Bill was christmas shopping. He spent 1/2 of his money on a gift for his sister and 1/3 on a gift for his brother. Later that day he spent 2/3 of his remaining money on a present for his mom. Bill had $20 left. How much money did Bill have to start with?

1/2 + 1/3 = 5/6

2/3 of the remaining 1/6 is 1/9
That means that 1/18 is left
x/18 = 20
x = 360

To find out how much money Bill had to start with, we can work backwards using the information given.

Let's assume that the amount of money Bill had to start with is represented by the variable 'x'.

According to the problem, Bill spent 1/2 of his money on a gift for his sister and 1/3 on a gift for his brother. This means that he spent a total of (1/2)x + (1/3)x on these two gifts.

After buying these gifts, he had 2/3 of his remaining money left. This means that his remaining money was (2/3)(x - [(1/2)x + (1/3)x]).

Finally, we are told that Bill had $20 left. So, we can set up the equation:

(2/3)(x - [(1/2)x + (1/3)x]) = 20

To solve this equation and find the value of 'x' (the amount of money Bill had to start with), we can follow these steps:

Step 1: Simplify the expression inside the brackets:
[(1/2)x + (1/3)x] = (3/6)x + (2/6)x = (5/6)x

Step 2: Distribute the (2/3) to the expression inside the brackets:
(2/3)(x - [(1/2)x + (1/3)x]) = (2/3)(x - (5/6)x) = (2/3)(x - (5/6)x) = (2/3)((6/6)x - (5/6)x) = (2/3)(x/6) = (2/9)x

Step 3: Substitute the simplified expressions into the equation:
(2/9)x = 20

Step 4: Multiply both sides of the equation by 9/2 to isolate the variable 'x':
[(2/9)x][(9/2)] = (20)(9/2)
x = 180/2
x = 90

Therefore, Bill had $90 to start with.