Abel, Ben and Chris had 448 tarts altogether. Abel gave $\frac{1}{7}$ of his tarts to Ben and $\frac{2}{5}$ of his tarts to Chris. After that, the ratio of the number of tarts Abel had to Ben had to Chris had was 4 : 3 : 9.

1. Find the total fraction of tarts Abel gave away

$\frac{1}{7} + \frac{2}{5} = \frac{5}{35} + \frac{14}{35} = \frac{19}{35}$

2. Find the fraction of tarts Abel had left

$1 - \frac{19}{35} = \frac{16}{35}$

1. Find the value of 1 unit in the final ratio using the constant total

Total units in the end = $4 + 3 + 9 = 16$
Since the total number of tarts remains unchanged,
$16 \text{ units} = 448$
$1 \text{ unit} = 448 \div 16 = 28$

2. Find Abel's initial number of tarts

Abel's tarts in the end = $4 \times 28 = 112$
From part (a), Abel had $\frac{16}{35}$ of his tarts left.
$\frac{16}{35}$ of Abel's tarts = $112$
Abel's tarts at first = $112 \div 16 \times 35 = 245$

3. Find the number of tarts Abel gave to Ben and Chris

Tarts given to Ben = $\frac{1}{7} \times 245 = 35$
Tarts given to Chris = $\frac{2}{5} \times 245 = 98$

4. Find Ben and Chris's initial number of tarts and their difference

Ben's tarts in the end = $3 \times 28 = 84$
Ben's tarts at first = $84 - 35 = 49$

Chris's tarts in the end = $9 \times 28 = 252$
Chris's tarts at first = $252 - 98 = 154$

Difference = $154 - 49 = 105$