Question 551101
Let {{{ a }}} = gallons of 20% solution needed
Let {{{ b }}} = gallons of 65% solution needed
{{{ .2a }}} = gallons of acid in 20% solution
{{{ .65b }}} = gallons of acid in 65% solution
given:
(1) {{{ a + b =120 }}}
(2) {{{ ( .2a + .65b ) / 120 = .5 }}}
-------------------------
(2) {{{ .2a + .65b = .5*120 }}}
(2) {{{ .2a + .65b = 60 }}}
(2) {{{ 20a + 65b = 6000 }}}
Multiply both sides of (1) by {{{ 20 }}}
and subtract (1) from (2)
(2) {{{ 20a + 65b = 6000 }}}
(1) {{{ -20a - 20b = -2400 }}}
{{{ 45b = 3600 }}}
{{{ b = 80 }}}
and, since
(1) {{{ a + b =120 }}}
(1) {{{ a + 80 =120 }}}
(1) {{{ a = 120 - 80 }}}
(1) {{{ a = 40 }}}
40 gallons of 20% solution are needed
80 gallons of 65% solution are needed
check answer:
(2) {{{ ( .2*40 + .65*80 ) / 120 = .5 }}}
(2) {{{ ( 8 + 52 ) / 120  = .5 }}}
(2) {{{ 60 / 120 = .5 }}}
(2) {{{ 60 = .5*120 }}}
(2) {{{ 60 = 60 }}}
OK