We always liked poking around Grandpa's attic whenever we had a family reunion. We found all sorts of neat stuff up there. Once we found a bunch of baseball cards, so Grandpa said, "Just divide 'em up among all the grandchildren." There were 5040 cards in all, so each of us got a lot of cards. But then we remembered that the Yakliches, who had five of the grandchildren, hadn't arrived yet. So each of those of us present had to give up 75 cards so that all the grandchildren would have the same number of cards. How many grandchildren does grandpa have?
I have to use the guess and check method for this problem. I know by using algebra and setting up the problem there are going to be 21 grandchildren. I need help setting up the guess and check method to solve the problem.
I know what we will be guessing is the number of grandchildren but how do I use the rest of the information to solve this problem, and how do I set it up?
Let the number of grandchildren initially present is $x$. So the total number of grandchildren is $x+5$.
In the first attempt of distribution, $5040$ cards were divided equally among $x$ grandchildren.
Let $\large\frac{5040}{x}=y$. ...(I)
In the next attempt of distribution, $5040$ cards were divided among $x+5$ with each of the $x$ grandchildren having 75 cards less from the first attempt of distribution.
So, $\large\frac{5040}{x+5}=y-75$ ...(II)
Solve equation (I) and (II) to get the answer.