Question 1048832
Let {{{ g }}} = Gary's age now
Let {{{ h }}} = Harry's age now
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With these types of problems, I start at
the end of a statement.
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"when Gary’s age was half the sum of their present ages."
Gary's age was {{{ ( g + h )/2 }}}
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"as Harry was when Gary’s age was half the sum of their present ages."
Harry was what was Gary's age then plus the difference in
their ages ( I'm assuming Harry is the older one )
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Harry's age was {{{ ( g + h ) / 2 + ( h - g ) }}}
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"when Gary is twice as old as Harry was when Gary’s 
age was half the sum of their present ages."
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This is when Gary is:
{{{ 2*( ( g+h )/2 + ( h-g ) ) }}}
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"as Harry will be when Gary is twice as old as Harry was 
when Gary’s age was half the sum of their present ages. 
This means I have to add the difference in their ages again
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Harry will be:
{{{ 2*( ( g+h )/2 + ( h-g ) ) + ( h-g ) }}}
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I am told that Gary is presently this age, so I can say:
(1) {{{ g = 2*( ( g+h )/2 + ( h-g ) ) + ( h-g ) }}}
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Reading the 2nd statement:
"when Harry was half the age he will be 10 years from now."
Harry was:
{{{ (1/2)*( h + 10 ) }}}
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"as Gary was when Harry was half 
the age he will be 10 years from now."
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Gary was the younger, so I will subtract the difference in their ages
{{{ (1/2)*( h + 10 ) - ( h-g ) }}}
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This is Harry's age now, so I can say:
(2) {{{ h = (1/2)*( h+10 ) - ( h-g ) }}}
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(1) {{{ g = 2*( ( g+h )/2 + ( h-g ) ) + ( h-g ) }}}
(1) {{{ g = ( g+h ) + 2*( h - g ) + ( h-g ) }}}
(1) {{{ g = g + h + 2h - 2g + h - g }}}
(1) {{{ g = -2g + 4h }}}
(1) {{{ 3g = 4h }}}
(1) {{{ h = (3/4)*g }}}
( It seems that Harry was actually younger )
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(2) {{{ h = (1/2)*( h+10 ) - ( h-g ) }}}
(2) {{{ 2h = h + 10 -2*( h-g ) }}}
(2) {{{ h = 10 - 2h + 2g }}}
(2) {{{ 3h = 2g + 10 }}}
(2) {{{ 2g = 3h - 10 }}}
(2) {{{ g = (3/2)*h - 5 }}}
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By substitution:
(1) {{{ h = (3/4)*(( 3/2)*h - 5 ) }}}
(1) {{{ 4h = 3*(3/2)*h - 20 }}}
(1) {{{ 8h = 9h - 40 }}}
(1) {{{ h = 40 }}}
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Harry is 40
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check:
(2) {{{ g = (3/2)*h - 5 }}}
(2) {{{ g = (3/2)*40 - 5 }}} 
(2) {{{ g = 60 - 5 }}}
(2) {{{ g = 55 }}}
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You can go back to the start and try
to prove this step by step. I'm too
tired. Hope I got it right!
Get a 2nd opinion if you like.