Question 927618
Let {{{ R[w] }}} = rate of working for a woman in jobs / hr
Let {{{ R[m] }}} = rate of working for a man in jobs / hr
Let {{{ R[g] }}} = rate of working for a girl in jobs / hr
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Add rates of working to get rate working together ( generally )
(3) {{{ R[m] = 1/4 - R[g] }}}
(2) {{{ R[w] + 2R[g] = 1/2 }}}
(3) {{{ R[m] + R[g] = 1/4 }}}
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This is 3 equations and 3 unknowns, so it's solvable
(3) {{{ R[m] = 1/4 - R[g] }}}
Substitute this into (1)
(1) {{{ 1/4 - R[g] + R[w] = 1/2 }}}
(1) {{{ R[w] - R[g] = 1/4 }}}
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Subtract (1) from (2)
(3) {{{ R[m] = 1/4 - R[g] }}}
(1) {{{ -R[w] + R[g] = -1/4 }}}
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{{{ 3R[g] = 1/4 }}}
{{{ R[g] = 1/12 }}}
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and, since:
(3) {{{ R[m] = 1/4 - R[g] }}}
(3) {{{ R[m] = 1/4 - 1/12 }}}
(3) {{{ R[m] = 3/12 - 1/12 }}}
(3) {{{ R[m] = 1/6 }}}
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(1) {{{ R[w] - R[g] = 1/4 }}}
(1) {{{ R[w] - 1/12 = 1/4 }}}
(1) {{{ R[w] = 3/12 + 1/12 }}}
(1) {{{ R[w] = 1/3 }}}
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1 woman takes 3 hrs
1 man takes 6 hrs
1 girl takes 12 hrs
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check:
(3) {{{ R[m] = 1/4 - R[g] }}}
(3) {{{ 1/6 = 1/4 - 1/12 }}}
(3) {{{ 2/12 = 3/12 - 1/12 }}}
(3) {{{ 2/12 = 2/12 }}}
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(2) {{{ R[w] + 2R[g] = 1/2 }}}
(2) {{{ 1/3 + 2*(1/12) = 1/2 }}}
(2) {{{ 4/12 + 2/12 = 6/12 }}}
(2) {{{ 6/12 = 6/12 }}}
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You can check the other one