Question 844222
{{{x}}}= number of mangoes each friend should get if divided fairly.
{{{3x}}}= number of mangoes stolen.
Friend  number 1, reaches agreed meeting point, takes {{{x}}} mangoes and runs away.
There are {{{2x}}} mangoes left at the meeting point.
Friend number 2, finds {{{2x}}} mangoes.
Not knowing that friend number 1 has already taken {{{1/3}}} of the stolen mangoes,
friend number 2 tries to figure out how to divide {{{2x}}} into 3 equal portions.
{{{2x}}} cannot be divided evenly, but setting aside {{{1}}} of the mangoes,
{{{2x-1}}} can be evenly divided into 3 equal portions of {{{(2x-1)/3}}} .
Friend number 2 decides that the other 2 friends will be satisfied with {{{(2x-1)/3}}} mangoes,
takes {{{(2x-1)/3}}} plus the {{{1}}} mango set aside,
and happily leaves with {{{(2x-1)/3+1}}} mangoes.
Coincidentally, the amount that friend number 2 took is exactly {{{x}}} .
{{{highlight((2x-1)/3+1=x)}}} is our equation.
By lucky coincidence, friends 1 and 2 took their fair share of {{{x}}} mangoes each, and there are {{{x}}} mangoes left for friend number 3.
{{{(2x-1)/3+1=x}}}
Subtracting {{{1}}} from both sides of the equal sign, we get
{{{(2x-1)/3=x-1}}}
Multiplying times {{{3}}} both sides of the equal sign, we get
{{{2x-1=3(x-1)}}}
Applying the distributive property to the {{{3(x-1)}}} we get
{{{2x-1=3x-3}}}
Adding {{{3}}} to both sides of the equal sign, we get
{{{2x-1+3=3x-3+3}}} --> {{{2x+2=3x}}}
Subtracting {{{2x}}} from both sides of the equal sign, we get
{{{2x-2x+2=3x-2x}}} --> {{{2=(3-2)x}}} --> {{{2=1*x}}} --> {{{highlight(x=2)}}}
So {{{3x=3*2=highlight(6)}}}.
Friend number 1 had taken {{{6}}} mangoes when he had to run away because the garden's owner was approaching.
Friend number 1 took {{{6/3=2}}} mangoes, leaving {{{6-2=4}}} mangoes left for the other two friends.
Friend number 2, assuming that those {{{4}}} mangoes was all that had been stolen, realized that {{{4}}} could not be divided evenly among the 3 of them. Friend number 2, figured that it would be fair for each friend to get 1 mango, and there would be another {{{1}}} mango left.
So friend number 2 took {{{2}}} mangoes (the 1 mango "fair share" plus the leftover mango), leaving {{{4-2=2}}} mangoes behind.
Friend number 3 took those {{{2}}} mangoes, assuming that the other two had already taken their share.