Milan puts \( \frac{1}{4} \) of her lawn-mowing money in savings and uses \( \frac{1}{2} \) of the remaining money to pay back her sister. If she has \( \$ 15 \) left, how much did she have at first?

Let's denote the amount of money Milan had at first as \( x \). First, she saves \( \frac{1}{4} \) of her money: \[ \text{savings} = \frac{1}{4}x \] The remaining amount after savings will be: \[ \text{remaining} = x - \frac{1}{4}x = \frac{3}{4}x \] Next, she uses \( \frac{1}{2} \) of the remaining money to pay back her sister: \[ \text{payment to sister} = \frac{1}{2} \left( \frac{3}{4}x \right) = \frac{3}{8}x \] After making this payment, the amount of money she has left is: \[ \text{left} = \frac{3}{4}x - \frac{3}{8}x \] To combine these, we need a common denominator which is 8: \[ \frac{3}{4}x = \frac{6}{8}x \] Now, substitute: \[ \text{left} = \frac{6}{8}x - \frac{3}{8}x = \frac{3}{8}x \] It's given that she has \( \$15 \) left, so we set up the equation: \[ \frac{3}{8}x = 15 \] To solve for \( x \), we can multiply both sides by \( \frac{8}{3} \): \[ x = 15 \times \frac{8}{3} = 15 \times \frac{8}{3} = 40 \] Thus, the amount of money Milan had at first is \( \boxed{40} \).