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2) [tex]\( X, Y \)[/tex] and [tex]\( Z \)[/tex] shared [tex]\( £68 \)[/tex] so that for every [tex]\( £1 \)[/tex] that [tex]\( Z \)[/tex] gets, [tex]\( Y \)[/tex] gets [tex]\( £2 \)[/tex] and for every [tex]\( £3 \)[/tex] that [tex]\( Y \)[/tex] gets, [tex]\( X \)[/tex] gets [tex]\( £4 \)[/tex]. How much does [tex]\( Y \)[/tex] get?

Sagot :

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.
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