The problem is as follows:
Mark has $\overline{ab}$ marbles and Ferdinand has $\overline{cde}$ marbles. Find the total of marbles which both have knowing that the product of the number of marbles they have is the ninth part of $\overline{abcde}$.
The alternatives in my book are as follows:
$\begin{array}{ll} 1.&\textrm{237 marbles}\\ 2.&\textrm{133 marbles}\\ 3.&\textrm{126 marbles}\\ 4.&\textrm{128 marbles}\\ 5.&\textrm{132 marbles}\\ \end{array}$
How exactly are these numbers being asked found?. What I've attempted to do was to write the question into an equation.
Since it mentions that the product between the numbers is:
$$\overline{ab} \times \overline{cde}= \frac{1}{9} \overline{abcde}$$
And they ask to find $E=\overline{ab}+\overline{cde}$
This is assuming that the intended number system is on base 10.
But that's it. There's where I'm stuck. The only other thing which I could spot was that $c$ cannot be zero as well $a$. But from there I don't know how to use the hint which they have referring to being the ninth part of $\overline{abcde}$ does it exist some sort of identity or thing which to use to use this clue?.
Since I am lost into this question, can someone explain this in a step by step approach and the most detailed as possible to find the answer?. The part most confusing to me was to reduce the available options within the numbers to get the right combination to match what it is being asked.
More importantly, does it exist an approach to get the answer by finding individually first the digits involved instead of getting the requested answer first?.
Let $\overline{ab} = x$ and $\overline{cde} = y$. Then, we want to find $x + y$ given $xy = \frac{1000x + y}{9}$. We have:
$$9xy = 1000x + y$$
$$9xy - 1000x - y = 0$$
$$81xy - 9000x - 9y = 0$$
$$81xy - 9000x - 9y + 1000 = 1000$$
$$(9x - 1)(9y - 1000) = 1000$$
Now, since $9x-1$ and $9y-1$ are integers, we can find factor pairs of $1000$ and solve for $x$ and $y$. (Keep in mind here that $x < 100$ and $y < 1000$.) We find:
$$(9x - 1)(9y -1000) = 125(8)$$
$$9x -1 = 125, 9y - 1000 = 8$$
$$9x = 126, 9y = 1008$$
$$x = 14, y = 112$$
$$x + y = \boxed{126}$$
An alternate, shorter solution (that may not work with different answer choices):
Note that $\overline{abcde}\equiv a + b + c + d +e\equiv 0\pmod{9}$ because we are working in base $10$. Then, $\overline{ab} + \overline{cde} = 10a + b + 100c + 10d + e \equiv a + b + c + d+e \equiv 0\pmod{9}$. Thus, the correct answer must be divisible by $9$. Note that the only answer choice divisible by $9$ is $126$, so this must be the correct answer.