Question

if the numerator of a a fraction is increased by 2 and it's denominator is decreased by 1,then it becomes2/3.
if the numerator is increased by 1 and denominator increased by 2,then it becomes 1/3
find the fraction
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Answer 1

EXPLANATION.

• GIVEN

Let the numerator be = x

Let the denominator be = y

CASE = 1.

If the numerator of a fraction is increased by

2 and it's denominator is decreased by 1

it becomes = 2/3

=> x + 2 / y - 1 = 2/3

=> 3 ( x + 2 ) = 2 ( y - 1 )

=> 3x + 6 = 2y - 2

=> 3x - 2y = -8 ......(1)

CASE = 2.

if the numerator is increased by 1 and

denominator increased by 2

it becomes = 1/3

=> x + 1 / y + 2 = 1/3

=> 3 ( x + 1 ) = 1 ( y + 2 )

=> 3x + 3 = y + 2

=> 3x - y = -1 .......(2)

From equation (1) and (2) we get,

=> 3x - 2y = -8 .....(1)

=> 3x - y = -1 ......(2)

multiply equation (1) by 1

multiply equation (2) by 2

we get,

=> 3x - 2y = -8

=> 6x - 2y = -2

we get,

=> -3x = -6

=> x = 2

put the value of x = 2 in equation (1)

we get,

=> 3(2) - 2y = -8

=> 6 - 2y = -8

=> -2y = -14

=> y = 7

Therefore,

original fraction =x/y = 2/7

Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• If the numerator of a a fraction is increased by 2 and it's denominator is decreased by 1,then it becomes ⅔.

• If the numerator is increased by 1 and denominator increased by 2,then it becomes ⅓.

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The original fraction.

$\bf{\underline{\underline\green{Solution:-}}}$

Let the numerator be x

And denominator be y

According to the 1st condition:-

Numerator is increased by 2.

The numerator = x + 2

The denominator is decreased by 1.

The denominator = y - 1

The fraction becomes ⅔.

$\rm \longrightarrow \dfrac{x + 2}{y - 1} = \dfrac{2}{3}$

By cross multiplication:-

$\rm \longrightarrow {3(x + 2)} = {2(y - 1)}$

$\rm \longrightarrow {3x + 6} = {2y -2}$

$\rm \longrightarrow {3x - 2y} = { -2 - 6}$

$\rm \longrightarrow {3x - 2y} = { -8.....(i)}$

According to 2nd condition:-

Numerator is increased by 1

The numerator = x + 1

Denominator is increased by 2.

The denominator = y + 2

The fraction becomes ⅓.

$\rm \longrightarrow \dfrac{x + 1}{y +2} = \dfrac{1}{3}$

By cross multiplication:-

$\rm \longrightarrow {3(x + 1)} = {1(y + 2)}$

$\rm \longrightarrow {3x + 3} = {y + 2}$

$\rm \longrightarrow {3x - y} = { 2 - 3}$

$\rm \longrightarrow {3x - y} = { -1} .....(ii)$

Adding equation (i) and (ii)

• 3x - 2y = -8

• 3x - y = -1.

Eliminate 3x we get.

➜ -2y + y = -8 + 1

➜ -y = -7

➜ y = 7

Substituting y = 7 in equation (ii)

➜ 3x - y = -1

➜ 3x - 7 = -1

➜ 3x = - 1 +7

➜ 3x = 6

➜ x = 2

Since:

Numerator= x = 2

Denominator = y = 7

Therefore:-

$\underline{ \boxed{ \rm \purple{ \therefore The \: fraction \: = \frac{2}{7} }}}$