Solution:

(i) (4x² + xy): (3xy – y²) = 12: 5

We can write it as

(4x² + xy)/ (3xy – y²) = 12/ 5

By cross multiplication

20x² + 5xy = 36xy – 12y²

20x² + 5xy – 36xy + 12y² = 0

20x² – 31xy + 12y² = 0

Now divide the entire equation by y²

20x²/y² – 31xy/y² + 12y²/y² = 0

So we get

20 (x/y)² – 31 (x/y) + 12 = 0

20 (x/y)² – 15(x/y) – 16 (x/y) + 12 = 0

Taking common terms

5 (x/y) [4 (x/y) – 3] – 4 [4 (x/y) – 3] = 0

[4 (x/y) – 3] [5 (x/y) – 4] = 0

Here 4 (x/y) – 3 = 0

4 (x/y) = 3

So we get x/y = ¾

Similarly 5 (x/y) – 4 = 0

5 (x/y) = 4

So we get x/y = 4/5

Now dividing by y

(x + 2y)/ (2x + y) = (x/y + 2)/ (2 x/y + 1)

(a) If x/y = 3/4, then

= (x/y + 2)/ (2 x/y + 1)

Substituting the values

= (3/4 + 2)/ (2 × 3/4 + 1)

By further calculation

= 11/4/ (3/2 + 1)

= 11/4/ 5/2

= 11/4 × 2/5

= 11/10

So we get

(x + 2y): (2x + y) = 11: 10

(b) If x/y = 4/5 then

(x + 2y)/ (2x + y) = [x/y + 2]/ [2 x/y + 1]

Substituting the value of x/y

= [4/5 + 2]/ [2 × 4/5 + 1]

So we get

= 14/5/ [8/5 + 1]

= 14/5/ 13/5

= 14/5 × 5/13

= 14/13

We get

(x + 2y)/ (2x + y) = 11/10 or 14/13

(x + 2y): (2x + y) = 11: 10 or 14: 13

(ii) y (3x – y): x (4x + y) = 5: 12

It can be written as

(3xy – y²)/ (4x² + xy) = 5/12

By cross multiplication

36xy – 12y² = 20x² + 5xy

20x² + 5xy – 36xy + 12y² = 0

20x² – 31xy + 12y² = 0

Divide the entire equation by y²

20x²/y² – 31 xy/y² + 12y²/y² = 0

20(x²/y²) – 31 (xy/y²) + 12 = 0

We can write it as

20(x²/y²) – 15 (x/y) – 16 (x/y) + 12 = 0

Taking common terms

5 (x/y) [4 (x/y) – 3] – 4 [4 (x/y) – 3] = 0

[4 (x/y) – 3] [5 (x/y) – 4] = 0

Here

4 (x/y) – 3 = 0

So we get

4 (x/y) = 3

x/y = 3/4

Similarly

5 (x/y) – 4 = 0

So we get

5 (x/y) = 4

x/y = 4/5

(a) x/y = 3/4

We know that

(x² + y²): (x + y)² = (x² + y²)/ (x + y)²

Dividing both numerator and denominator by y²

= (x²/y² + y²/y²)/ [1/y² (x + y)²]

= (x²/ y² + 1) (x/y + 1)²

Substituting the value of x/y

= [(3/4)² + 1]/ [3/4 + 1]²

By further calculation

= (9/16 + 1)/ (7/4)²

So we get

= 25/16/ 49/16

= 25/16 × 16/49

= 25/49

So we get

(x² + y²): (x + y)² = 25: 49

(b) x/y = 4/5

We know that

(x² + y²): (x + y)² = (x² + y²)/ (x + y)²

Dividing both numerator and denominator by y²

= (x²/y² + y²/y²)/ [1/y² (x + y)²]

= (x²/ y² + 1) (x/y + 1)²

Substituting the value of x/y

= [(4/5)² + 1]/ [4/5 + 1]²

By further calculation

= (16/25 + 1)/ (9/5)²

So we get

= 41/25/ 81/25

= 41/25 × 25/81

= 41/81

So we get

(x² + y²): (x + y)² = 41: 81