Expert: Surajit Basu - 9/4/2009

Question
Ok, I was helping my daughter with her homework the other night.  She had one problem I could not figure out, and have since had 4 people at my place of employment (a bank) try to help me and we cannot figure this out.  Her homework has been turned in, of course with a question mark on this question, and graded, so I am not obtaining any answers for her, it is for me..lol.  I can't take ....I've worked for 3 days to figure this out.


Jane goes into a bakery and buys donuts.  She takes the donuts home and gives 1/2 of the total donuts she bought to her mother, plus and additional 1/2 of a donut, then she gives half of the total remaining donuts to her sister, plus an additional 1/2 a donut.  She then gives her friend 1/2 of her remaining donuts, plus an additional half a donut.  She is now left with 1/4 of the total of what she originally bought.  How many donuts did she buy?

I've tried 1 2 and 3 dozen, I've tried 6 donutes and 10 donuts and 9 1/2 donuts...I cannot figure this out.

Answer
i shall assume that you shall in futre get several such "jane"s in your daughter's homework, so i shall try to guide you with a general way to solve these problems.

Let's do a simpler problem.
Jane goes into a bakery and buys donuts.  She gives her friend 1/2 of her remaining donuts, plus an additional half a donut.  She is now left with 1/4 of the total of what she originally bought.  How many donuts did she buy?

We dont know how many such doughnuts were there. Suppose it is x.

How many does she give her friend? 1/2 of x + 1/2.
How many does she have left? x - ( what she gave to her friend) i.e.
x - (1/2 of x + 1/2)
i.e. x - 1/2 of x - 1/2, i.e. 1/2 of x - 1/2.

This is 1/4 of the total of what she originally bought,

So, 1/2 of x - 1/2 = 1/4 of x.
i.e. 1/2 of x - 1/4 of x = 1/2
i.e.1/4 of x = 1/2
i.e x = 2.



Now, lets do this with the given problem.

Jane goes into a bakery and buys donuts.  She takes the donuts home and gives 1/2 of the total donuts she bought to her mother, plus and additional 1/2 of a donut, then she gives half of the total remaining donuts to her sister, plus an additional 1/2 a donut.  She then gives her friend 1/2 of her remaining donuts, plus an additional half a donut.  She is now left with 1/4 of the total of what she originally bought.  How many donuts did she buy?

We dont know how many such doughnuts were there. Suppose it is Y.

How many does she give her mother? 1/2 of Y + 1/2.
How many does she have left? Y - ( what she gave to her mother) i.e.
Y - (1/2 of Y + 1/2)
i.e. Y - 1/2 of Y - 1/2, i.e. 1/2 of Y - 1/2.

Now, she gives half of the total remaining donuts to her sister, plus an additional 1/2 a donut.
How many does she give her sister? 1/2 of (1/2 of Y - 1/2) + 1/2
i.e. 1/4 of Y - 1/4 + 1/2
i.e. 1/4 of Y +1/4
How many does she have left? (1/2 of Y - 1/2) - ( what she gave to her sister)
i.e.(1/2 of Y - 1/2) - (1/4 of Y +1/4)
i.e. 1/2 of Y - 1/2 - 1/4 of Y - 1/4
i.e. 1/4 of Y - 3/4

She then gives her friend 1/2 of her remaining donuts, plus an additional half a donut.  
How many does she give her friend? 1/2 of (1/4 of Y - 3/4) + 1/2
i.e. 1/8 of Y - 3/8 + 1/2
i.e. 1/8 of Y + 1/8
How many does she have left? (1/4 of Y - 3/4) - ( what she gave to her friend)
i.e. (1/4 of Y - 3/4) - (1/8 of Y + 1/8)
i.e. 1/4 of Y - 3/4 - 1/8 of Y - 1/8)
i.e. 1/8 of Y - 3/4 - 1/8
i.e. 1/8 of Y - 7/8


This is 1/4 of the total of what she originally bought,

    1/8 of Y - 7/8 = 1/4 of Y
i.e. 1/8 of Y - 1/4 of Y = 7/8
i.e. -1/8 of Y = 7/8
i.e. Y = -7!!!


so, there you are.
that is why you can't solve it!
it is absurd.