Let's break down the problem step-by-step to find how much £Y gets.
1. Understanding the Ratios:
- For every £1 that Z gets, Y gets £2. Therefore, the ratio of Z's amount to Y's amount is [tex]\(1:2\)[/tex].
- For every £3 that Y gets, X gets £4. Therefore, the ratio of Y's amount to X's amount is [tex]\(3:4\)[/tex].
2. Expressing Ratios Uniformly:
- Let Z's portion be [tex]\(1x\)[/tex].
- Accordingly, Y's portion will be [tex]\(2x\)[/tex] because Y gets twice as much as Z.
- For every £3 that Y gets (£2x), X gets £4. Hence, X's portion is [tex]\( \frac{4}{3} \times 2x = \frac{8}{3}x \)[/tex].
3. Combining the Ratios:
- Combine the portions we have:
- Z's share: [tex]\(1x\)[/tex]
- Y's share: [tex]\(2x\)[/tex]
- X's share: [tex]\(\frac{8}{3}x\)[/tex]
- The total combined portion in terms of [tex]\(x\)[/tex] is:
[tex]\[
1x + 2x + \frac{8}{3}x = \left(1 + 2 + \frac{8}{3}\right)x = \left(\frac{3}{3} + \frac{6}{3} + \frac{8}{3}\right)x = \frac{17}{3}x
\][/tex]
4. Finding the Total Amount:
- The total amount shared is £68.
- We set up the equation:
[tex]\[
\frac{17}{3}x = 68
\][/tex]
- Solving for [tex]\(x\)[/tex]:
[tex]\[
x = 68 \div \frac{17}{3} = 68 \times \frac{3}{17} = 4 \times 3 = 12
\][/tex]
- So, [tex]\(x = 12\)[/tex].
5. Calculating Y's Share:
- Recall that Y's portion is [tex]\(2x\)[/tex].
- Substituting the value of [tex]\(x = 12\)[/tex]:
[tex]\[
Y's\ share = 2x = 2 \times 12 = 24
\][/tex]
Thus, Y gets £24.
