How long was Lee's holiday?

Question

 Lee spent $1200 on his holiday. If he had spent $50 less per day, he could have stayed an extra two days. How long was his holiday?

Answer 1

Let x = number of days

     y = amount spent each day

 

Then xy = 1200 and (x+2)(y-50) = 1200

 

        So, xy + 2y - 50x - 100 = 1200

 

            1200 + 2y - 50x - 100 = 1200

 

                       2y - 50x = 100

 

                         y - 25x = 50

 

                                 y = 25x + 50

 

Since xy = 1200, we have x(25x + 50) = 1200

 

                                       25x² + 50x - 1200 = 0

 

                                           x² + 2x - 48 = 0

 

                                           (x+8)(x-6) = 0

 

x = -8 or x = 6.  Since x can't be negative, x = 6days

 

Answer 2

let d=# of days

let m=amount of money spent per day

d*m=1200

d=1200/m

(d+2)(m-50)=1200

dm+2m-50d-100=1200

substitute 1200/m for d

(1200/m)(m)+2m-(50)(1200/m)-100=1200

1200+2m-(60,000/m)=1300

multiply both sides by m

1200m+2m²-60,000=1300m

2m²+1200m-1300m-60,000=0

2m²-100m-60,000=0

divide both sides by 2

m²-50m-30,000=0

factor the polynomial by factoring 30,000 or use the quadratic equation

(50±√[2500+120,000])/2

(50±√122,500)/2

(50±350)/2

400/2=200

m=$200

1200/200=6 days was the length of his holiday

if you factor 30,000 you get...

30,000=3*2*2*2*2*5*5*5*5=(3*2*5*5)(2*2*2*5*5)=150*200 and 200-150=50, the coefficient of the middle term