Question

Jeri finds a pile of money with at least $\$200$. If she puts $\$50$ of the pile in her left pocket, gives away $\frac{1}{3}$ of the rest of the pile, and then puts the rest in her right pocket, she'll have more money than if she instead gave away $\$200$ of the original pile and kept the rest. What are the possible values of the number of dollars in the original pile of money? (Give your answer as an interval.)

Answer 1

Let x be the amount in dollars. After putting $50 in her left pocket and giving away one-third of the remaining, she keeps
50 + (2/3)(x − 50) = (2/3)x + 50/3.
If instead she gave away $200 she would keep x − 200. The condition is
(2/3)x + 50/3 > x − 200.
Multiply by 3: 2x + 50 > 3x − 600 ⇒ x < 650.
Since x ≥ 200, the possible values are 200 ≤ x < 650. (As integers: 200, 201, …, 649.)