Question 722406
Let {{{ r[a] }}} = Aaron's rate of painting in houses/day
Let {{{ r[b] }}} = Brenda's rate of painting in houses/day
Let {{{ r[c] }}} = Charlie's rate of painting in houses/day
Let {{{ r[d] }}} = Danita's rate of painting in houses/day
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Add rates to get the rate working together
(1) {{{ r[a] + r[b] + r[c] = 1/3 }}}
(2) {{{ r[a] + r[d] = 1/6 }}}
(3) {{{ r[c] + r[d] = 1/8 }}}
(4) {{{ r[b] + r[c] + r[d] = 1/5 }}}
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This is 4 equations and 4 unknowns, so it should be solvable
Subtract (4) from (1)
(1) {{{ r[a] + r[b] + r[c] = 1/3 }}}
(4) {{{ -r[d] - r[b] - r[c] = -1/5 }}}
(4) {{{ r[a] - r[d] = 5/15 - 3/15 }}}
(4) {{{ r[a] - r[d] = 2/15 }}}
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Add this result to (2)
(2) {{{ r[a] + r[d] = 1/6 }}}
(4) {{{ r[a] - r[d] = 2/15 }}}
{{{ 2r[a] = 5/30 + 4/30 }}}
{{{ 2r[a] = 1/30 }}}
{{{ r[a] = 1/60 }}}
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and, since
(2) {{{ r[a] + r[d] = 1/6 }}}
(2) {{{ 1/60 + r[d] = 1/6 }}}
(2) {{{ r[d] = 10/60 - 1/60 }}}
(2) {{{ r[d] = 9/60 }}}
(2) {{{ r[d] = 3/20 }}}
You can finish- have to go