document.write( "Question 1116185: The senior class at a high school had a banquet that cost $150. If there had been one 5 persons less, the cost would have been $1.00 more. how many persons were at the banquet? \n" ); document.write( "

Algebra.Com's Answer #731034 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "If a solution without formal algebra is acceptable, then by far the easiest way to find the answer to the problem is by trial and error.

\n" ); document.write( "The number of people is certainly going to be a whole number; and the cost per person is very likely to be a whole number. So look for ways to write 150 as the product of two whole numbers and find two pairs that satisfy the conditions of the problem.

\n" ); document.write( "Pairs of whole numbers with a product of 150:

\n" ); document.write( " 1 150
\n" ); document.write( " 2 75
\n" ); document.write( " 3 50
\n" ); document.write( " 5 30
\n" ); document.write( " 6 25
\n" ); document.write( "10 15

\n" ); document.write( "The pairs 5*30 and 6*25 satisfy the conditions of the problem: 30 people cost $5 each; 25 people cost $6 each -- 5 fewer people increases the cost for each person by $1.

\n" ); document.write( "Here is an algebraic solution....

\n" ); document.write( "Let x be the number of people and y the cost per person. Then
\n" ); document.write( "(1) $\"xy+=+150\"$
\n" ); document.write( "(2) $\"%28x-5%29%28y%2B1%29+=+150\"$

\n" ); document.write( "Solve (1) for y:
\n" ); document.write( " $\"y+=+150%2Fx\"$

\n" ); document.write( "Substitute into (2):
\n" ); document.write( " $\"%28x-5%29%28150%2Fx%2B1%29+=+150\"$
\n" ); document.write( " $\"150%2Bx-750%2Fx-5+=+150\"$
\n" ); document.write( " $\"x-5-750%2Fx+=+0\"$
\n" ); document.write( " $\"x%5E2-5x-750+=+0\"$
\n" ); document.write( " $\"%28x-30%29%28x%2B25%29+=+0\"$
\n" ); document.write( " $\"x+=+30\"$ or $\"x+=+-25\"$

\n" ); document.write( "Of course reject the negative answer; so the number of people is x = 30.

\n" ); document.write( "That was a whole lot of work; and in the end you still had to use some trial and error to find the 30 and 25 to factor the quadratic equation.... \n" ); document.write( "

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