Fraction becomes 9/11, If 2 is Added to Both the Numerator and Denominator. If 3 is Added to Both the Numerator and the Denominator, it Becomes 5/6. Find the Fraction. | Bzziii.com

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A fraction becomes `9/11`, if 2 is added to both the numerator and denominator. If 3 is added to both the numerator and the denominator, it becomes `5/6`. Find the fraction.







Let Numerator be x & Denominator be y

So, Fraction is = `x/y`

Given that,

If 2 is added to both the numerators and denominator, fraction becomes `9/11`

`"Numerators + 2"/"Denominator + 2"`= `9/11`

`"x+ 2"/"y + 2"`= `9/11`

11 (`x + 2`) = 9 (`y + 2`)

11`x` + 22 = 9`y` + 18

11`x` - 9`y` = 11 - 22

11`x` - 9`y` = -4 .......(1)


Also, Given that if 3 is added to both the numerator and the denominator, fraction become `5/6`

`"Numerators + 3"/"Denominator + 3"`= `5/6`

`"x+ 3"/"y + 3"`= `5/6`

6 (`x + 3`) = 5 (`y + 3`)

6`x` + 18 = 5`y` + 15

6`x` - 5`y` = 15 - 18

6`x` - 5`y` = - 3 .......(2)


Hence, our equations are

11`x` - 9`y` = -4 .......(1)
6`x` - 5`y` = - 3 .......(2)


From equation (1)

11`x` - 9`y` = -4

11x = 9`y` - 4

x = `9y - 4/11`


Putting `x`'s value on equation (2)

6`x` - 5`y` = - 3

6`x` - 5`y`+ 3 = 0

6 ( `9y - 4/11`) - 5`y`+ 3 = 0


Multiply both sides by 11

11 `\times``( 69y - 4)/11` - 5`y` `\times`11+ 3 `\times`11= 0 `\times`11

6(9y-4) - 55y + 33 = 0

6(9y) 6(4) - 55y + 33 = 0

54y - 24 - 55y + 33 = 0

54y - 55y -24 + 33 = 0

-y + 9 = 0

-y = -9

y = 9


Putting `y`'s value on equation (1)

11`x` - 9`y` = -4

11`x` - 9 (9) = -4

11`x` - 81 = -4

11`x` = -4 + 81

11`x` = 77

`x` = `77/11`

`x` = 7


Therefore, `x` = 7 and `y` = 9

Numerator (x) = 7 and

Denominator (y) = 9

Hence, Original Fraction = `"Numerators"/"Denominator"`

`x/y` = `7/9`

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