Zara and Irene had a total of 40 beads. Irene gave ¹₆ of her beads to Zara. In return, Zara gave ¹₃ of the total number of beads that she had to Irene. In the end, each girl had the same number of beads. How many beads did Zara have at first?

	Zara	Irene	Total
Before 1	?	6 u	40
Change 1	+ 1 u	- 1 u
After 1	30	5 u	40
Before 2	3 p	5 u	40
Change 2	- 1 p	+ 1 p
After 2	2 p (20)	20	40

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

Number of beads that Irene and Zara each had in the end is the same.

Number of beads that Zara had in the end
= 40 ÷ 2
= 20

Number of beads that Zara had in the end = 2 p

2 p = 20

1 p = 20 ÷ 2 = 10

Number of beads that Zara had after receiving some beads from Irene
= 3 p
= 3 x 10
= 30

Number of beads that Irene had after giving to Zara
= 40 - 30
= 10

5 u = 10

1 u = 10 ÷ 5 = 2

Number of beads that Irene had at first
= 6 u
= 6 x 2
= 12

Number of beads that Zara had at first
= 40 - 12
= 28

Answer(s): 28