After 30 minutes into a race, Ivan has run ⁶₁₃ of the route while Caden has covered only ⁴₁₃ of the distance. Ivan runs at the same average speed throughout the race and Caden’s average speed is 98 m/min slower than Ivan’s. If Caden wants to finish the race at the same time as Ivan, 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 Ivan ran more than Caden in first 30 min
= 30 x 98
= 2940 m

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

²₁₃ of the entire distance = 2940 m
¹₁₃ of the entire distance = 2940 ÷ 2 = 1470 m

Distance that Caden ran in the first 30 min
= 4 x 1470
= 5880 m

Caden's average speed
= 5880 ÷ 30
= 196 m/min

Ivan's average speed
= 196 + 98
= 294 m/min

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

¹₁₃ of the entire distance = 1470 m
⁷₁₃ of the entire distance = 7 x 1470 = 10290 m

Duration that Ivan would take to complete
= 10290 ÷ 294
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1470 = 13230 m  

Speed that Caden needs to take to complete distance at the same time as Ivan
= 13230 ÷ 35
= 378 m/min

Increase in Caden's average speed
= 378 - 196
= 182 m/min

Percentage increase in Caden's average speed
= ¹⁸²₁₉₆ x 100%
≈ 93%

Answer(s): 93%