The model is incorrect. Because she spent 2/9 of her remaining $ on the file and pen, not 2/9 of all her $.

Let her remaining $ be 9 units.

She spent 2/9 of remaining $ on file and pen → 2 units.

So the $ that was left in the end
= 9 units - 2 units
= 7 units



Question says she had ½ the sum of her money left.

This means that ½ the sum of money
= 7 units

The sum of money = 7 units x 2 = 14 units.



So amount spent on the bag = 14 units - file and pen - money left at the end

= 14 units - 2 units - 7 units
= 5 units

5 units = $40

1 unit = $40 ÷ 5 = $8

Her sum of money
= 14 units
= 14 x $8
= $ 112

Fraction of money spent on bag
= $40/$112
= 5/14

OR

Simply, fraction of money spent on bag
= 5 units / 14 units
= 5/14

b)

Cost of file and pen
= 2 units
= 2 x $8
= $16


Cost of file was thrice cost of pen.

So cost of file → 3 parts
Cost of pen → 1 part
Total → 4 parts

$16 = 4 parts

Cost of pen = 1 part
= $16 ÷ 4
= $4

OR

Cost of pen = $16 x ¼
= $4