Brandon, Ivan and George had a total of 45 pens. The ratio of Ivan's pens to George's pens was 4 : 5 at first. Brandon and Ivan each gave away ¹₂ of their pens. Given that the three boys had 30 pens left, how many pens did Brandon have in the end?

	Brandon	Ivan	George	Total
Comparing Ivan and George at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	45
Change	- 1 p	-1x2 = - 2 u		- 15
After	1 p	1x2 = 2 u	5 u	30

Total number of pens that Brandon and Ivan gave away
= 45 - 30
= 15

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

1 p + 2 u = 45 - 30
1 p + 2 u = 15
1 p = 15 - 2 u --- (1)

1 p + 2 u + 5 u = 30
1 p + 7 u = 30
1 p = 30 - 7 u --- (2)

(1) = (2)
15 - 2 u = 30 - 7 u
7 u - 2 u = 30 - 15
7 u - 2 u = 15
5 u = 15

1 u = 15 ÷ 5 = 3

Substitute 1 u = 3 into (1).
1 p = 15 - 2 u
1 p = 15 - 2 x 3
1 p = 15 - 6

1 p = 9

Number of pens that Brandon had in the end
= 1 p
= 9

Answer(s): 9