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# (i) If (4×2 + xy): (3xy – y2) = 12: 5, find (x + 2y): (2x + y). (ii) If y (3x – y): x (4x + y) = 5: 12. Find (x2 + y2): (x + y)2.

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Question no.12  From ML aggarwal book, class10, chapter 7, ratio and proportion

(i) we have given (4×2 + xy): (3xy – y2) = 12: 5, and we have to find (x + 2y): (2x + y).

(ii)also If y (3x – y): x (4x + y) = 5: 12.

Then we have to Find (x2 + y2): (x + y)2.

Ques no. 12

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1. 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

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