Emily and Betty had a total of 72 beads. Betty gave ¹₃ of her beads to Emily. In return, Emily gave ¹₇ of the total number of beads that she had to Betty. In the end, each girl had the same number of beads. How many beads did Emily have at first?

	Emily	Betty	Total
Before 1	?	3 u	72
Change 1	+ 1 u	- 1 u
After 1	42	2 u	72
Before 2	7 p	2 u	72
Change 2	- 1 p	+ 1 p
After 2	6 p (36)	36	72

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

Number of beads that Betty and Emily each had in the end is the same.

Number of beads that Emily had in the end
= 72 ÷ 2
= 36

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

6 p = 36

1 p = 36 ÷ 6 = 6

Number of beads that Emily had after receiving some beads from Betty
= 7 p
= 7 x 6
= 42

Number of beads that Betty had after giving to Emily
= 72 - 42
= 30

2 u = 30

1 u = 30 ÷ 2 = 15

Number of beads that Betty had at first
= 3 u
= 3 x 15
= 45

Number of beads that Emily had at first
= 72 - 45
= 27

Answer(s): 27