SOLUTION: 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 diff

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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
Answer by KMST(5328) About Me  (Show Source):
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= the number of children in the party.
n%2Ap=18
n-2= the number of children who could pay for the trip.
p%2B0.10= the amount paid by each child who could pay for the trip.
%28n-2%29%28p%2B0.10%29=18
n%2Ap%2B0.10n-2p-0.20=18
Substituting 18=n%2Ap for n%2Ap , we get
18%2B0.10n-2p-0.20=18
0.10n-2p-0.20=0
0.10n=2p%2B0.20
10%280.10n%29=10%282p%2B0.20%29
n=20p%2B2
Substituting the expression 20p%2B2 for n in n%2Ap=18 , we get
%2820p%2B2%29%2Ap=18
20p%5E2%2B2p=18
%281%2F2%29%2820p%5E2%2B2p%29=%281%2F2%29%2818%29
10p%5E2%2Bp=9
10p%5E2%2Bp-9=0
That is a quadratic equation that can be easily solved by factoring.
As for all quadratic equations, it can also be solved by "completing the square", or by using the quadratic formula.
To solve by factoring, we look at the coefficients: 10 , 1 , and -9 .
We look for pairs of factors that multiply to yield
10%28-9%29=-90 and add up to 1
90 can be written as
1-90=90 , 2%2A45=90 , 3%2A30=90 , 5%2A18=90 , 6%2A15=90 , and 9%2A10=90 .
Making one factor positive and another negative,
we find that 10 and -9 are the pair we are looking for, because
10%2A%28-9%29=-90 and 10%2B%28-9%29=1 .
We re-wrte 10p%5E2%2Bp-9=0 as 10p%5E2%2B10p-9p-9=0 ,
and factor by parts to get
10p%28p%2B1%29-9%28p%2B1%29=0 and %2810p-9%29%28p%2B1%29=0 .
The factored form of the equation tells us that one solution comes from
10p-9=0<--->10p=9<--->p=9%2F10<--->highlight%28p=0.90%29 .
Substituting that value in n%2Ap=18 we get
n%280.90%29=180<--->n=18%2F0.90<--->highlight%28n=20%29
The other solution to the equation is p%2B1=0<--->p=-1 ,
which does not make sense as a per child price/cost for the trip,
so it is not a solution to the problem.