Question 818401
it's a work problem that is divided
into 3 parts:
(1) painting done by both
(2) short period of no painting
(3) painting done by Pat alone
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Let {{{ t[1] }}} = the amount of time in hours
from noon to when they started their
10 minute argument
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Let {{{ t[2] }}} = the amount of time in hours
from the end of the 10 minute argument
to 2:25 PM
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Let {{{ x }}} = the fraction of the job that they
both get done in the 1st time period
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{{{ 1 - x }}} = the fraction of the job that
Pat does in the 3rd time period
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For the 1st time period, I can say:
{{{ 1/3 + 1/4 = x/t[1] }}}
Multiply both sides by {{{ 12 }}}
{{{ 4 + 3 = 12*( x/t[1]) }}}
{{{ x/t[1] = 7/12 }}}
(a) {{{ t[1] = (12/7)*x }}}
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Fot the 3rd time period:
{{{ 1/4 = ( 1-x ) / t[2] }}}
Multiply both sides by {{{ 4*t[2] }}}
{{{ t[2] = 4*( 1-x ) }}}
(b) {{{ t[2] = 4 - 4x }}}
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So far I have 2 equations and 3 unknowns.
The 3rd equation is:
{{{ t[1] + 1/6 + t[2] = 2 + 5/12 }}}
( note that I converted minutes to hours )
{{{ t[1] + t[2] = 2 + 5/12 - 2/12  }}}
(c) {{{ t[1] + t[2] = 27/12 }}}
Substitute (a) and (b) into (c)
(c) {{{ (12/7)*x + 4 - 4x = 9/4 }}}
Multiply both sides by {{{ 28 }}}
(c) {{{ 48x + 112 - 112x = 63 }}}
(c) {{{ 64x = 112 - 63 }}}
(c) {{{ 64x = 49 }}}
(c) {{{ x = 49/64 }}}
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(a) {{{ t[1] = (12/7)*x }}}
(a) {{{ t[1] = (12/7)*(49/64) }}}
(a) {{{ t[1] = 84/64 }}}
(a) {{{ t[1] = 21/16 }}}
and
(b) {{{ t[2] = 4 - 4x }}}
(b) {{{ t[2] = 4 - 4*(49/64) }}}
(b) {{{ t[2] = 4*( 1 - 49/64 ) }}}
(b) {{{ t[2] = 4*( 15/64 ) }}}
(b) {{{ t[2] = 15/16 }}}
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check:
{{{ t[1] + 1/6 + t[2] = 2 + 5/12 }}}
{{{ 21/16 + 1/6 + 15/16 = 2 + 5/12 }}}
{{{ 36/16 = 2 + 5/12 - 2/12 }}}
{{{ 36/16 = 27/12 }}}
{{{ 9/4 = 9/4 }}}
OK
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Noon plus {{{ t[1] }}} = {{{ 21/16 }}}
{{{ 21/16 = 16/16 + 5/16 }}}
{{{ (5/16)*60 = 18.75 }}}
{{{ .75*60 = 45 }}}
The argument began at 1:18:45
Unless I made a mistake- check the work