Jeri finds a pile of money with at least 200. If she puts $50$ of the pile in her left pocket, gives away $\frac{1}{8}$ 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.)

1 answer

Let \( x \) be the amount of money in the original pile.

First, we examine the case where Jeri puts $50 in her left pocket and gives away \( \frac{1}{8} \) of the rest of the pile.

  1. After putting $50 in her left pocket, she has \( x - 50 \) left.

  2. She gives away \( \frac{1}{8} \) of this remaining amount, which is

    \[ \frac{1}{8}(x - 50). \]

  3. The amount she gives away results in her keeping

    \[ x - 50 - \frac{1}{8}(x - 50) = x - 50 - \frac{x - 50}{8} = x - 50 - \frac{x}{8} + \frac{50}{8}. \]

    Simplifying this, we have:

    \[ x - 50 - \frac{x}{8} + \frac{50}{8} = \frac{8x - 400 - x + 50}{8} = \frac{7x - 350}{8}. \]

    Thus, after putting $50 in her left pocket and giving away \( \frac{1}{8} \) of the remaining money, she has:

    \[ 50 + \frac{7x - 350}{8}. \]

Next, we consider the second case where she gives away $200 instead of \( \frac{1}{8} \) of the remaining money.

  1. She keeps \( x - 200 \).

Now, we set up the inequality since she wants the first scenario to yield more money than the second scenario:

\[ 50 + \frac{7x - 350}{8} > x - 200. \]

To simplify this inequality, we start by eliminating the fraction. Multiply everything by 8:

\[ 8 \cdot 50 + 7x - 350 > 8(x - 200). \]

This simplifies to:

\[ 400 + 7x - 350 > 8x - 1600. \]

Reorganizing gives:

\[ 400 - 350 + 1600 > 8x - 7x, \] \[ 1650 > x, \] or more conveniently,

\[ x < 1650. \]

Now we need to consider the lower limit on \( x \), which is given in the problem as \( x \geq 200 \).

Combining these inequalities, we find:

\[ 200 \leq x < 1650. \]

Thus, the possible values of \( x \) form the interval:

\[ \boxed{[200, 1650)}. \]