Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem22

Level 5&6 Video Solutions 2015 problem22

Video Transcription

Question number 22. Jane bought some books. For the first book, she paid half of her money and $1 more. For the second book, she paid half of the remaining money and $2 more.  Finally, for the third book, she paid half of the remaining money and $3 more, thus spending all of her money. How much did the three books cost altogether?  Asking how much the three books cost altogether and knowing that Jane ended up spending all of her money is the same as asking for the amount of money she had to begin with.  And in transaction 3, after buying the third book, she did spend all of her money. That is the important clue in this problem.  So we can follow the transactions backwards to recover the original amount of money. For the third transaction, what we have is the following.  Let's say that before buying the third book, Jane had the $3 and she would spend half of her money and that would account for all of her money.  So x representing here the amount of money she had before making the purchase is a variable we can solve for.  Let's move the 1 half to the other side. We have 3 is equal to 1 half x or x is equal to 6. So Jane had exactly $6 before buying the third book or the cost of the third book was $6.  Then we can follow the logic for the second transaction. And we can say that before buying the second book, Jane had... what did she have?  She would have the $6 that the third book cost plus the $2 more that she ended up spending on the second book and that accounted for half the money remaining at that point.  Let me use y now to note that. And we can solve for that amount pretty easily. It comes out to be 16. So the second book cost exactly $16.  And finally to begin with, we can calculate how much money Jane had. She would have had the $16 spent on the second book plus the $1 more  and that finally accounted for half of the money she had to purchase books with. Let me use z to denote that.  And again we can pretty easily solve for z just as before and obtain 34 as the answer. So Jane had exactly $34 to spend on books.  And since she did end up spending all of her money, she must have spent $34 to purchase the three books and that is going to be choice C, $34.

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