Question 131188
Let Bob's current age = {{{b}}}
Let John's current age = {{{j}}}
Let  half the sum of their present ages = {{{k = (1/2)*(b + j)}}}
Let  half John's age ten years from now = {{{m = (j + 10)/2}}}
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The problem could now be restated:
"Bob is as old as John will be when Bob is twice as old as John was when Bob's age was {{{k}}}.  John is as old as Bob was when John was {{{m}}}
Find {{{b}}} and {{{j}}}"
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Assume John is older than Bob. At any time, the difference in
their ages will be {{{j - b}}}
When Bob was {{{k}}}, John was {{{k + (j - b)}}}
Twice that age is {{{2*(k + j - b)}}}
When Bob is that age, John will be {{{(j-b) + 2*(k + j - b)}}}
It says Bob is this old
{{{b = (j-b) + 2*(k + j - b)}}}
{{{b = j - b + 2k + 2j - b}}}
{{{3b = 3j + 2k}}}
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When John was {{{m}}}, Bob was {{{m - (j - b)}}}
It says that John is this age
{{{j = m - j - b}}}
{{{2j = m - b}}}
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The equations I have now are:
{{{k = (1/2)*(b + j)}}}
{{{m = (j + 10)/2}}}
{{{3b = 3j + 2k}}} 
{{{2j = m - b}}}
-----------------------------
{{{k = b/2 + j/2}}}
{{{2k = b + j}}}
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{{{m = j/2 + 5}}}
{{{2m = j + 10}}}
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{{{2j = m - b}}}
{{{4j = 2m - 2b}}}
{{{2m = 4j + 2b}}}
substituting,
{{{j + 10 = 4j + 2b}}}
{{{3j = 10 - 2b}}}
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{{{3b = 3j + 2k}}} 
{{{3b = 10 - 2b + 2k}}}
{{{5b = 10 + 2k}}}
{{{k = (1/2)*(b + j)}}}
{{{2k = b + j}}}
substituting,
{{{5b = 10 + b + j}}}
{{{4b = 10 + j}}}
{{{j = 4b - 10}}}
-----------------------------
{{{2j = m - b}}}
{{{m = (j + 10)/2}}}
{{{2j = (j+10)/2 - b}}}
{{{4j = j + 10 - 2b}}}
substituting,
{{{4(4b - 10) = 4b - 10 + 10 - 2b}}}
{{{16b - 40 = 2b}}}
{{{14b = 40}}}
{{{b = 20/7}}}
{{{j = 4b - 10}}}
{{{j = 80/7 - 70/7}}}
{{{j = 10/7}}}
Bob's age is 20/7, John's age is 10/7
check:
{{{k = (1/2)*(b + j)}}}
{{{m = (j + 10)/2}}}
{{{3b = 3j + 2k}}} 
{{{2j = m - b}}}
-------------------------
{{{k = (1/2)*(30/7)}}}
{{{k = 15/7}}}
{{{m = (10/7 + 70/7)/2}}}
{{{m = 40/7}}}
{{{3b = 3j + 2k}}}
{{{3*(20/7) = 3*(10/7) + 2*(15/7)}}}
{{{60/7 = 30/7 + 30/7}}}
{{{60/7 = 60/7}}}
OK
{{{2j = m - b}}}
{{{2*(10/7) = 40/7 - 20/7}}}
{{{20/7 = 20/7}}}
OK
Unless I messed up along the way, that's it