document.write( "Question 750071: If 8 students are seated on each bench ,3 benches are left over .If 5 students are seated on each bench,12 students are left over How many benches students are there? \n" ); document.write( "

Algebra.Com's Answer #456337 by KMST(5328) $\"\"$ $\"About$

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AS A SYSTEM OF EQUATIONS PROBLEM:
\n" ); document.write( " $\"n\"$ = number of students
\n" ); document.write( " $\"b\"$ = number of benches
\n" ); document.write( "\"If 5 students are seated on each bench, 12 students are left over\" translates as
\n" ); document.write( " $\"n=5b%2B12\"$
\n" ); document.write( "\"If 8 students are seated on each bench, 3 benches are left over\" cound be translated as
\n" ); document.write( " $\"n=8b-24\"$ if we are creative enough to think that the 3 empty benches could have housed $\"3%2A8=24\"$ extra students.
\n" ); document.write( "Otherwise, we could translate \"If 8 students are seated on each bench, 3 benches are left over\" literally as
\n" ); document.write( " $\"n%2F8%2B3=b\"$ since $\"n%2F8\"$ is the number of benches that are occupied.
\n" ); document.write( "Of course, the two equations are equivalent:
\n" ); document.write( " $\"n%2F8%2B3=b\"$ --> $\"n%2F8=b-3\"$ --> $\"n=8b-3%2A8\"$ --> $\"n=8b-24\"$
\n" ); document.write( " $\"system%28n=8b-24%2Cn=5b%2B12%29\"$ --> $\"system%288b-24=5b%2B12%2Cn=5b%2B12%29\"$ --> $\"system%288b-5b=12%2B24%2Cn=5b%2B12%29\"$ --> $\"system%283b=36%2Cn=5b%2B12%29\"$ --> $\"system%28highlight%28b=12%29%2Cn=5b%2B12%29\"$ --> $\"system%28highlight%28b=12%29%2Cn=5%2A12%2B12%29\"$ --> $\"system%28highlight%28b=12%29%2Chighlight%28n=72%29%29\"$
\n" ); document.write( "
\n" ); document.write( "AS A GUESS-AND-CHECK/DIVISIBILITY PROBLEM:
\n" ); document.write( "Since all students fit into a number $\"b\"$ of benches with $\"8\"$ students on each bench, the number of students $\"8b\"$ is a multiple of $\"8\"$ .
\n" ); document.write( "Sitting $\"5\"$ students per bench you fill all the benches and have 12 students left over.
\n" ); document.write( "
\n" ); document.write( "You could start guessing and checking from there, but it may be time consuming. You could have 4 or more benches, and 8, 16, 24, 32, 40, ... students. You can try each number, or skip some. Making a table can help. It would be better to reduce the number of choices to try.
\n" ); document.write( "
\n" ); document.write( "Guessing away without further thought:
\n" ); document.write( "You could imagine having 4 benches. With 8 students, you can fill 1 bench with 8 students and would have 3 benches left. But with a total of 4 benches, sitting 5 on each bench you need $\"5%2A4%2B12=32\"$ students to fill the benches and have 12 students left over. It must be more than 8 students.
\n" ); document.write( "With 5 benches, you can fill 2 benches with 8 students each (a total of 16 students), and would have 3 benches left. But 5 benches at 5 students per bench would sit 25 students, so you could not fill those 5 benches and have 12 students left over, so there are more than 5 benches and more than 16 students.
\n" ); document.write( "With 6 benches, you could fill 3 benches with 8 students each (24 student total), and would have 3 benches left. But 6 benches at 5 students per bench would sit 30 students, so 6 benches/24 students is still too low.
\n" ); document.write( "Let's skip 7 benches and go to 8 benches.
\n" ); document.write( "With 8 benches, you could fill 5 benches with 8 students each (40 students), and would have 3 benches left. But 8 benches at 5 students per bench would sit 40 students, with no student left over, so the numbers of benches/students at 8/40 are still too low.
\n" ); document.write( "Let's skip further and go to 13 benches.
\n" ); document.write( "With 13 benches/80 students, you could fill 10 benches with 8 students each and would have 3 benches left. But 31 benches at 5 students per bench would sit 65 students, with $\"80-65=15\"$ students left over, and that is too many.
\n" ); document.write( "The numbers of benches/students are less than 13/80, so we may try the next smaller guess, 12 benches/72 students.
\n" ); document.write( "With 12 benches/72 students, you could fill 9 benches with 8 students each and would have 3 benches left. With 12 benches, at 5 students per bench, you would sit 60 students, with $\"72-60=12\"$ students left over, and that is exactly what the problem asks for. So $\"highlight%2812%29\"$ benches and $\"highlight%2872%29\"$ students is the answer.
\n" ); document.write( "
\n" ); document.write( "ALTERNATIVELY, you can use the information to reduce the number of guesses to try to 4 benches/32 students, 12 benches/72 students, 4 benches/32 students, 20 benches/112 students, etc.
\n" ); document.write( "
\n" ); document.write( "One way:
\n" ); document.write( "Sitting $\"5\"$ students per bench you fill all the benches and have 12 students left.T hat means you could fill $\"2\"$ more benches with another $\"2%2A5=10\"$ students and have just $\"2\"$ students without a seat.
\n" ); document.write( "That tells you that the number of students is $\"2\"$ more than a multiple of $\"5\"$ . The multiples of 5 that are not odd rather than even (like 5,15, 15, 35, etc) would not work, because adding $\"2\"$ to those numbers will still give you and odd number for the total number of students, and the total number of students is a multiple of 8, so it must be even.
\n" ); document.write( "Now we know that the total number of students should be an even multiple of 5 (a multiple of 10) plus 2, and it must also be a multiple of 8.
\n" ); document.write( "The total number of students cannot be 12, or 22, or 42, or 52, or 62, or 82, or 92, or 102, or 122 (not multiples of 8), but we should try 32, 72, and 112.
\n" ); document.write( "
\n" ); document.write( "A craftier way (maybe too creative) to get at the same point:
\n" ); document.write( "If I gave you 8 more students, and 4 more benches you could have everyone comfortably seated at 5 students per bench.
\n" ); document.write( "With the 8 added students, the number of students would be a multiple of 8 and a multiple of 5, so it would be a multiple of 40.
\n" ); document.write( "The number of benches would be a multiple of 8 and the number of students would be a multiple of 40, as in 8/40, 16/80, 24,120.
\n" ); document.write( "That means that before being given the 4 extra benches/8 extra students you had
\n" ); document.write( " $\"8-4=4\"$ benches/ $\"40-8=32\"$ students, or
\n" ); document.write( " $\"16-4=12\"$ benches/ $\"80-8=72\"$ students, or
\n" ); document.write( " $\"24-4=20\"$ benches/ $\"120-8=112\"$ students, or ...
\n" ); document.write( "
\n" ); document.write( "At 8 students per bench,
\n" ); document.write( "32 students use $\"32%2F8=4\"$ benches, using all 4 benches with no bench left over (too few); and
\n" ); document.write( "112 students use $\"112%2F8=14\"$ benches, with $\"16-14=4\"$ benches left over (too many); and
\n" ); document.write( "72 students use $\"72%2F8=9\"$ benches, with $\"12-9=3\"$ benches left over (just right). \n" ); document.write( "

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