Sarah and Natalie had a total of 120 pens. Natalie gave ¹₃ of her pens to Sarah. In return, Sarah gave ¹₇ of the total number of pens that she had to Natalie. In the end, each girl had the same number of pens. How many pens did Sarah have at first?

	Sarah	Natalie	Total
Before 1	?	3 u	120
Change 1	+ 1 u	- 1 u
After 1	70	2 u	120
Before 2	7 p	2 u	120
Change 2	- 1 p	+ 1 p
After 2	6 p (60)	60	120

Since Natalie gave some pens to Sarah and Sarah then gave some pens to Natalie, it is an internal transfer of pens between the two girls. So, the total number of pens remains unchanged.

Number of pens that Natalie and Sarah each had in the end is the same.

Number of pens that Sarah had in the end
= 120 ÷ 2
= 60

Number of pens that Sarah had in the end = 6 p

6 p = 60

1 p = 60 ÷ 6 = 10

Number of pens that Sarah had after receiving some pens from Natalie
= 7 p
= 7 x 10
= 70

Number of pens that Natalie had after giving to Sarah
= 120 - 70
= 50

2 u = 50

1 u = 50 ÷ 2 = 25

Number of pens that Natalie had at first
= 3 u
= 3 x 25
= 75

Number of pens that Sarah had at first
= 120 - 75
= 45

Answer(s): 45