Rachel has 60 scones. she sells them to robbie, cameron, louis, tom and charlie in that order. each customer buys more scones than the last and the increase is the same number in each case. robbie and cameron’s combined total number of scones is three sevenths of the total of louis, tom and charlie. how many scones does each boy buy?

Robbie: 6
Cameron: 9
Louis: 12
Tom: 15
Charlie: 18

Explanation:

1) Assume that Rachel sold all the cones: 60

2) Call x the amount that Robbie bought and d the constant increase on the purchse.

3) That means that the number of cones bought by each one is:

Robbie:            x
Cameron:         x + d
Louis:               x + d + d = x + 2d
Tom:                 x + 2d + d = x + 3d
Charlie             x + 3d + d = x + 4d

The sum of that is: x + (x + d) + (x + 2d) + (x + 3d) + (x + 4d) = 5x + 10d = 60

5x + 10d  = 60 may be simplified by dividing each side by 5 =>  x + 2d = 12.

Equation (1) x + 2d = 12

4) Robbie and Cameron’s combined total number of scones is three sevenths of the total of Louis, Tom and Charlie

=> x + (x + d) = [3/7] [ (x + 2d) + (x + 3d) + (x + 4d) ]

=> 2x + d = [3/7] (3x + 9d)

=> 7(2x + d) = 3 (3x + 9d)

=> 14x + 7d = 9x + 27d

=> 14x - 9x = 27d - 7d

=> 5x = 10d

=> x = 2d

Equation (2) x = 2d

5) Now you just have to solve this simple system of equations:

x + 2d = 12

x = 2d

Subitituing x for 2d in the first equation: x + x = 12 => 2x = 12 => x = 6

Now replace x = 6 in any  of the two equations and you get the value of d.

For example, from x = 2d => d = x  / 2 = 6 /2 = 3.

6) Finally, calcualte the number of cones of each:

                                                                           Result
Robbie:            x                                                6
Cameron:         x + d                                          6 + 3 = 9
Louis:               x + d + d = x + 2d                      9 + 3 = 12
Tom:                 x + 2d + d = x + 3d                   12 + 3 = 15
Charlie             x + 3d + d = x + 4d                    15 + 3 = 18
                                                               
Total                                                                   6 + 9 + 12 + 15 + 18 = 60