document.write( "Question 966201: The total cost of taking a party of children to cape town was R18 , and the children were expected to pay equal amounts.As two of them could not pay , the others made up the difference by each paying 10c extra. how many children were in the party \n" ); document.write( "

Algebra.Com's Answer #590553 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"p\"$ = the price/cost per child, in R, based on all children paying the same amount.
\n" ); document.write( " $\"n\"$ = the number of children in the party.
\n" ); document.write( " $\"n%2Ap=18\"$
\n" ); document.write( " $\"n-2\"$ = the number of children who could pay for the trip.
\n" ); document.write( " $\"p%2B0.10\"$ = the amount paid by each child who could pay for the trip.
\n" ); document.write( " $\"%28n-2%29%28p%2B0.10%29=18\"$
\n" ); document.write( " $\"n%2Ap%2B0.10n-2p-0.20=18\"$
\n" ); document.write( "Substituting $\"18=n%2Ap\"$ for $\"n%2Ap\"$ , we get
\n" ); document.write( " $\"18%2B0.10n-2p-0.20=18\"$
\n" ); document.write( " $\"0.10n-2p-0.20=0\"$
\n" ); document.write( " $\"0.10n=2p%2B0.20\"$
\n" ); document.write( " $\"10%280.10n%29=10%282p%2B0.20%29\"$
\n" ); document.write( " $\"n=20p%2B2\"$
\n" ); document.write( "Substituting the expression $\"20p%2B2\"$ for $\"n\"$ in $\"n%2Ap=18\"$ , we get
\n" ); document.write( " $\"%2820p%2B2%29%2Ap=18\"$
\n" ); document.write( " $\"20p%5E2%2B2p=18\"$
\n" ); document.write( " $\"%281%2F2%29%2820p%5E2%2B2p%29=%281%2F2%29%2818%29\"$
\n" ); document.write( " $\"10p%5E2%2Bp=9\"$
\n" ); document.write( " $\"10p%5E2%2Bp-9=0\"$
\n" ); document.write( "That is a quadratic equation that can be easily solved by factoring.
\n" ); document.write( "As for all quadratic equations, it can also be solved by \"completing the square\", or by using the quadratic formula.
\n" ); document.write( "To solve by factoring, we look at the coefficients: $\"10\"$ , $\"1\"$ , and $\"-9\"$ .
\n" ); document.write( "We look for pairs of factors that multiply to yield
\n" ); document.write( " $\"10%28-9%29=-90\"$ and add up to $\"1\"$
\n" ); document.write( " $\"90\"$ can be written as
\n" ); document.write( " $\"1-90=90\"$ , $\"2%2A45=90\"$ , $\"3%2A30=90\"$ , $\"5%2A18=90\"$ , $\"6%2A15=90\"$ , and $\"9%2A10=90\"$ .
\n" ); document.write( "Making one factor positive and another negative,
\n" ); document.write( "we find that $\"10\"$ and $\"-9\"$ are the pair we are looking for, because
\n" ); document.write( " $\"10%2A%28-9%29=-90\"$ and $\"10%2B%28-9%29=1\"$ .
\n" ); document.write( "We re-wrte $\"10p%5E2%2Bp-9=0\"$ as $\"10p%5E2%2B10p-9p-9=0\"$ ,
\n" ); document.write( "and factor by parts to get
\n" ); document.write( " $\"10p%28p%2B1%29-9%28p%2B1%29=0\"$ and $\"%2810p-9%29%28p%2B1%29=0\"$ .
\n" ); document.write( "The factored form of the equation tells us that one solution comes from
\n" ); document.write( " $\"10p-9=0\"$ <---> $\"10p=9\"$ <---> $\"p=9%2F10\"$ <---> $\"highlight%28p=0.90%29\"$ .
\n" ); document.write( "Substituting that value in $\"n%2Ap=18\"$ we get
\n" ); document.write( " $\"n%280.90%29=180\"$ <---> $\"n=18%2F0.90\"$ <---> $\"highlight%28n=20%29\"$
\n" ); document.write( "The other solution to the equation is $\"p%2B1=0\"$ <---> $\"p=-1\"$ ,
\n" ); document.write( "which does not make sense as a per child price/cost for the trip,
\n" ); document.write( "so it is not a solution to the problem. \n" ); document.write( "

\n" );