Ethan, Neave and Howard had some marbles in the ratio of 5 : 6 : 11. Howard gave 50% of his marbles to Ethan and Neave. As a result, Ethan had the same number of marbles as Neave and Howard had 44 fewer marbles than Ethan.

1. What was the percentage increase in Ethan's marbles after receiving from Howard?
2. How many marbles did Neave have at first?

Ethan	Neave	Howard	Total
5 u	6 u	11 u	22 u
+ 5.5 u		- 5.5 u
8.25 u	8.25 u	5.5 u	22 u

(a)
Number of marbles that Howard gave away
= 50% x 11
= ⁵⁰₁₀₀ x 11
= 5.5 u

Number of marbles that Ethan and Neave each had in the end
= (22 u - 5.5 u) ÷ 2
= 16.5 u ÷ 2
= 8.25 u

Increase in the number of marbles that Ethan had
= 8.25 u - 5 u
= 3.25 u

Percentage increase in Ethan's marbles after receiving from Howard
= ^3.25₅ x 100%
= 65%

(b)
8.25 u = 5.5 u + 44
8.25 u - 5.5 u = 44
2.75 u = 44

1 u = 44 ÷ 2.75 = 16

Number of marbles that Neave had at first
= 6 u
= 6 x 16
= 96

Answer(s): (a) 65%; (b) 96