Question 1143846
THE ALGEBRA CLASS SOLUTION:
We have two unknown numbers, so we need two variables.
We are given two facts, so we can make two equations.
Choosing variables is an art, but practice makes you a better artist.
My variables:
Josh starts with {{{4n}}} marbles (n is a positive integer), and {{{n}}} is the number of marbles Josh gives May.
May starts with {{{M}}} marbles.
Fist Fact: "After Josh gave 1/4 of his marbles to May,
May had 3 times as many marbles as him."
My first equation draft: {{{M+n=3(3n)}}} .
(I can simplify that equation to {{{M=8n}}} ).
Second Fact: "May then gave Josh 12 marbles.
In the end, each of them had an equal number of marbles."
My second equation draft: {{{M+n-12=3n+12}}} .
(I can simplify that equation to {{{M-2n=24}}} ).
At that point, I solve fornr {{{n}}} first,
substituting {{{8n}}} for {{{M}}} in my second equation. 
Then, I use the {{{n}}} value found to find {{{M}}} using the first equation.


THE FIFTH GRADER SOLUTION:
The fifth grader uses trial and error, tabulating results of each try.
Josh starts with a number of marbles that is a multiple of 4.
There are three points in the story:
the start, when Josh has {{{J1}}} marbles and May has {{{M1}}} ,
the midpoint, when they have {{{J2}}} and {{{M2}}} marbles,
and the end, when they have {{{J3}}} and {{{M3}}} marbles.
It could be {{{J1=4 }}} , or 8, or 12, or 16, and so on.
That would be the first numbered entered on the table.
After he gives 1/4 of his marbles to May, Josh has {{{J2}}} marbles,
and May has {{{M2}}} marbles, and those are the next entries.
The fifth grader does not worry about {{{M1}}} ,
because he realizes try #1 did not work. 
He just keeps calculating until he gets to both kids having
the same number {{{J3=M3}}} of marbles at the end.
Here is the fifth grader's table:
{{{matrix(6,6,J1,M1,J2,M2,J3,M3,
4,"__",3,9,15,"??!!",
8,"__",6,18,18,6,
12,"__",9,27,21,15,
16,"__",12,36,24,24,
20,"__",15,45,27,33)}}}
Looking at the line that end in "24  24" ,
the fifth grader realizes that
if May had {{{M2=36}}} marbles after receiving {{{4}}} marbles from Josh,
she must have started with {{{M1=36-4=highlight(32)}}} .