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

Solution:(i) (4x

^{2}+ xy): (3xy – y^{2}) = 12: 5We can write it as

(4x

^{2}+ xy)/ (3xy – y^{2}) = 12/ 5By cross multiplication

20x

^{2}+ 5xy = 36xy – 12y^{2}20x

^{2}+ 5xy – 36xy + 12y^{2}= 020x

^{2}– 31xy + 12y^{2}= 0Now divide the entire equation by y

^{2}20x

^{2}/y^{2}– 31xy/y^{2}+ 12y^{2}/y^{2}= 0So we get

20 (x/y)

^{2}– 31 (x/y) + 12 = 020 (x/y)

^{2}– 15(x/y) – 16 (x/y) + 12 = 0Taking 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 – y

^{2})/ (4x^{2}+ xy) = 5/12By cross multiplication

36xy – 12y

^{2}= 20x^{2}+ 5xy20x

^{2}+ 5xy – 36xy + 12y^{2}= 020x

^{2}– 31xy + 12y^{2}= 0Divide the entire equation by y

^{2}20x

^{2}/y^{2}– 31 xy/y^{2}+ 12y^{2}/y^{2}= 020(x

^{2}/y^{2}) – 31 (xy/y^{2}) + 12 = 0We can write it as

20(x

^{2}/y^{2}) – 15 (x/y) – 16 (x/y) + 12 = 0Taking 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

(x

^{2}+ y^{2}): (x + y)^{2}= (x^{2}+ y^{2})/ (x + y)^{2}Dividing both numerator and denominator by y

^{2}= (x

^{2}/y^{2}+ y^{2}/y^{2})/ [1/y^{2}(x + y)^{2}]= (x

^{2}/ y^{2}+ 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

(x

^{2}+ y^{2}): (x + y)^{2}= 25: 49(b) x/y = 4/5

We know that

(x

^{2}+ y^{2}): (x + y)^{2}= (x^{2}+ y^{2})/ (x + y)^{2}Dividing both numerator and denominator by y

^{2}= (x

^{2}/y^{2}+ y^{2}/y^{2})/ [1/y^{2}(x + y)^{2}]= (x

^{2}/ y^{2}+ 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

(x

^{2}+ y^{2}): (x + y)^{2}= 41: 81