Question 1033571
Let the original number of red ones = {{{ r }}}
Let the original number of yellow ones = {{{ y }}}
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(1) {{{ r/y = 3/4 }}}

(2) {{{ ( r-3 ) / ( y - 6 ) = 4/5 }}}
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(1) {{{ r = (3/4)*y }}}
and
(2) {{{ r - 3 = (4/5)*( y - 6 ) }}}
Substitute (1) into (2)
(2) {{{ (3/4)*y - 3 = (4/5)*( y - 6 ) }}}
Multiply both sides by {{{ 20 }}}
(2) {{{ 15y - 60 = 16*( y - 6 ) }}}
(2) {{{ 15y - 60 = 16y - 96 }}}
(2) {{{ y = 36 }}}
and
(1) {{{ r = (3/4)*y }}}
(1) {{{ r = (3/4)*36 }}}
(1) {{{ r = 27 }}}
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36 yellow beans were in the dish
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check:
(1) {{{ r/y = 3/4 }}}
(1) {{{ 27/36 = 3/4 }}}
(1) {{{ ( 9/9 )*( 3/4 ) = 3/4 }}}
(1) {{{ 3/4 = 3/4 }}}
and
(2) {{{ ( r-3 ) / ( y - 6 ) = 4/5 }}}
(2) {{{ ( 27-3 ) / ( 36 - 6 ) = 4/5 }}}
(2) {{{ 24 / 30 = 4/5 }}}
(2) {{{ ( 6/6 )*( 4/5 ) = 4/5 }}}
(2) {{{ 4/5 = 4/5 }}}
OK