How many boxes of each size did he use?

A dealer packages marbles in two different box sizes. One size holds 5 marbles and the other size holds 12 marbles. If the dealer packaged 99 marbles and used more than 10 boxes, how many boxes of each size did he use?

Let A represent number of boxes holding 5 marbles and B represent number of boxes holding 12 marbles.

=> 5A+12B = 99
& A+B > 10

A = \(\dfrac{99-12B}{5}\)

For A to be a whole number 99-12B should be divisible by 5. For any number to be divisible by 5, the units digit should be 5 or 0. 

i.e; 12B should have 4 or 9 in units place as 9-4=5 or 9-9=0
As no multiple of 12 will have 9 in units place, 12B should have 4 in units place.
=> B can be 2 as 12B=24 or B can be 7 as 12B=84

If B=2, A = \(\dfrac{99-12\cdot 2}{5}\) = \(\dfrac{75}{5}\)=15

If B=7, A = \(\dfrac{99-12\cdot 7}{5}\) = \(\dfrac{15}{5}\)=3

As A+B > 10 => A=15, B=2

The dealer used 15 boxes of 5 marbles size and 2 boxes of 12 marbles size.