Diana and Gwen had a total of 24 marbles. Gwen gave ¹₄ of her marbles to Diana. In return, Diana gave ¹₃ of the total number of marbles that she had to Gwen. In the end, each girl had the same number of marbles. How many marbles did Diana have at first?

	Diana	Gwen	Total
Before 1	?	4 u	24
Change 1	+ 1 u	- 1 u
After 1	18	3 u	24
Before 2	3 p	3 u	24
Change 2	- 1 p	+ 1 p
After 2	2 p (12)	12	24

Since Gwen gave some marbles to Diana and Diana then gave some marbles to Gwen, it is an internal transfer of marbles between the two girls. So, the total number of marbles remains unchanged.

Number of marbles that Gwen and Diana each had in the end is the same.

Number of marbles that Diana had in the end
= 24 ÷ 2
= 12

Number of marbles that Diana had in the end = 2 p

2 p = 12

1 p = 12 ÷ 2 = 6

Number of marbles that Diana had after receiving some marbles from Gwen
= 3 p
= 3 x 6
= 18

Number of marbles that Gwen had after giving to Diana
= 24 - 18
= 6

3 u = 6

1 u = 6 ÷ 3 = 2

Number of marbles that Gwen had at first
= 4 u
= 4 x 2
= 8

Number of marbles that Diana had at first
= 24 - 8
= 16

Answer(s): 16