CPhill

Adam, Ben, Cathy and Dana share some lollies. First Adam takes 3/5 of the lollies, then Ben takes 1/4 of the remainder, and then Cath takes1/3 of what is left. Dana gets all the rest. What fraction of the original number of lollies does Dana get ???

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After Adam takes 3/5 of them there are 2/5 left

Ben takes 1/4 of these, so that means that there are (3/4)(2/5) left = 6/20 = 3/10 left

Cath takes 1/3 of these, so there are (2/3)(3/10) l= 2/10  left for Dana

Check

Adam gets 3/5 = 6/10

Ben gets (1/4)(2/5)  = 1/10

Cath  gets (1/3)(3/10)  = 1/10

Dana gets 2/10

And these all sum to 1

I'm going to answer this question....but I'm assuming an even spacing of the points  {I actually don't know if this makes a difference, or not  !!!!!  }

When the number of points is even, it is quite complicated to determine the number of intersecting points.

When the number of points is odd.....number theory shows that no more than two diagonals ever intersect at any one point !!!.... and we have the following "formula" for the number of intersections when the number of points is odd:

C(n, 4)  where n is the number of odd points on the circle

For instance, when n = 5, we have  C(5,4) = 5

When n = 7, we have  C(7,4)  = 35

So, when n= 9,  we have C(9,4) =126 points of intersection

BTW......

With 6 points, there are 13 points of intersection

By the "formula," 6 points should give us 15 points of intersection, but note that 3 diagonals actually meet in the center....thus...two "pair-wise" intersections are "lost"

And with 8 points, there are 47 points of intersection .... (if I counted correctly  !!!)

The number of possible hands of drawing any 6 cards from 60 = C(60,6) = 50063860

And from these possibilities we want to draw  2 red cards from 15 and 4 blue cards from 45

So, this gives us

C(15.2) * C(45, 4) = 15644475

So

15644475 / 50063860   = anout 31.25%

BJ.....I'm not sure what is wanted here.....notice that

sin^2x sin^2x   =

sin^4x

Or....this could equal

(1-cos^2x)(1-cos^2x)  =

1 - 2cos^2x + cos^4x

Other forms are possible......but the first answer of sin^4x   seems the most simple......(???)

4L = 4000ml  .....and 9% is sugar = 360ml

And 500ml added to this = 4500ml

So  360/4500 = 8% sugar in the new solution

-2x + 8 + 5x > 2x + 1    simplify

3x + 8 > 2x + 1             subtract 2x and 8 from both sides

x > -7   

5x + 7 ≤ 8x - 3 + 2x    simplify

5x + 7 ≤ 10x - 3         add 3 to both sides....subtract 5x from both sides

10 ≤ 5x       divide both sides by 5

2 ≤ x

Thanks, TayJay.....LOL, rosala...!!!!

To be precisely correct we need to include -2√5 as a solution, too.