What’s the *fastest* way to solve this problem?

During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per hour. She ran and biked a total of 200 miles that month, and she biked for twice as many hours as she ran. What is the total number of miles that Morgan biked during the month?
A) 80 B) 100 C) 120 D) 160

Fastest way for me is to note that she bikes exactly twice as fast as she runs, and she bikes for twice as many hours. So the miles she covers biking should be four times as many as she covers running. Because we know the miles add up to 200, choice D is the answer: 160 + 40 = 200, and 160 = 4(40).

That same insight isn’t going to arrive in everyone’s brain at the same speed, though (it probably would arrive in my brain at a different speed early in the morning on a stressful test day) which is why having a built-in strategy like backsolving for questions like this is super useful.

If you assume C is the answer, you assume she biked 120 miles, and therefore ran 80 miles. Running 80 miles at 5 miles per hour would take 16 hours; biking 120 miles at 10 miles per hour would take 12 hours. Does that match what the question said about how long she biked vs. how long she ran? Nope. The question said she was biking for longer than she was running, and even at 120 miles biking she still spent longer running.

Therefore, gotta go with D, the only choice that has her biking for even more miles. 🙂