After 15 minutes into a race, Sam has run ⁵₁₁ of the route while Caden has covered only ²₁₁ of the distance. Sam runs at the same average speed throughout the race and Caden’s average speed is 60 m/min slower than Sam’s. If Caden 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 Caden in first 15 min
= 15 x 60
= 900 m

Fraction of the entire distance that was covered by Sam more than Caden in the first 15 min
= ⁵₁₁ - ²₁₁
= ³₁₁

³₁₁ of the entire distance = 900 m
¹₁₁ of the entire distance = 900 ÷ 3 = 300 m

Distance that Caden ran in the first 15 min
= 2 x 300
= 600 m

Caden's average speed
= 600 ÷ 15
= 40 m/min

Sam's average speed
= 40 + 60
= 100 m/min

Fraction of the journey that Sam needed to run to complete the race
= 1 - ⁵₁₁
= ⁶₁₁

¹₁₁ of the entire distance = 300 m
⁶₁₁ of the entire distance = 6 x 300 = 1800 m

Duration that Sam would take to complete
= 1800 ÷ 100
= 18 min

Fraction of the journey that Sam needed to run to complete the race
= 1 - ²₁₁
= ⁹₁₁

⁹₁₁ of the entire distance = 9 x 300 = 2700 m  

Speed that Caden needs to take to complete distance at the same time as Sam
= 2700 ÷ 18
= 150 m/min

Increase in Caden's average speed
= 150 - 40
= 110 m/min

Percentage increase in Caden's average speed
= ¹¹⁰₄₀ x 100%
≈ 275%

Answer(s): 275%