The problem consists of two parts. First, we need to find an expression for the amount of money Mary has more than Diana, given that Mary has $(3x - 4y)$ and Diana has $(2y - x)$. Second, we are given that Mary has $12 and Diana has $8, and we need to find the values of $x$ and $y$.

1. Problem Description

The problem consists of two parts. First, we need to find an expression for the amount of money Mary has more than Diana, given that Mary has $(3x - 4y)$ and Diana has $(2y - x)$. Second, we are given that Mary has $12 and Diana has$8, and we need to find the values of $x$ and $y$.

2. Solution Steps

(i) To find the amount of money Mary has more than Diana, we subtract the amount Diana has from the amount Mary has:

$(3x - 4y) - (2y - x)$.

Distribute the negative sign:

$3x - 4y - 2y + x$.

Combine like terms:

$3x + x - 4y - 2y = 4x - 6y$.

So, Mary has $(4x - 6y)$ more than Diana.

(ii) We are given that Mary has $12 and Diana has$

8. Thus we have the following equations:

$3x - 4y = 12$ (1)

$2y - x = 8$ (2)

We can multiply equation (2) by 3 to eliminate x:

$3(2y - x) = 3(8)$

$6y - 3x = 24$ (3)

Now we have the system of equations:

$3x - 4y = 12$ (1)

$-3x + 6y = 24$ (3)

Add equations (1) and (3):

$(3x - 4y) + (-3x + 6y) = 12 + 24$

$2y = 36$

$y = \frac{36}{2}$

$y = 18$

Substitute $y = 18$ into equation (2):

$2(18) - x = 8$

$36 - x = 8$

$x = 36 - 8$

$x = 28$

3. Final Answer

(i) The simplified expression for the amount of money that Mary has more than Diana is $4x - 6y$.

(ii) The values of $x$ and $y$ are $x = 28$ and $y = 18$.