To determine how much £Y receives from the £68 total, follow this detailed, step-by-step solution:
1. Understanding the Ratios:
- Let [tex]\( Z \)[/tex] get an amount of £ [tex]\( k \)[/tex].
- [tex]\( Y \)[/tex] gets twice the amount of [tex]\( Z \)[/tex]. Therefore [tex]\( Y \)[/tex] gets [tex]\( 2k \)[/tex].
- For every £3 that [tex]\( Y \)[/tex] gets, [tex]\( X \)[/tex] gets £4. To express [tex]\( X \)[/tex] in terms of [tex]\( k \)[/tex], start with [tex]\( Y \)[/tex]'s amount:
- [tex]\( Y \)[/tex]'s amount: [tex]\( 3k \)[/tex]
- If [tex]\( Y \)[/tex] gets £3k, then [tex]\( X \)[/tex] gets [tex]\( \frac{4}{3}Y \)[/tex]:
- [tex]\( X \)[/tex] gets: [tex]\( \frac{4}{3}(2k) = \frac{8k}{3} \)[/tex].
2. Setting up the Total Amount:
Calculate the total sum of [tex]\( X \)[/tex], [tex]\( Y \)[/tex], and [tex]\( Z \)[/tex]:
[tex]\[
X + Y + Z = \frac{8k}{3} + 2k + k
\][/tex]
3. Combine Like Terms:
Combine all terms into a single common denominator:
[tex]\[
\frac{8k}{3} + 2k + k = \frac{8k}{3} + \frac{6k}{3} + \frac{3k}{3} = \frac{17k}{3}
\][/tex]
4. Equating to the Total Amount:
We know the total amount shared is £68. So, equate this sum to £68:
[tex]\[
\frac{17k}{3} = 68
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
To find [tex]\( k \)[/tex], multiply both sides by 3 to clear the fraction:
[tex]\[
17k = 68 \times 3 \implies 17k = 204
\][/tex]
Now, divide by 17:
[tex]\[
k = \frac{204}{17} \implies k = 12
\][/tex]
6. Find the Amount Y Receives:
[tex]\( Y \)[/tex]'s amount is calculated as [tex]\( 2k \)[/tex]:
[tex]\[
Y = 2k = 2 \times 12 = 24
\][/tex]
Hence, [tex]\( Y \)[/tex] receives £24.
Thus, the amount that [tex]\( Y \)[/tex] gets is £24.
