Gabriel, Archie and Ken had some cards in the ratio of 4 : 5 : 9. Ken gave 50% of his cards to Gabriel and Archie. As a result, Gabriel had the same number of cards as Archie and Ken had 45 fewer cards than Gabriel.

1. What was the percentage increase in Gabriel's cards after receiving from Ken?
2. How many cards did Archie have at first?

Gabriel	Archie	Ken	Total
4 u	5 u	9 u	18 u
+ 4.5 u		- 4.5 u
6.75 u	6.75 u	4.5 u	18 u

(a)
Number of cards that Ken gave away
= 50% x 9
= ⁵⁰₁₀₀ x 9
= 4.5 u

Number of cards that Gabriel and Archie each had in the end
= (18 u - 4.5 u) ÷ 2
= 13.5 u ÷ 2
= 6.75 u

Increase in the number of cards that Gabriel had
= 6.75 u - 4 u
= 2.75 u

Percentage increase in Gabriel's cards after receiving from Ken
= ^2.75₄ x 100%
= 68.75%

(b)
6.75 u = 4.5 u + 45
6.75 u - 4.5 u = 45
2.25 u = 45

1 u = 45 ÷ 2.25 = 20

Number of cards that Archie had at first
= 5 u
= 5 x 20
= 100

Answer(s): (a) 68.75%; (b) 100