How many students are now participating in this giving project?

Question

Some other students and you are buying a gift for your fantastic mathematics teacher. The gift costs $2,010.00

Unfortunately, when the cost per student is announced, ten students realize they cannot afford it, and they have to decline the opportunity to give. The remaining students now each have to pay $16.75 more than they originally planned. How many students are now participating in this giving project?

Answer 1

x=# of students

y=cost per student

xy=2010

(x-y)(y+16.75)=2010

xy-10y+16.75x-167.5=2010

xy-10y+16.75x-167.5-2010=0

xy-10y+16.75x-2177.5=0

from xy=2010 you get y=2010/x

substitute

x(2010/x)-10(2010/x)+16.75x-2177.5=0

2010-(20,100/x)+16.75x-2177.5=0

multiply both sides by x

2010x-20,100+16.75x²-2177.5x=0

combine like terms

16.75x²-167.5x-20,100=0

multiply both sides by 4 to eliminate the decimals

67x²-670x-80,400=0

divide both sides by 67

x²-10x-1200=0  (if you don't see that 40*30=1200 and 40-30=10, then factor 1200 and proceed from there)

1200=2*2*2*2*3*5*5

regroup the factors

1200=(2*2*2*5)*(2*3*5)=40*30 and 40-30=10, the coefficient of the middle term

factor

(x-40)(x+30)=0

x-40=0

x=40 original students who were going to pay $50.25 each

40-10=30 students now are participating in the project and each will pay $67

30*$67=$2010

40*$50.25=$2010