

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

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

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

Answer(s): 93%