Question

[Maths]
Topic: Ratio And Proportion

(i) If (4x² + 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.​

Answer 1

Answer:

★ Question no 1 :

i) If (4x² + xy) : (3xy - y²) = 12 : 5, find the value of (x + 2y) : (2x + y)

Solution :-

➲ (4x² + xy) : (3xy - y²) = 12 : 5

So, we can write as :

↦ 4x² + xy/3xy - y² = 12/5

By doing cross multiplication we get,

↦ 5(4x² + xy) = 12(3xy - y²)

↦ 20x² + 5xy = 36xy - 12y²

↦ 20x² + 5xy - 36xy + 12y² = 0

↦ 20x² - 31xy + 12y² = 0

By doing middle term break we get,

↦ 20x² - (16 + 15)xy + 12y² = 0

↦ 20x² - 16xy - 15xy + 12y² = 0

↦ 4x(5x - 4y) - 3y(5x - 4y) = 0

↦ (4x - 3y)(5x - 4y) = 0

↦ 4x - 3y = 0

↦ 4x = 3y

➦ x = 3y/4

And,

↦ 5x - 4y = 0

↦ 5x = 4y

➦ x = 4y/5

Now,

➤ In case I :

⇒ x = 3y/4

Then,

↦ (x + 2y) : (2x + y)

↦ x + 2y/2x + y

↦ { 3y/4 + 2y}/{2(3y/4) + y}

↦ (3y + 8y)/4 / (6y/4 + y)

↦ 11y/4 / 6y + 4y/4

↦ 11y/4 / 10y/4

➠ 11/10

Again,

➤ In case II :

⇒ x = 4y/5

Then,

↦ (x + 2y) : (2x + y)

↦ x + 2y/2x + y

↦ { 4y/5 + 2y}/{2(4y/5) + y}

↦ (4y + 10y)/5 / (8y/5 + y)

↦ 14y/5 / 8y + 5y/5

↦ 14y/5 / 13y/5

➠ 14/13

∴ The value of (x + 2y) : (2x + y) is 11/10 and 14/13.

______________________

★ Question No 2 :

ii) If y(3x - y) : x(4x + y) = 5 : 12, then find the value of (x² + y²) : (x + y)²

Solution :-

➲ y(3x - y) : x(4x + y) = 5 : 12

So, we can write as :

↦ y(3x - y)/x(4x + y) = 5/12

By doing cross multiplication we get,

↦ 12y(3x - y) = 5x(4x + y)

↦ 36xy - 12y² = 20x² + 5xy

↦ 20x² + 5xy - 36xy + 12y² = 0

↦ 20x² - 31xy + 12y² = 0

↦ 20x² - (16 + 15)xy + 12y² = 0

↦ 20x² - 16xy - 15xy + 12y² = 0

↦ 4x(5x - 4y) - 3y(5x - 4y) = 0

↦ (4x - 3y)(5x - 4y) = 0

↦ 4x - 3y = 0

↦ 4x = 3y

➦ x = 3y/4

And,

↦ 5x - 4y = 0

↦ 5x = 4y

➦ x = 4y/5

Now,

➤ In case I :

⇒ x = 3y/4

↦ (x² + y²) : (x + y)²

↦ (x² + y²)/(x + y)²

↦ { (3y/4)² + y² } / { (3y/4 + y)² }

↦ ( 9y²/16 + y²)/(3y + 4y/4 + y²)

↦ (25y²/16) / (7y/4)²

↦ (25y²/16) / 49y²/16

➠ 25/49

Again,

➤ In case II :

⇒ x = 4y/5

↦ (x² + y²) : (x + y)²

↦ (x² + y²)/(x + y)²

↦ { (4y/5)² + y² } / { (4y/5 + y)² }

↦ (16y²/25 + y²)/(4y + 5y/5 + y²)

↦ (41y²/25) / (9y/5)²

↦ (41y²/25) / 81y²/25

➠ 41/81

∴ The value of (x² + y²) : (x + y)² is 25/49 and 41/81.