起初，克里斯托弗和海伦一共有 1044 个橡皮擦。克里斯托弗把他所有橡皮擦的 $\frac{4}{5}$ 给了他的弟弟。海伦把她所有橡皮擦的 $\frac{5}{8}$ 给了她的妹妹。最后，海伦的橡皮擦比克里斯托弗多 150 个。 起初克里斯托弗有多少个橡皮擦？ 答：___

1. Express the remaining fractions of erasers for Christopher and Helen

Christopher gave away $\frac{4}{5}$, so he had $1 - \frac{4}{5} = \frac{1}{5}$ left.
Helen gave away $\frac{5}{8}$, so she had $1 - \frac{5}{8} = \frac{3}{8}$ left.

2. Set up equations based on the given conditions

Let Christopher's initial erasers be $C$ and Helen's be $H$.
We know:

1. $C + H = 1044$
2. $\frac{3}{8}H - \frac{1}{5}C = 150$

3. Solve the simultaneous equations for C

From (1), $H = 1044 - C$.
Substitute into (2):
$\frac{3}{8}(1044 - C) - \frac{1}{5}C = 150$
$391.5 - \frac{3}{8}C - \frac{1}{5}C = 150$
$391.5 - 150 = \left(\frac{3}{8} + \frac{1}{5}\right)C$
$241.5 = \left(\frac{15}{40} + \frac{8}{40}\right)C$
$241.5 = \frac{23}{40}C$
$C = 241.5 \times \frac{40}{23} = 420$