Rachel has 60 scones. She sells them to Robbie, Cameron, Louis, Tom and Charlie in that order. Each customer buys more scones then 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?

It is given in the statement that the increase in number of scones is same from Robbie till Charlie.
Let us assume that Robbie buy x scones and every boy buys y more scones than the previous ones.
The number of scones for each of the boy can be written as:

Robbie = x
Cameron = x + y
Louis = x + 2y
Tom =  x + 3y
Charlie = x + 4y

Total number of scones = 60

So,
[tex]x + (x+y) + (x+2y) + (x+3y) + (x+4y) = 60 \\ 5x+10y=60 \\ 5(x+2y)=60 \\ x+2y=12 \\ x=12-2y[/tex]

Robbie and Cameron's combined total number of scones is three sevenths of the total of Louis, Tom and Charlie. Mathematically this can be stated as:
[tex]x+(x+y)= \frac{3}{7}(x+2y+x+3y+x+4y) \\ 2x+y= \frac{3}{7}(3x+9y) \\ 14x+7y=9x+27y \\ 14x-9x=27y-7y \\ 5x=20y \\ x=4y [/tex]

Using Solving the two equations simultaneously:

[tex]x=12-2y \\ x=4y \\ -\ \textgreater \ 4y=12-2y \\ 6y=12 \\ y=2 \\ x=4(2) \\ x = 8 [/tex]

This means, Robbie buys 8 scones,
Cameron buys 8 + 2 = 10 scones,
Louis buys 8 + 4 = 12 scones,
Tom buys 8 + 6 = 14 scones,
and Charlie buys 8 + 8 = 16 scones