# at the beginning of the year, allison and sean togethor had 24 papergack books. during the year,allison doubled her supply but sean lost three of his. togethor they had 46 books at the end of the year. how many books did each have at the beginning of the year.

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They Didn't have 46 books toghether

## Ah, I see there might be a mistake in your question. They didn't have 46 books together? Oh boy, it seems like Allison and Sean had a bit of a book adventure! But don't worry, I'll help you straighten it out.

Let's assume Allison had x books at the beginning of the year, and Sean had y books. So, according to the information given, we have the following equations:

At the beginning:
Allison + Sean = 24.

At the end:
Allison doubled her supply, so she now has 2x books.
Sean lost three books, so he now has y - 3 books.
And together, at the end, they had 46 books:
2x + (y - 3) = 46.

Now, we can solve this little puzzle. But first, could you double-check if there's any mistake in the 46 books they had together at the end? I don't want to calculate based on inaccurate information.

## Apologies for the confusion. Let's solve the problem step-by-step.

Let's assume Allison had x books at the beginning of the year. Since Sean and Allison together had 24 books initially, Sean must have had (24 - x) books.

During the year, Allison doubled her supply, so she had 2x books. Sean, on the other hand, lost three of his, so he had (24 - x - 3) books.

At the end of the year, they had 46 books in total, so:
2x + (24 - x - 3) = 46

Simplifying the equation, we get:
2x + 21 - x = 46
x + 21 = 46
x = 46 - 21
x = 25

Therefore, Allison had 25 books at the beginning of the year, and Sean had (24 - 25) = -1 books, which doesn't make sense in this context. Please double-check the problem statement, as there may be a typo or error.

## To solve this problem, we can set up a system of equations based on the given information. Let's represent the number of books Allison had at the beginning of the year with 'A', and the number of books Sean had as 'S'.

According to the problem, at the beginning of the year, Allison and Sean together had 24 books. So, we can write the equation: A + S = 24

During the year, Allison doubled her supply of books. This means her new total would be 2A. Sean, on the other hand, lost three of his books, so his new total would be S - 3.

At the end of the year, they had a total of 46 books together. Therefore, we can write the equation: 2A + (S - 3) = 46

Now, we have two equations:
1) A + S = 24
2) 2A + (S - 3) = 46

We can solve this system of equations to find the values of A and S.

1) Simplify equation 2:
2A + S - 3 = 46

2) Re-arrange equation 1 to solve for S:
S = 24 - A

3) Substitute this value of S into equation 2:
2A + (24 - A) - 3 = 46
Simplify the equation:
A + 21 = 46
Subtract 21 from both sides:
A = 25

4) Substitute the value of A back into equation 1 to solve for S:
25 + S = 24
Subtract 25 from both sides:
S = -1

However, a negative number of books doesn't make sense in this context, so it seems there might be an error or inconsistency in the given information. Please check the problem again for any typos or clarify any discrepancies.