Justin, Reggie and Lee had a total of 102 pens. The ratio of Reggie's pens to Lee's pens was 4 : 9 at first. Justin and Reggie each gave away ¹₂ of their pens. Given that the three boys had 78 pens left, how many pens did Justin have at first?

	Justin	Reggie	Lee	Total
Comparing Reggie and Lee at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	102
Change	- 1 p	-1x2 = - 2 u		- 24
After	1 p	1x2 = 2 u	9 u	78

Total number of pens that Justin and Reggie gave away
= 102 - 78
= 24

The number of pens that Reggie had at first is repeated.  Make the number of pens that Reggie had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 102 - 78
1 p + 2 u = 24
1 p = 24 - 2 u --- (1)

1 p + 2 u + 9 u = 78
1 p + 11 u = 78
1 p = 78 - 11 u --- (2)

(1) = (2)
24 - 2 u = 78 - 11 u
11 u - 2 u = 78 - 24
11 u - 2 u = 54
9 u = 54

1 u = 54 ÷ 9 = 6

Substitute 1 u = 6 into (1).
1 p = 24 - 2 u
1 p = 24 - 2 x 6
1 p = 24 - 12

1 p = 12

Number of pens that Justin had at first
= 2 p
= 2 x 12
= 24

Answer(s): 24