N is the TOTAL amount of money they saved.

Tim saved (2/7)N and Sam saved (5/7)N per the original ratios.

After they saved $100 each, the total is N+200

(1/3)(N+200) and (2/3)(N+200) are the new ratios

However, (2/7)N + 100 = (1/3)(N+200)

6N + 2100 = 7(N+200)

6N + 2100 = 7N + 1400

700 = N

So they had $700 saved ---> Tim saved $200 and Sam saved $500

THEN

**Tim had saved $300 **while Sam saved $600 which in fact is 1 to 2 ratio