Question 785763
Let {{{ R[v] }}} = Vic's rate of working in ( jobs ) / ( days )
Let {{{ R[a] }}} = Alou's rate of working in ( jobs ) / ( days )
( 1 job ) / ( 15 days ) = {{{ 1/15 }}}
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(1) {{{ R[v] + R[a] = 1/15 }}}
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What fraction of the job does Vic get done in {{{ 10 }}} days?
He does {{{ 10R[v] }}} of the job
Then there is {{{ 1 - 10R[v] }}} of the job left to do
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Alou does this in {{{ 18 }}} days
(2) {{{ R[a] = ( 1 - 10R[v] ) / 18 }}}
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(1) {{{ R[a] = 1/15 - R[v] }}}
Substitute (1) into (2)
(2) {{{ 1/15 - R[v] = ( 1 - 10R[v] ) / 18 }}}
Multiply both sides by {{{ 18 }}}
(2) {{{ 18/15 - 18R[v] = 1 - 10R[v] }}}
(2) {{{ 6/5 - 18R[v] = 1 - 10R[v] }}}
Multiply both sides by {{{ 5 }}}
(2) {{{ 6 - 90R[v] = 5 - 50R[v] }}}
(2) {{{ 40R[v] = 1 }}}
(2) {{{ R[v] = 1/40 }}}
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And, since
(1) {{{ R[v] + R[a] = 1/15 }}}
(1) {{{ 1/40 + R[a] = 1/15 }}}
Multiply both sides by {{{ 40*15 }}}
(1) {{{ 15 + 40*15*R[a] = 40 }}}
(1) {{{ 40*15*R[a] = 25 }}}
(1) {{{ 8*15*R[a] = 5 }}}
(1) {{{ 8*3*R[a] = 1 }}}
(1) {{{ R[a] = 1/24 }}}
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Vic takes 40 days working alone
Alou takes 24 days working alone
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check:
(2) {{{ R[a] = ( 1 - 10R[v] ) / 18 }}}
(2) {{{ 1/24 = ( 1 - 10/40 ) / 18 }}}
Multiply both sides by {{{ 24 }}}
(2) {{{ 1 = (24/18)*( 3/4) }}}
Multiply both sides by {{{ 18*4 }}}
(2) {{{ 18*4 = 24*3 }}}
(2) {{{ 72 = 72 }}}
OK
(1) {{{ R[v] + R[a] = 1/15 }}}
(1) {{{ 1/40 + 1/24 = 1/15 }}}
(1) {{{ 24*15 + 40*15 = 24*40 }}}
(1) {{{ 15*( 24 + 40 ) = 960 }}}
(1) {{{ 15*64 =  960 }}}
(1) {{{ 960 = 960 }}}
OK