To solve this problem, we need to understand the ratio in which N68 is divided among X, Y, and Z based on the given conditions. Let's break it down step by step:
1. Understand the Ratios:
- For every 1 unit that Z gets, Y gets 2 units.
- For every 3 units that Y gets, X gets 4 units.
2. Express in Common Terms:
- Let the amount that Z gets be denoted as [tex]\(z\)[/tex].
- Since Y gets twice the amount Z gets, Y will have [tex]\(2z\)[/tex].
- For X, since X gets four-thirds of the amount Y gets (because for every 3 units Y gets, X gets 4 units):
[tex]\[ X = \frac{4}{3}Y = \frac{4}{3} \times 2z = \frac{8}{3}z \][/tex]
3. Total Sum:
- Now, sum the amounts X, Y, and Z get:
[tex]\[ z + 2z + \frac{8}{3}z \][/tex]
[tex]\[ z + 2z + \frac{8}{3}z = \frac{3z}{3} + \frac{6z}{3} + \frac{8z}{3} = \frac{17z}{3} \][/tex]
4. Calculate the Total Parts:
- The total amount to be shared is N68. Therefore, we have:
[tex]\[ \frac{17z}{3} = 68 \][/tex]
5. Solve for [tex]\(z\)[/tex]:
- To isolate [tex]\(z\)[/tex], multiply both sides by 3:
[tex]\[ 17z = 68 \times 3 \][/tex]
[tex]\[ 17z = 204 \][/tex]
[tex]\[ z = \frac{204}{17} \][/tex]
[tex]\[ z = 12 \][/tex]
6. Calculate the Amount for Y:
- Since Y gets twice the amount Z gets:
[tex]\[ Y = 2z \][/tex]
[tex]\[ Y = 2 \times 12 \][/tex]
[tex]\[ Y = 24 \][/tex]
Thus, the amount Y gets is N24.
