AMC10 2008 10A, Problem 15

Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?

Solution: Set the time Ian traveled as $I$, and set Han's speed as $H$. Therefore, Jan's speed is $H+5.$ We get the following equation for how much Han is ahead of Ian: $H+5I = 70.$ The expression for how much Jan is ahead of Ian is: $2(H+5)+10I.$ This simplifies to: $2H+10+10I.$ However, this is just $2(H+5I)+10.$ Substitute, from the first equation, $H+5I$ as $70.$ Therefore, the answer is $140 + 10$, which is $150$, or $\fbox{150}$

Answer: 150