Question

a fraction becomes 9/11, if 2is added to both the numerator and denominator if, 3is added to both the numerator and denominator if become 5/6.find the fraction.​

Answer 1

GIVEN :-

• Fraction becomes 9/11 when 2 is added to both numerator and denominator.
• Fraction becomes 5/6 when 3 is added to both numerator and denominator.

TO FIND :-

• The fraction.

SOLUTION :-

Let the fraction be x/y.

x is numerator and y is denominator.

♦ Fraction becomes 9/11 when 2 is added to both numerator and denominator.

$\\ \implies \sf \: \dfrac{x + 2}{y + 2} = \dfrac{9}{11} \\ \\ \implies \sf \: 11(x + 2) = 9(y + 2) \\ \\ \implies\sf \: 11x + 22 = 9y + 18 \\ \\ \implies \sf \: 11x - 9y = 18 - 22 \\ \\ \implies\sf \: 11x - 9y = - 4$

Multiplying whole equation by 5,

$\\ \underline{\sf \: 55x - 45y = - 20 }\: \: \: \: \: \: \: \: - - (1)$

♦ Fraction becomes 5/6 when 3 is added to both numerator and denominator.

$\\ \implies \sf \: \dfrac{x + 3}{y + 3} = \dfrac{5}{6} \\ \\ \implies\sf \: 6(x + 3) = 5(y + 3) \\ \\ \implies \sf \: 6x + 18 = 5y + 15 \\ \\ \implies\sf \: 6x - 5y = 15 - 18 \\ \\ \implies \sf \: 6x - 5y = - 3$

Multiplying whole equation by 9,

$\\ \underline{ \sf \: 54x - 45y = - 27} \: \: \: \: \: \: - - (2)$

Now , subtracting equation (2) by equation (1),

→55x - 45y - (54x - 45y) = -20 - (-27)

→55x - 45y - 54x + 45y = -20 + 27

→x = 7

Now , substituting value of x=7 in equation (2),

→ 54(7) - 45y = -27

→ 378 - 45y = -27

→ -45y = -27 - 378

→ -45y = -405

→ y = -405/-45

→ y = 9

Hence , x = 7 and y = 9.

$\\ \sf \: fraction \to \dfrac{x}{y} \\ \\ \therefore \underline{\boxed{\frak{ \: fraction \to \: \dfrac{7}{9} }}}$

Answer 2

Given

• A fraction becomes 9/11, if 2 is added to both the numerator and denominator
• A fraction becomes 5/6, if 3 is added to both the numerator and denominator

Explanation

Let the Fraction be x/y

1st Condition

➟ x+2/y+2 = 9/11

➟ 11x+22 = 9y+18

➟ 11x - 9y = 18 - 22

➟ 11x - 9y = -4 ___(1)

2nd condition

Fraction be x/y

➟ x+3/y+3 = 5/6

➟ 6x+18 = 5y+15

➟ 6x - 5y = 15 - 18

➟ 6x - 5y = -3 ___(2)

❖ After Equalising Both Equations (1) and (2) as:-

11x - 9y = -4 __(1)

6x - 5y = -3 __(2)

❖ We have to Multiply Eq. (1) with 5 and Eq. (2) with 9 to Equal both Fractions :-

55x - 45y = -20

54x - 45y = -27

- + +

____________

x = 7

❖ Now, We can Find value of y by putting value of x in Any Equation:-

➟ 11x - 9y = -4

➟ 11×7 - 9y = -4

➟ 77 + 4 = 9y

➟ 81 = 9y

➟ y = 9

Hence,

• x = 7
• y = 9
• The Fraction will be x/y = 7/9 .