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

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

Number of marbles that Usha and Fiona each had in the end is the same.

Number of marbles that Fiona had in the end
= 120 ÷ 2
= 60

Number of marbles that Fiona had in the end = 6 p

6 p = 60

1 p = 60 ÷ 6 = 10

Number of marbles that Fiona had after receiving some marbles from Usha
= 7 p
= 7 x 10
= 70

Number of marbles that Usha had after giving to Fiona
= 120 - 70
= 50

2 u = 50

1 u = 50 ÷ 2 = 25

Number of marbles that Usha had at first
= 3 u
= 3 x 25
= 75

Number of marbles that Fiona had at first
= 120 - 75
= 45

Answer(s): 45