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

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

Number of beads that Usha and Barbara each had in the end is the same.

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

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

6 p = 36

1 p = 36 ÷ 6 = 6

Number of beads that Barbara had after receiving some beads from Usha
= 7 p
= 7 x 6
= 42

Number of beads that Usha had after giving to Barbara
= 72 - 42
= 30

2 u = 30

1 u = 30 ÷ 2 = 15

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

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

Answer(s): 27