Opal, Emma and Gwen each had some balls and decided to play a game with their balls. In round 1, Opal lost ¹₃ of her balls to Emma. In round 2, Emma lost ¹₃ of her total number of balls to Gwen. In round 3, Gwen lost ¹₃ of her total number of balls to Opal. In the end, Opal, Emma and Gwen had 119, 90 and 50 balls respectively. How many balls did Emma have at first?

	Opal	Emma	Gwen
		88
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	135
Before 2	94	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	75
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Opal, Emma and Gwen in the end	119	90	50

Working backwards.

2 p = 50

1 p = 50 ÷ 2 = 25

At the start of Round 3:
Number of balls that Gwen had
= 3 p
= 3 x 25
= 75

2 u = 90

1 u = 90 ÷ 2 = 45

At the start of Round 2:
Number of balls that Emma had
= 3 u
= 3 x 45
= 135

At the start of Round 2:
Number of balls that Opal had
= 119 - 25
= 94

2 b = 94

1 b = 94 ÷ 2 = 47

At the start of Round 1:
Number of balls that Emma had
= 135 - 1 b
= 135 - 47
= 88

Answer(s): 88