Neave, Justin and Fred had a total of 135 pencils. The ratio of Justin's pencils to Fred's pencils was 6 : 5 at first. Neave and Justin each gave away ¹₂ of their pencils. Given that the three boys had 95 pencils left, how many pencils did Neave have at first?

	Neave	Justin	Fred	Total
Comparing Justin and Fred at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	135
Change	- 1 p	-1x3 = - 3 u		- 40
After	1 p	1x3 = 3 u	5 u	95

Total number of pencils that Neave and Justin gave away
= 135 - 95
= 40

The number of pencils that Justin had at first is repeated.  Make the number of pencils that Justin had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 135 - 95
1 p + 3 u = 40
1 p = 40 - 3 u --- (1)

1 p + 3 u + 5 u = 95
1 p + 8 u = 95
1 p = 95 - 8 u --- (2)

(1) = (2)
40 - 3 u = 95 - 8 u
8 u - 3 u = 95 - 40
8 u - 3 u = 55
5 u = 55

1 u = 55 ÷ 5 = 11

Substitute 1 u = 11 into (1).
1 p = 40 - 3 u
1 p = 40 - 3 x 11
1 p = 40 - 33

1 p = 7

Number of pencils that Neave had at first
= 2 p
= 2 x 7
= 14

Answer(s): 14