Mary had a number of cookies. after eating one, she gave half the remainer to her sister. after eating another cookie, she gave half of what was left to her brother. mary now had only five cookies left. how many cookies did she start with?

She originally started with 23. 

We can tell this by working the equation backwards. Since she has 5 left and she had just given half to her brother, we know she had 10 a moment ago. 

Before that, Mary had eaten a cookie. So we add 1 to the total and now have 11.

Before eating that cookie, she has given half to her sister, which would give her 22. 

And before giving half to her sister, she had eaten the first one, which would give us 23 to start with. 

The first digit is in the middle school 100s place