Question 835192
He is saying that
(1) {{{ t = n*u }}}
where {{{ t }}} is the tens digit of his age,
{{{ u }}} is the units digit of his age, and
{{{ n }}} is a whole number
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(2) {{{ t - u = (1/2)*( t + u ) }}}
Substitute (1) into (2) 
(2) {{{ n*u - u = (1/2)*( n*u + u ) }}}
(2) {{{ ( n-1 )*u = (1/2)*( n+1 )*u }}}
(2) {{{ 2u*( n-1 ) = u*( n+1 ) }}}
(2) {{{ 2*( n-1 ) = n + 1 }}}
(2) {{{ 2n - 2 = n + 1 }}}
(2) {{{ n = 3 }}}
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(1) {{{ t = 3u }}}
{{{ u }}} can only be 1,2, or 3 because if {{{ u = 4 }}},
then the age exceeds 2 digits
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Suppose {{{ u = 1 }}} and {{{ t = 3 }}}
(2) {{{ t - u = (1/2)*( t + u ) }}}
(2) {{{ 3 - 1 = (1/2)*( 3 + 1 ) }}}
(2) {{{ 2 = (1/2)*4 }}}
(2) {{{ 2 = 2 }}}
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Suppose {{{ u = 2 }}} and {{{ t = 6 }}}
(2) {{{ t - u = (1/2)*( t + u ) }}}
(2) {{{ 6 - 2 = (1/2)*( 6 + 2 ) }}}
(2) {{{ 4 = (1/2)*8 }}}
(2) {{{ 4 = 4 }}}
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Suppose {{{ u = 3 }}} and {{{ t = 9 }}}
That makes him 93 and too old to teach
so, Mr Hughes is either 31 or 62