Question

A fraction becomes 1/3 when 2 is subtracted from numerator and it becomes 1/2 when 1 is subtracted from denominator. Find the fraction.
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Answer 1

Given:-

Fraction is $\sf{\dfrac{1}{3}}$ = -2

Fraction is $\sf{\dfrac{1}{2}}$ = -1

⠀

Solution:-

$\tt\longmapsto{\dfrac{x - 2}{y} =\dfrac{1}{3}}$

$\tt\longmapsto{3(x - 2) = y}$

$\tt\longmapsto{3x - 6 = y}$

$\tt\longmapsto{3x - y = 6}$⠀⠀.........(i)

⠀

$\tt\longmapsto{\dfrac{x}{y - 1} =\dfrac{1}{2}}$

$\tt\longmapsto{2x = y - 1}$

$\tt\longmapsto{2x - y = -1}$⠀⠀.........(ii)

⠀

From (i) and (ii)

$\tt\longmapsto{3x - y = 6}$

$\tt\longmapsto{2x - y = -1}$

$\tt\longmapsto{x = 7}$

⠀

★ Putting the value of x in the eq. (i)

$\tt\longmapsto{3x - y = 6}$

$\tt\longmapsto{3(7) - y = 6}$

$\tt\longmapsto{21 - y = 6}$

$\tt\longmapsto{-y = 6 -21}$

$\tt\longmapsto{-y = -15}$

$\tt\longmapsto{y = 15}$

∴ Fraction = $\tt\longmapsto{\dfrac{7}{15}}$

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Answer 2

Given:-

Fraction is \sf{\dfrac{1}{3}}

3

1

= -2

Fraction is \sf{\dfrac{1}{2}}

2

1

= -1

⠀

Solution:-

\tt\longmapsto{\dfrac{x - 2}{y} =\dfrac{1}{3}}⟼

y

x−2

=

3

1

\tt\longmapsto{3(x - 2) = y}⟼3(x−2)=y

\tt\longmapsto{3x - 6 = y}⟼3x−6=y

\tt\longmapsto{3x - y = 6}⟼3x−y=6 ⠀⠀.........(i)

⠀

\tt\longmapsto{\dfrac{x}{y - 1} =\dfrac{1}{2}}⟼

y−1

x

=

2

1

\tt\longmapsto{2x = y - 1}⟼2x=y−1

\tt\longmapsto{2x - y = -1}⟼2x−y=−1 ⠀⠀.........(ii)

⠀

From (i) and (ii)

\tt\longmapsto{3x - y = 6}⟼3x−y=6

\tt\longmapsto{2x - y = -1}⟼2x−y=−1

\tt\longmapsto{x = 7}⟼x=7

⠀

★ Putting the value of x in the eq. (i)

\tt\longmapsto{3x - y = 6}⟼3x−y=6

\tt\longmapsto{3(7) - y = 6}⟼3(7)−y=6

\tt\longmapsto{21 - y = 6}⟼21−y=6

\tt\longmapsto{-y = 6 -21}⟼−y=6−21

\tt\longmapsto{-y = -15}⟼−y=−15

\tt\longmapsto{y = 15}⟼y=15

∴ Fraction = \tt\longmapsto{\dfrac{7}{15}}⟼

15

7

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