Rael, Vincent and Archie had a total of 101 coins. The ratio of Vincent's coins to Archie's coins was 4 : 3 at first. Rael and Vincent each gave away ¹₂ of their coins. Given that the three boys had 67 coins left, how many coins did Rael have at first?

	Rael	Vincent	Archie	Total
Comparing Vincent and Archie at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	101
Change	- 1 p	-1x2 = - 2 u		- 34
After	1 p	1x2 = 2 u	3 u	67

Total number of coins that Rael and Vincent gave away
= 101 - 67
= 34

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

1 p + 2 u = 101 - 67
1 p + 2 u = 34
1 p = 34 - 2 u --- (1)

1 p + 2 u + 3 u = 67
1 p + 5 u = 67
1 p = 67 - 5 u --- (2)

(1) = (2)
34 - 2 u = 67 - 5 u
5 u - 2 u = 67 - 34
5 u - 2 u = 33
3 u = 33

1 u = 33 ÷ 3 = 11

Substitute 1 u = 11 into (1).
1 p = 34 - 2 u
1 p = 34 - 2 x 11
1 p = 34 - 22

1 p = 12

Number of coins that Rael had at first
= 2 p
= 2 x 12
= 24

Answer(s): 24