Gillian and Diana had a total of 144 beads. Diana gave ¹₅ of her beads to Gillian. In return, Gillian gave ¹₇ of the total number of beads that she had to Diana. In the end, each girl had the same number of beads. How many beads did Gillian have at first?

	Gillian	Diana	Total
Before 1	?	5 u	144
Change 1	+ 1 u	- 1 u
After 1	84	4 u	144
Before 2	7 p	4 u	144
Change 2	- 1 p	+ 1 p
After 2	6 p (72)	72	144

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

Number of beads that Diana and Gillian each had in the end is the same.

Number of beads that Gillian had in the end
= 144 ÷ 2
= 72

Number of beads that Gillian had in the end = 6 p

6 p = 72

1 p = 72 ÷ 6 = 12

Number of beads that Gillian had after receiving some beads from Diana
= 7 p
= 7 x 12
= 84

Number of beads that Diana had after giving to Gillian
= 144 - 84
= 60

4 u = 60

1 u = 60 ÷ 4 = 15

Number of beads that Diana had at first
= 5 u
= 5 x 15
= 75

Number of beads that Gillian had at first
= 144 - 75
= 69

Answer(s): 69