Question 759859
Let {{{ a }}} = gallons of 20% yellow paint needed
Let {{{ b }}} = gallons of 45% yellow paint needed
{{{ .2a }}} = gallons of yellow paint in 20% yellow paint
{{{ .45b }}} = gallons of yellow paint in 45% yellow paint
-----------------------------------------------
(1) {{{ a + b = 50 }}}
(2) {{{ ( .2a + .45b ) / 50 = .25 }}}
----------------------------
(2) {{{ .2a + .45b = .25*50 }}}
(2) {{{ .2a + .45b = 12.5 }}}
(2) {{{ 20a + 45b = 1250 }}}
Multiply both sides of (1) by {{{ 20 }}}
and subtract (1) from (2)
(2) {{{ 20a + 45b = 1250 }}}
(1) {{{ -20a - 20b = -1000 }}}
-------------------------
{{{ 25b = 250 }}}
{{{ b = 10 }}}
and, since
(1) {{{ a + b = 50 }}}
(1) {{{ a + 10 = 50 }}}
(1) {{{ a = 40 }}}
40 gallons of 20% yellow paint are needed
10 gallons of 45% yellow paint are needed
check:
(2) {{{ ( .2a + .45b ) / 50 = .25 }}}
(2) {{{ ( .2*40 + .45*10 ) / 50 = .25 }}}
(2) {{{ ( 8 + 4.5 ) / 50 = .25 }}}
(2) {{{ 12.5 = .25*50 }}}
(2) {{{ 12.5 = 12.5 }}}
OK