Question 879989
Let the rate of the elder brother = {{{ r[1] }}}
Let the rate of the younger brother = {{{ r[2] }}}
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The elder brother has to have the faster rate
in order to catch up with the younger brother
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Also, when they walk in opposite directions,
they must be walking towards each other, or
else they would never meet
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Let {{{ d }}} = distance the elder brother has
to travel to catch younger brother
Walking in same direction:
Equation for younger brother:
(1) {{{ d - 20 = r[2]*10 }}}
Equation for elder brother:
(2) {{{ d = r[1]*10 }}}
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Walking towards each other:
Think of this as the elder brother moving
at the sum of their speeds and the 
younger brother not moving
(3) {{{ 20 = ( r[1] + r[2] )*2 }}}
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Substitute (2) into (1)
(1) {{{ 10r[1] - 20 = 10r[2] }}}
(1) {{{ 10*( r[1] - r[2] ) = 20 }}}
(1) {{{ r[1] - r[2] = 2 }}}
and
(3) {{{ r[1] + r[2] = 10 }}}
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Add (1) and (3)
{{{ 2r[1] = 12 }}}
{{{ r[1] = 6 }}}
and, since
(3) {{{ r[1] + r[2] = 10 }}}
(3) {{{ 6 + r[2] = 10 }}}
(3) {{{ r[2] = 4 }}}
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The eldest brother's rate is 6 km/hr
The younger brother's rate is 4 km/hr
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check:
(2) {{{ d = r[1]*10 }}}
(2) {{{ d = 60 }}} km
and
(1) {{{ d - 20 = r[2]*10 }}}
(1) {{{ d = 4*10 + 20 }}}
(1) {{{ d = 60 }}} km
OK