document.write( "Question 551376: One group of workers can do a job in 8days. After this group has worked 3 days, another group joins it and together they complete the job in 3 more days. In what time could the second group have done the job alone? Please, thank you~ :) \n" ); document.write( "
Algebra.Com's Answer #359691 by Theo(13342)\"\" \"About 
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group 1 can do the job in 8 days.
\n" ); document.write( "this group works for 3 days.
\n" ); document.write( "then a second group joins it and they complete the job in 3 more days.
\n" ); document.write( "the formula is rate * time = units
\n" ); document.write( "the number of unis is equal to 1 (the job).
\n" ); document.write( "the rate of the first group is 1/8 of the job per day because they can complete the whole job in 8 days.
\n" ); document.write( "the rate of the second group is 1/x of the job per day because they can complete the whole job in x days (x is what we're trying to find).
\n" ); document.write( "the first group finishes 3/8 of the job in 3 days.
\n" ); document.write( "this means that 5/8 of the job remains to be done.
\n" ); document.write( "when they work together, the rates are additive, so we get:
\n" ); document.write( "(1/8 + 1/x) * 3 = 5/8
\n" ); document.write( "this is because it takes them 3 more days to finish 5/8 of the job.
\n" ); document.write( "simpllify this equation to get:
\n" ); document.write( "3/8 + 3/x = 5/8
\n" ); document.write( "multiply both sides of this equation by 8*x to get:
\n" ); document.write( "3*8*x/8 + 3*8*x/x = 5*8*x/8
\n" ); document.write( "simplify this to get:
\n" ); document.write( "3*x + 3*8 = 5*x
\n" ); document.write( "subtract 3*x from both sides of this equation to get:
\n" ); document.write( "3*8 = 5*x - 3*x
\n" ); document.write( "simplify to get:
\n" ); document.write( "24 = 2*x
\n" ); document.write( "divide both sides of this equation by 2 to get:
\n" ); document.write( "x = 12
\n" ); document.write( "it appears that this second group could complete the job by itself in 12 days.
\n" ); document.write( "let's see if that holds true.
\n" ); document.write( "since they can complete the job in 12 days, this means they can complete 1/12 of the job in 1 day.
\n" ); document.write( "that's their rate.
\n" ); document.write( "we go back to the original equation.
\n" ); document.write( "the first group completes 3/8 of the job in 3 days.
\n" ); document.write( "5/8 of the job remains.
\n" ); document.write( "they work with the second group to complete the remaining 5/8 of the job in 3 days.
\n" ); document.write( "their combined rates are 1/8 and 1/12.
\n" ); document.write( "our equation becomes:
\n" ); document.write( "(1/8 + 1/12) * 3 = 5/8
\n" ); document.write( "we simplify to get:
\n" ); document.write( "3/8 + 3/12 = 5/8
\n" ); document.write( "since 2/8 is equivalent to 9/24 and 3/12 is equivalent to 6/24 and 5/8 is equivalent to 15/24, our equation becomes:
\n" ); document.write( "9/24 + 6/24 = 15/24 which simplifies to:
\n" ); document.write( "15/24 = 15/24
\n" ); document.write( "this confirms the value of x is good and the fact that the second group, working alone, would be able to complete the job in 12 days.
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