Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will

Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money?

Let w be the number of weeks that go by for saving/spending.

Set up Barney's balance equation, B(w). Spending means we subtract
B(w) = Initial Amount - spend per week * w weeks
B(w) = 450 - 3w

Set up Betty's balance equation, B(w). Saving means we add
B(w) = Initial Amount + savings per week * w weeks
B(w) = 120 + 8w

The same amount of money means both of their balance equations B(w) are equal. So we set Barney's balance equal to Betty's balance and solve for w:
450 - 3w = 120 + 8w

Add 3w to each side to isolate w:
450 - 3w + 3w = 120 + 8w + 3w

Cancelling the 3w on the left side, we get:
450 = 120 + 11w

Rewrite to have constant on the right side:
11w + 120 = 450

Subtract 120 from each side:
11w + 120 - 120 = 450 - 120

Cancelling the 120's on the left side, we get:
11w = 330

To solve for w, we divide each side by 11
11w/11 = 330/11

Cancelling the 11's on the left side, we get:
w = 30