Liz received some money for her birthday. On Monday, she spent half of the money on a new video game; on Tuesday, she spent a third of what was left on a shirt; and on Wednesday, she spent a fourth of what was left on a new book. If she has $15 remaining. How much money did Liz receive for her birthday?

If Liz had received $360, her spending would have been too high. On Monday, she would spend $180, leaving her with $180. On Tuesday, spending a third of that ($60) would leave her with $120. Finally, spending a fourth of $120 ($30) would leave her with $90, not $15. Thus, $360 is not a possible amount.

With an initial amount of $180, Liz's expenses would again not align with her remaining balance. After spending $90 on Monday, she would have $90 left. If she spent a third of this ($30) on Tuesday, she would have $60 remaining. Then, spending a fourth of $60 ($15) would leave her with $45. This does not match the $15 she has left, making $180 incorrect.

Starting with $120, Liz's calculations also do not yield the correct remaining balance. After spending $60 on Monday, she would have $60 left. Spending a third of that ($20) on Tuesday would leave her with $40. Finally, spending a fourth of $40 ($10) would leave her with $30, which does not equal the $15 she has left.