Question 287596
First, let's put this in a form where we can see what we're looking for:<ul>
<li>Hannah, Mark and Tom have a combined age of 68. So:
H + M + T = 68<br>
<li>Hannah is twice as old as the difference between Mark and Tom's age. 
H = 2(|M - T|)<br>
<li>Mark and Tom's combined age is 16 times Hannah's age.
M + T = 16(H) <br>
<li>Mark is older than Tom.</ul>
Now we have a series of equations that we can solve:
H + M + T = 68
H = 2(M - T)
M + T = 16(H)<br>
The 1st thing that I see is the similarity between the 1st & 3rd equations:
{{{ H + M + T = 68 }}}
{{{ M + T = 16(H)}}} so
{{{ (H + M + T)- H = 68 - H }}} Subtract H from both sides, giving us:
{{{ M + T = 68 - H }}}<br> and from the 3rd equation we know that:
{{{ M + T = 16(H)}}} so
{{{ 16(H) = 68 - H }}} Add H to each side and:
{{{ 17(H) = 68 }}}<br> Divide both sides by 17 and we find that:
<b>H = 4 So Hannah is 4 years old.</b> <br>
{{{ M + T = 68 - H }}} So {{{ M + T = 64 }}}
From the 4th clue we know that Mark is older than Tom, and from the 2nd equation we know that:
{{{ 4/2 = (2(M - T))/2 }}} so {{{ 2 = M - T }}}we now know that the difference between Mark & Tom's age is <b>2</b> years<br>
So taking our {{{ M + T = 64 }}}, we will substitute Tom's age + 2 for Marks age (and reverse their order, just to clarify things):
{{{ T + T + 2 = 64 }}} = {{{ 2T + 2 = 64 }}} Subtract 2 from each side, then divide each side: <br>
{{{ 2T/2 = 62/2 }}} so <b>Tom is 31</b> years old, making <b>Mark = 33</b> <br>
Substitute those numbers back into the equations to check, reread the problem to make sure you're answering the right question <i>and bob's your uncle!</i>