MATHS PUZZLE # 6 FIRST LEG (Second Year)

(from the 8th to the 15th of November 2010)

THE GIFT

A group of friends decided to buy a present whose price is a number between 175 and 195 euros. In the last moment two friends decided not to participate, so the rest of the group had to pay one more euro. Which is the price of the present if we know it is an integer?

You can send the solution to www.Fernando.Escuin@iescampanar.com

SOLUTION
For factorial decomposition of the numbers between 170 and 195 only 180 is a product of two different forms xy = (x-2) (y + 1) (180 = 20 x 9 = 18 x 10). So 180 is the solution.
Where x = number of friends and y = euros by person.
In a different way,
We have the following equations system:
P= xy and P= (x -2) ( y+ 1) → xy = xy + x – 2y – 2 → x = 2 + 2y
xy = P → (2 + 2y ) y = P → 2 y2 + 2y – P = 0
Where P is the cost of the gift.
The discriminant of this second grade equation Δ = 4 + 8P has to a rational number. This is possible if P=180 → Δ = 38 → y = (-2 + 38 )/4 = 9 and x = 180/9 = 20.