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jugi can paint the room in 5 weeks.
\n" ); document.write( "this means that jugi can paint the room at a rate of 1/5 of the room per week.
\n" ); document.write( "mac can paint the same room in 7 weeks.
\n" ); document.write( "this means that mac can paint the same room at a rate of 1/7 of the room per week.
\n" ); document.write( "they work together for 2 weeks and then mac finishes the room.
\n" ); document.write( "working together means their rates are additive.
\n" ); document.write( "this means that jugi's rate plus mac's rate will be equal to 1/5 + 1/7.
\n" ); document.write( "add these 2 rates together as follows:
\n" ); document.write( "1/5 * 7/7 = 7/35
\n" ); document.write( "1/7 * 5/5 = 5/35
\n" ); document.write( "their rates now have a common denominator and they can be added together.
\n" ); document.write( "you get 7/35 + 5/35 = 12/35
\n" ); document.write( "that's their rate working together.
\n" ); document.write( "for 2 weeks they are working at the combined rate of 12/35 of the room per week.
\n" ); document.write( "this means that, after 2 weeks, they have completed 2 * 12/35 of the room which means they have completed 24/35 of the room.
\n" ); document.write( "that means that they have 35 - 24 = 11/35 of the room left to complete.
\n" ); document.write( "this is because 11/35 plus 24/35 = 1 which represents the whole room.
\n" ); document.write( "so mac is left alone and mac works at the rate of 1/7 of the room per week.
\n" ); document.write( "the formula used is R * T = Q
\n" ); document.write( "R = rate
\n" ); document.write( "T = time
\n" ); document.write( "Q = quantity
\n" ); document.write( "The rate is equal to 1/7 of the room per week which is the rate that mac paints the room.
\n" ); document.write( "the time is what we want to find.
\n" ); document.write( "the quantity is the 11/35 of the room that still needs to be painted.
\n" ); document.write( "the formula becomes:
\n" ); document.write( "1/7 * T = 11/35
\n" ); document.write( "multiply both sides of this equation by 7/1 and you get:
\n" ); document.write( "7/1 * 1/7 * T = 7/1 * 11/35
\n" ); document.write( "simplify to get:
\n" ); document.write( "T = 77/35
\n" ); document.write( "it will take mac an addition 77/35 weeks to finish the room.
\n" ); document.write( "let's see how that works out.
\n" ); document.write( "they work together for 2 weeks.
\n" ); document.write( "mac works at the rate of 1/7 of the room per week.
\n" ); document.write( "jugi works at the rate of 1/5 of the room per week.
\n" ); document.write( "in the 2 weeks:
\n" ); document.write( "mac has finished 2/7 of the room.
\n" ); document.write( "jugi has finished 2/5 of the room.
\n" ); document.write( "to add these 2 fractions together you need to find the common denominator.
\n" ); document.write( "the common denominator is 35.
\n" ); document.write( "2/7 is equivalent to 10/35
\n" ); document.write( "2/5 is equivalent to 14/35
\n" ); document.write( "add 10/35 and 14/35 and you will get 24/35 which is the fraction of the room that is completed in 2 weeks.
\n" ); document.write( "now we get to mac finishing the rest of the room.
\n" ); document.write( "he works at the rate of 1/7 of the room per week and he works for 77/35 weeks.
\n" ); document.write( "the formula is still R * T = Q
\n" ); document.write( "R = 1/7
\n" ); document.write( "T = 77/35
\n" ); document.write( "formula becomes 1/7 * 77/35 = Q
\n" ); document.write( "we solve for Q to get Q = 1/7 * 77/35 = 11/35
\n" ); document.write( "jugi and mac, working together finished 24/35 of the room.
\n" ); document.write( "mac working alone finished 11/35 of the room.
\n" ); document.write( "24/35 added to 11/35 equals 35/35 of the room which is equal to 1 which means the whole room has been painted.
\n" ); document.write( "it took mac 77/35 weeks to finish the room by himself.
\n" ); document.write( "2 weeks + 77/35 weeks is equal to 70/35 + 77/35 which is equal to 147/35 weeks.
\n" ); document.write( "convert that into a mixed fraction and you get 4 and 7/35 weeks which can be further simplified to 4 and 1/5 weeks.\r
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\n" ); document.write( "\n" ); document.write( "the whole problem involves the use of the formula of R * T = Q
\n" ); document.write( "R = rate of work
\n" ); document.write( "T = hours or weeks or whatever represents the amount of time to do the work.
\n" ); document.write( "Q represents the quantity consumed or produced.
\n" ); document.write( "in this case the quantity produced is equal to 1 painted room.
\n" ); document.write( "when they work together, their rates are additive.\r
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