Question 323455
In words:
(final liters of dye)/(final liters of solution) = 50%
Let {{{a}}} = liters of 40% solution needed
Let {{{b}}} = liters of 70% solution needed
given:
(1) {{{a + b = 120}}}
liters of dye in {{{a}}} = {{{.4a}}}
liters of dye in {{{b}}} = {{{.7b}}}
liters of dye in final solution = {{{.5*120 = 60}}}
(2) {{{.4a + .7b = 60}}}
Multiply both sides of (1) by {{{4}}}
Multiply both sides of (2) by {{{10}}}
Subtract (1) from (2)
(2) {{{4a + 7b  = 600}}}
(1) {{{-4a - 4b = -480}}}
{{{3b = 120}}}
{{{b = 40}}}
and, since
{{{a + b = 120}}}
{{{a = 120 - 40}}}
{{{a = 80}}}
80 liters of 40% solution and 40 liters of 70% solution are needed
check:
(2) {{{.4a + .7b = 60}}}
(2) {{{.4*80 + .7*40 = 60}}}
{{{32 + 28 = 60}}}
{{{60 = 60}}}
OK