One day at a family reunion we found a bunch of baseball cards in Grandpa's basement. Grandpa said we could split them among all the grandkids. There were 5040 cards in all, so each of us got a lot of cards. but then we remembered that the Paxtons, who had four of the grandchildren, hadn't arrived yet. That meant each of us present had to give up 42 cards, so that all the grandchildren would have the same number of cards.

How many grandchildren does Grandpa have?

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Let

$x$ equal the number of grandchildren.

The Paxtons, in total, have:

$42(x-4)$ cards (since each grandchild, except the 4 Paxtons, gave up 42 cards)

Each indiviual Paxton has:

$10.5(x-4)$, or $10.5x-42$ cards.

Since each Paxton has the same number of cards as any other grandchild:

$10.5x-42 = 5040/x$.

Then simple algebra gives:

$10.5x^2-42x-5040 = 0$

when the quadratic formula gives:

$x = 24, -20$

Result:

Obviously, Grandpa has -20 grandchildren.

(This is kidding. It's 24.)

Grandpa has a total of:

24 grandchildren.

Because

$5040/20=252$ is the number of cards each grandkid got originally.
$5040/24=210$ is the number of cards each grandkid should get after the additional four grandkids arrived.
If each grandkid gives 42 cards away that's a total of $42*20=840$ cards. $840/4=210$.

The answer is:

24

because:

Let $n$ be the number of original grandchildren. Therefore each of these received $\frac{5040}{n}$ cards. After giving away $42$ cards, they each now have $\frac{5040}{n+4}$ cards, so $\frac{5040}{n}-42=\frac{5040}{n+4}$. Cancelling by $42$ leaves $\frac{120}{n}-1=\frac{120}{n+4}$. So both $n$ and $n+4$ must be divisors of $120$, and the quotients must differ by $1$. The divisors of $120$ are $1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120$, and the only two that satisfy the rule are $20$ and $24$. So $n=20$ and $n+4=24$ is the answer.

How many grandchildren does Grandpa have?

He has 24 grandchildren in all (made up of 4 Paxtons, and 20 non-Paxtons).

Originally,

the 20 non-Paxton grandchildren divided 5,040 cards among themselves so that each had 252 cards.

But when the Paxton kids arrived,

Each of the 20 non-Paxton kids gave up 42 cards each (for a total of 840 cards given up), and those 840 cards were redistributed to the four Paxton kids.

So that finally:

Each non-Paxton kid ended up with 252-42 = 210 cards, and each Paxton kid ended up with 840/4 = 210 cards. In the end, each of the 24 grandchildren ended up with 210 cards each.

Fun puzzle!

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